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The following is the raw text received: --- { "slug": "order-emergence-principles", "outlineId": "sye3v0", "outline": "sye3v0-1.0 Introduction: The Conceptual Crisis in Physics and the Call for a Generative Ontology\nsye3v0-1.1 Prevailing materialist ontology limitations\nsye3v0-1.1.1 Physical matter and energy as fundamental primitives\nsye3v0-1.1.2 Existence within a pre-defined spacetime\nsye3v0-1.2 Connection between E=mc² and E=hf\nsye3v0-1.2.1 Equivalence reveals deep connection\nsye3v0-1.2.2 Identity ω=m in natural units\nsye3v0-1.2.3 Mass as manifestation of fundamental oscillation and information processing\nsye3v0-1.3 Unresolved questions challenging materialist view\nsye3v0-1.3.1 Origin and specificity of physical laws and constants\nsye3v0-1.3.1.1 Why these particular laws?\nsye3v0-1.3.1.2 Why specific constant values (fine-tuning)?\nsye3v0-1.3.1.3 Are they arbitrary or intrinsic?\nsye3v0-1.3.2 Quantum non-locality and the measurement problem\nsye3v0-1.3.2.1 Entanglement (instantaneous, non-local correlations)\nsye3v0-1.3.2.2 Measurement problem (superposition to definite outcome)\nsye3v0-1.3.3 The hard problem of consciousness\nsye3v0-1.3.3.1 Subjective experience irreducible to objective processes\nsye3v0-1.3.4 The nature of spacetime and gravity\nsye3v0-1.3.4.1 General Relativity struggles with quantization and singularities\nsye3v0-1.3.4.2 Is spacetime fundamental or emergent?\nsye3v0-1.3.5 The unification challenge (GR and QM incompatibility)\nsye3v0-1.3.6 Accounting for dark matter and dark energy\nsye3v0-1.4 Suggestion of incomplete foundational understanding\nsye3v0-1.4.1 Mistaking emergent phenomena for foundational elements\nsye3v0-1.5 Necessity for re-evaluation and alternative ontologies\nsye3v0-1.5.1 Shift from "things" to dynamic processes and emergent patterns\nsye3v0-1.5.2 Explain how complexity, structure, laws arise intrinsically\nsye3v0-1.6 Autaxys framework introduced\nsye3v0-1.6.1 Derived from Greek auto (self) and taxis (order/arrangement)\nsye3v0-1.6.2 Principle of intrinsic self-ordering, self-arranging, self-generating\nsye3v0-1.6.3 Reality as a computational process\nsye3v0-1.6.4 Laws as emergent solutions to intrinsic tension\nsye3v0-1.6.5 Cosmic algorithm perpetually running itself into existence\nsye3v0-1.7 Relational Calculus (formal mathematical description)\nsye3v0-2.0 Symmetry in Physics\nsye3v0-2.1 Definition of symmetry\nsye3v0-2.1.1 Symmetries of states\nsye3v0-2.1.1.1 State |ψ symmetric under unitary transformation U\nsye3v0-2.1.1.2 Transformed state U|ψ identical up to phase factor eiϕ\nsye3v0-2.1.1.3 Physical states correspond to rays in Hilbert space\nsye3v0-2.1.1.4 Symmetry observed relative to a different, non-symmetric system\nsye3v0-2.1.2 Symmetries of Hamiltonians\nsye3v0-2.1.2.1 Quantum operator A invariant under U if U†AU = A or [U, A] = 0\nsye3v0-2.1.2.2 Unitary transformation U is symmetry of Hamiltonian H if [U, H] = 0\nsye3v0-2.1.2.3 Expectation value of symmetric H invariant under U\nsye3v0-2.1.2.4 Eigenstates of H are eigenstates of U†HU\nsye3v0-2.1.2.5 If H has eigenstate |ψ with eigenvalue Eψ, U|ψ is also eigenstate with Eψ\nsye3v0-2.1.2.6 Non-trivial implication: degenerate state exists for non-symmetric eigenstate\nsye3v0-2.1.2.7 Symmetries can be defined in Lagrangian formulation\nsye3v0-2.2 Noether's theorem\nsye3v0-2.2.1 Conserved quantities\nsye3v0-2.2.1.1 Time evolution determined by Hamiltonian\nsye3v0-2.2.1.2 Time evolution operator U(t) = eiHt\nsye3v0-2.2.1.3 Symmetry transformation U commutes with H, implies U commutes with U(t)\nsye3v0-2.2.1.4 Unitary U = eiQ for Hermitian operator Q (symmetry generator)\nsye3v0-2.2.1.5 [Q, U(t)] = 0 implies expectation value of Q is conserved in time\nsye3v0-2.2.1.6 Eigenstate of Q evolves to eigenstate of Q with same eigenvalue\nsye3v0-2.2.1.7 Any unitary symmetry U corresponds to observable Q (conserved quantity)\nsye3v0-2.2.1.8 Exercise 1.1 (Exponential of an operator)\nsye3v0-2.2.2 Continuity equations\nsye3v0-2.2.2.1 Continuous symmetry transformations parametrised by α\nsye3v0-2.2.2.2 Each continuous symmetry associated with current jν(x,t)\nsye3v0-2.2.2.3 Current obeys local conservation law ∂νjν = 0\nsye3v0-2.2.2.4 Discrete symmetries have global constant of motion Q, no local continuity equation\nsye3v0-2.2.2.5 Q can be written as integral over local density ρ(x,t): Q = ∫dDx ρ(x,t)\nsye3v0-2.2.2.6 ∂νjν = 0 is equivalent to ∂tQ(t) + ∫dSn jn(x,t) = 0\nsye3v0-2.2.2.7 Assuming jn falls off at spatial infinity, ∂tQ = 0\nsye3v0-2.2.2.8 Direct relation between symmetry and continuity equation is Noether's theorem\nsye3v0-2.2.3 Proving Noether's theorem\nsye3v0-2.2.3.1 Applies to theories with continuous symmetry described by Lagrangian L\nsye3v0-2.2.3.2 Fields φa(x) = φa(x,t)\nsye3v0-2.2.3.3 Action S = ∫dtdDx L\nsye3v0-2.2.3.4 Infinitesimal symmetry transformations φa(x) → φa(x) + δsφa(x)\nsye3v0-2.2.3.5 Variation of Lagrangian δsL = ∂L/∂φa δsφa + ∂L/∂(∂νφa) δs(∂νφa)\nsye3v0-2.2.3.6 δsL = ∂νKν for some function Kν is symmetry of the action\nsye3v0-2.2.3.7 Using δs(∂νφa) = ∂ν(δsφa) and integration by parts: δsL = ∂ν(∂L/∂(∂νφa) δsφa) + (∂L/∂φa - ∂ν(∂L/∂(∂νφa))) δsφa\nsye3v0-2.2.3.8 For field configurations satisfying Euler-Lagrange equations, δsL = ∂ν(∂L/∂(∂νφa) δsφa)\nsye3v0-2.2.3.9 If δsL = ∂νKν is a symmetry, then ∂ν(∂L/∂(∂νφa) δsφa - Kν) = 0\nsye3v0-2.2.3.10 Defining Noether current jν = 1/α (∂L/∂(∂νφa) δsφa - Kν) = ∂L/∂(∂νφa) ∆sφa - 1/α Kν\nsye3v0-2.2.3.11 Theorem 1.1 (Noether's theorem): Continuous symmetry of local action corresponds to locally conserved current ∂νjν = 0\nsye3v0-2.2.4 Noether charge\nsye3v0-2.2.4.1 Q(t) = ∫dDx jt(x,t) is globally conserved quantity\nsye3v0-2.2.4.2 jt = ρ is Noether charge density\nsye3v0-2.2.4.3 Q(t) is independent of time\nsye3v0-2.2.4.4 Relation to symmetry transformation: [iαQ, φa(x)] = δsφa(x)\nsye3v0-2.2.4.5 Q is also called the generator of the symmetry\nsye3v0-2.2.4.6 φa(x) → φ'a(x) = φa(x) + iα[Q, φa(x)] ≈ eiαQφa(x)e-iαQ\nsye3v0-2.2.4.7 Q corresponds to observable Q obeying [Q, H] = 0 in Hamiltonian formalism\nsye3v0-2.2.4.8 ∂νjν = 0 holds for any state satisfying equations of motion\nsye3v0-2.2.4.9 If symmetry is spontaneously broken, continuity equation implies physically different things (Nambu-Goldstone modes)\nsye3v0-2.3 Examples of Noether currents and Noether charges\nsye3v0-2.3.1 Schrödinger field (complex scalar field ψ(x,t))\nsye3v0-2.3.1.1 Action S[ψ, ψ*]\nsye3v0-2.3.1.2 Euler-Lagrange equation looks like Schrödinger equation\nsye3v0-2.3.1.3 Treat ψ and ψ* independently, [ψ(x), ψ*(y)] = δ(x-y)\nsye3v0-2.3.1.4 Canonical momenta π = iψ*/2, π* = -iψ/2\nsye3v0-2.3.1.5 Continuous symmetry: phase rotations ψ(x) → e-iαψ(x)\nsye3v0-2.3.1.6 Global transformation (α does not depend on x)\nsye3v0-2.3.1.7 Infinitesimal variations ∆sψ = -iψ, ∆sψ* = iψ*\nsye3v0-2.3.1.8 Noether current jt = ψ*ψ, jn = iħ/2m ((∂nψ*)ψ - ψ*(∂nψ))\nsye3v0-2.3.1.9 Conserved Noether charge Q = ∫dDx ψ*ψ (total amplitude/norm)\nsye3v0-2.3.1.10 Conservation of norm is consequence of global phase rotation invariance\nsye3v0-2.3.1.11 jn is probability current\nsye3v0-2.3.2 Relativistic complex scalar field\nsye3v0-2.3.2.1 Action S (different form)\nsye3v0-2.3.2.2 Invariant under Lorentz transformations (Klein-Gordon equation)\nsye3v0-2.3.2.3 Same global phase rotation symmetry as Schrödinger field\nsye3v0-2.3.2.4 Canonical momentum π = ∂tψ*/c²\nsye3v0-2.3.2.5 Relativistic Noether charge Q = ∫dDx i/c² (ψ*∂tψ - ∂tψ*ψ)\nsye3v0-2.3.2.6 Conserved field normalisation in Klein-Gordon theory\nsye3v0-2.3.3 Spacetime translations\nsye3v0-2.3.3.1 Symmetry transformation on fields changing Lagrangian by total derivative\nsye3v0-2.3.3.2 Global spacetime translation xν → xν + αν\nsye3v0-2.3.3.3 Variation ∆νsφ(x) = ∂νφ\nsye3v0-2.3.3.4 Variation of action δsS = ∫dtdDx ∂νL = ∂ν(ανL) ≡ ∂νKν\nsye3v0-2.3.3.5 D+1 independent continuous symmetry transformations\nsye3v0-2.3.3.6 Relativistic canonical momenta πµ = ∂L/∂(∂µφ)\nsye3v0-2.3.3.7 D+1 Noether currents jµν = πµ∆νsφ - δµνL = πµ∂νφ - δµνL\nsye3v0-2.3.3.8 Tensor jµν contains D+1 currents, each with D+1 spacetime components\nsye3v0-2.3.3.9 Conserved global Noether charges Qt = ∫dDx jt t = ∫dDx πt∂tφ - L = ∫dDx H = H (Hamiltonian/energy)\nsye3v0-2.3.3.10 Qn = ∫dDx jt n = ∫dDx πt∂nφ (total momentum)\nsye3v0-2.3.3.11 Tensor of Noether currents is canonical energy-momentum tensor\nsye3v0-2.3.3.12 Energy/momentum conserved due to spacetime translation symmetry\nsye3v0-2.3.3.13 Derivation done without specifying action form\nsye3v0-2.3.3.14 Exercise 1.2 (SO(2) and U(1) symmetry)\nsye3v0-2.3.3.15 Action of two real fields A(x), B(x) invariant under SO(2) rotations\nsye3v0-2.3.3.16 Correspondence between phase rotations of complex scalar and rotations of 2-vector (isomorphism U(1) and SO(2))\nsye3v0-2.3.3.17 Exercise 1.3 (Noether's trick)\nsye3v0-2.3.3.18 Technique to obtain Noether current by making symmetry parameter spacetime dependent α → α(x)\nsye3v0-2.3.3.19 δS|O(∂α) = ∫dx jµ∂µα, jµ is Noether current\nsye3v0-2.4 Types of symmetry transformations\nsye3v0-2.4.1 Not all transformations susceptible to spontaneous symmetry breaking\nsye3v0-2.4.2 Classifications introduced historically\nsye3v0-2.4.3 Physical relevance of distinctions for SSB\nsye3v0-2.4.4 Brief account of different types\nsye3v0-2.4.5 Discrete versus continuous symmetries\nsye3v0-2.4.5.1 Discrete cannot be parametrised by continuous variable\nsye3v0-2.4.5.2 Noether's theorem applies only to continuous symmetries (requires infinitesimal transformations)\nsye3v0-2.4.5.3 Nambu-Goldstone modes and Mermin-Wagner theorem apply only to broken continuous symmetries\nsye3v0-2.4.5.4 More richness in breaking of continuous symmetries\nsye3v0-2.4.5.5 Discrete symmetry breaking examples (Ising model stability Section 2.7)\nsye3v0-2.4.6 Anti-unitary symmetries\nsye3v0-2.4.6.1 Transformations conserve inner product up to phase factor\nsye3v0-2.4.6.2 Satisfy <ψ|U†U|ψ'> = <ψ|ψ'>*\nsye3v0-2.4.6.3 Time reversal symmetry (reverses flow of time)\nsye3v0-2.4.6.4 Can be spontaneously broken (ferromagnets, superfluid helium-3 A-phase)\nsye3v0-2.4.6.5 Necessarily discrete\nsye3v0-2.4.7 Global symmetries versus local symmetries\nsye3v0-2.4.7.1 SSB occurs in systems with many microscopic degrees of freedom\nsye3v0-2.4.7.2 Global symmetry acts in same way on each constituent\nsye3v0-2.4.7.3 Examples in Section 1.3 were global symmetries\nsye3v0-2.4.7.4 Only global symmetries can be spontaneously broken\nsye3v0-2.4.7.5 Local symmetries act differently on different local degrees of freedom\nsye3v0-2.4.7.6 Not to be confused with gauge freedoms\nsye3v0-2.4.7.7 Easiest example: classical ideal gas of N particles, Hamiltonian H = Σi Pi²/2m invariant under Xi(t) → Xi(t) + ai (local translation)\nsye3v0-2.4.7.8 Extended to free field theory: relativistic complex field ψ(x), ψ(x) → ψ(x) + α(x) (local spacetime displacement)\nsye3v0-2.4.7.9 If α satisfies ∂²α = 0, adds boundary term to action\nsye3v0-2.4.7.10 Local symmetries cannot be spontaneously broken (Exercise 2.3)\nsye3v0-2.4.7.11 Do give rise to conserved charges\nsye3v0-2.4.7.12 Free complex scalar field: Noether currents related to δsψ = α(x) at each x\nsye3v0-2.4.7.13 Interpretation: Fourier components of ψ(x) individually conserved\nsye3v0-2.4.7.14 Interacting systems: many-body localisation (MBL) phase characterised by emergent local symmetries\nsye3v0-2.4.8 Active versus passive, and internal versus external symmetries\nsye3v0-2.4.8.1 Not useful for clarifying physical effects\nsye3v0-2.4.8.2 Active transformation: actual transformation of coordinates/fields\nsye3v0-2.4.8.3 Passive transformation: coordinate transformation/relabelling\nsye3v0-2.4.8.4 Physicist perspective: effect on system description is same\nsye3v0-2.4.8.5 Internal symmetry: concerns properties of fields other than spacetime coordinate (e.g., phase rotation)\nsye3v0-2.4.8.6 External/spacetime symmetry: involves transformation of spacetime coordinates (e.g., translations, rotations, dilatations, boosts)\nsye3v0-2.4.8.7 No fundamental difference between breaking global spacetime vs internal symmetry (observable effects result from transformation properties of physical fields)\nsye3v0-2.4.8.8 Specific physical effects may be special to certain types (e.g., fractional dispersion of NG modes for broken spatial symmetries)\nsye3v0-2.5 Gauge freedom\nsye3v0-2.5.1 Transformations leaving Lagrangian/Hamiltonian invariant but have no physical consequence\nsye3v0-2.5.2 Do not give rise to Noether currents/charges\nsye3v0-2.5.3 Cannot be spontaneously broken\nsye3v0-2.5.4 Purely mathematical property of models\nsye3v0-2.5.5 Relabelling your measuring rod\nsye3v0-2.5.5.1 Arbitrary choice of coordinate system origin\nsye3v0-2.5.5.2 Arbitrary choice of scale (e.g., temperature scales Celcius/Fahrenheit/Kelvin)\nsye3v0-2.5.5.3 Term "gauge invariance" coined by Hermann Weyl (eichinvarianz)\nsye3v0-2.5.5.4 Arbitrariness underlies more sophisticated gauge freedom forms\nsye3v0-2.5.6 Superfluous degrees of freedom\nsye3v0-2.5.6.1 Introduced to simplify mathematical description\nsye3v0-2.5.6.2 Often dynamic fields with own equations of motion, but not observable quantities\nsye3v0-2.5.6.3 Maxwell electromagnetism example\nsye3v0-2.5.6.4 Physical observables E and B fields\nsye3v0-2.5.6.5 E and B constrained by Faraday-Maxwell equation (∇ × E + ∂tB = 0) and ∇ · B = 0\nsye3v0-2.5.6.6 E and B written in terms of scalar V and vector A potentials: E = -∇V - ∂tA, B = ∇ × A\nsye3v0-2.5.6.7 Maxwell action SMaxw in terms of four-potential Aµ = (1/c V, A)\nsye3v0-2.5.6.8 Action invariant under gauge transformation Aν(x) → Aν(x) + ∂¯να(x) for arbitrary scalar field α(x)\nsye3v0-2.5.6.9 Transformation does not affect E and B\nsye3v0-2.5.6.10 Aµ has 4 components, E and B determined by 3; fourth component redundant\nsye3v0-2.5.6.11 Gauge transformation never affects predictions for physical observables\nsye3v0-2.5.6.12 Gauge transformations are not symmetries\nsye3v0-2.5.6.13 "Global gauge transformations" terminology outdated\nsye3v0-2.5.6.14 Structure of gauge transformations heavily relied on in elementary particle physics (gauge fields mediating interactions)\nsye3v0-2.5.6.15 Gauge freedom in Maxwell electromagnetism appears with physical symmetry (global U(1))\nsye3v0-2.5.6.16 In presence of charged matter, action Sint = ∫dtd3x eAνjν (jν is Noether current of global U(1))\nsye3v0-2.5.6.17 Sint invariant under gauge transformation Aν → Aν + ∂¯να only if ∂¯νjν = 0 (gauge fields couple only to conserved currents)\nsye3v0-2.5.6.18 Link between global symmetries and gauge freedom using Noether's second theorem\nsye3v0-2.5.6.19 Transformations leaving Lagrangian invariant depend on parameters αn(x) and derivatives ∂µαn(x)\nsye3v0-2.5.6.20 Constraints on possible field configurations regardless of equations of motion\nsye3v0-2.5.6.21 Example: gauge field Aµ coupled to complex scalar field ψ (Ginzburg-Landau theory for superconductivity)\nsye3v0-2.5.6.22 Local transformation on fields: δsψ = -iα(x)ψ, δsAµ = -ħ/e* ∂µα(x) (e* is charge of ψ)\nsye3v0-2.5.6.23 Euler-Lagrange equation Eψ=0\nsye3v0-2.5.6.24 Noether's second theorem states: Eψ(-iψ) + Eψ* (iψ*) = (-ħ/e*)∂µEAµ\nsye3v0-2.5.6.25 Holds whether or not equations of motion satisfied\nsye3v0-2.5.6.26 In classical physics, reduces to trivial equation for on-shell fields\nsye3v0-2.5.6.27 In quantum field theory, constrains off-shell configurations (Ward-Takahasi identities)\nsye3v0-2.5.6.28 Alternative understanding: if EAµ=0, then ∂µjµ=0 (current coupled to must be conserved)\nsye3v0-2.5.6.29 Holds regardless of whether ψ satisfies equations of motion\nsye3v0-2.5.6.30 Final point: global U(1) symmetry spontaneously broken, gauge freedom persists\nsye3v0-2.5.6.31 Coupling with gauge fields leads to Anderson-Higgs effect (Section 7.3)\nsye3v0-2.5.7 Distinguishing gauge freedom from symmetry\nsye3v0-2.5.7.1 Looking at action, local spacetime translations (Eq 1.33) and local gauge transformations (Eq 1.38) seem similar\nsye3v0-2.5.7.2 Former is symmetry (conserved Noether currents), latter is gauge freedom (superfluous degree of freedom)\nsye3v0-2.5.7.3 Method for identifying constraints leading to local gauge freedoms: Dirac treatment of Hamiltonian constraints\nsye3v0-2.5.7.4 Relations between canonical fields/momenta vanishing constitute constraints\nsye3v0-2.5.7.5 Implies redundant degrees of freedom\nsye3v0-2.5.7.6 Distinction for global transformations more subtle\nsye3v0-2.5.7.7 Example: many-body spin system, global spin rotation invariance\nsye3v0-2.5.7.8 Global spin rotation vs global rotation of coordinate system\nsye3v0-2.5.7.9 Relabelling coordinates is archetype of global gauge transformation (no physical implications, cannot be broken)\nsye3v0-2.5.7.10 Way out: symmetry defined with respect to a reference\nsye3v0-2.5.7.11 Magnetisation measurable only if interaction with external reference exists\nsye3v0-2.5.7.12 Example: coupling Lcoupling = -h · S(x) to uniform applied field h\nsye3v0-2.5.7.13 Total action not invariant under global rotation of S keeping h fixed (physical symmetry broken)\nsye3v0-2.5.7.14 Total action invariant under global rotation of S and h (global gauge transformation)\nsye3v0-2.5.7.15 Spontaneous symmetry breaking occurs when ferromagnetic state survives as |h|→0\nsye3v0-2.5.7.16 Broken symmetry is global spin-rotation relative to some reference\nsye3v0-2.5.7.17 Mathematically: reduction of SU(2)S × SU(2)h to diagonal subgroup SU(2)L+R\nsye3v0-2.5.7.18 Case even with infinitely weak external field, must necessarily be allowed to exist\nsye3v0-2.6 Symmetry groups and Lie algebras\nsye3v0-2.6.1 Mathematical structure underlying symmetry transformations\nsye3v0-2.6.2 Symmetry groups\nsye3v0-2.6.2.1 Symmetry transformations correspond to manipulations leaving state/action invariant\nsye3v0-2.6.2.2 Properties: combined effect is symmetry (product), identity transformation I exists, inverse transformation U⁻¹ exists\nsye3v0-2.6.2.3 Set of symmetry transformations {U} forms a group\nsye3v0-2.6.2.4 Continuous vs discrete, finite vs infinite group\nsye3v0-2.6.2.5 Abelian/commutative group: U1U2 = U2U1\nsye3v0-2.6.2.6 Non-Abelian/non-commutative group: not all elements commute\nsye3v0-2.6.2.7 Examples: trivial group e/I, cyclic group Zn/Cn, dihedral group Dn, discrete translation group ZD, translation group RD, SO(2), O(2), SO(3), O(3), SU(2), SU(3)\nsye3v0-2.6.2.8 Exercise 1.4 (Dihedral groups)\nsye3v0-2.6.2.9 Subgroup H of G: subset forming a group itself (multiplication closed)\nsye3v0-2.6.2.10 Subgroups appear in SSB discussion (transformations leaving system invariant after transition)\nsye3v0-2.6.2.11 Coset gH: set of all elements gh with h∈H\nsye3v0-2.6.2.12 Quotient set G/H: collection of all cosets\nsye3v0-2.6.2.13 Cosets classify broken-symmetry states (Section 2.5.1)\nsye3v0-2.6.2.14 Exercise 1.5 (Equivalence and Quotient sets)\nsye3v0-2.6.2.15 Exercise 1.6 (Subgroups of Dihedral groups)\nsye3v0-2.6.3 Lie groups and algebras\nsye3v0-2.6.3.1 Lie group: continuous group with differentiable manifold structure (smooth parameters)\nsye3v0-2.6.3.2 All continuous groups in notes are Lie groups\nsye3v0-2.6.3.3 Dimension of Lie group: number of variables to parametrise transformations\nsye3v0-2.6.3.4 Lie groups contain transformations infinitely close to identity\nsye3v0-2.6.3.5 Transformations expanded around identity: Uα = 1 + iαaQa + O(α²)\nsye3v0-2.6.3.6 Qa are Hermitian operators (symmetry generators)\nsye3v0-2.6.3.7 Generators determine transformations close to identity\nsye3v0-2.6.3.8 Generators Qa are conserved Noether charges\nsye3v0-2.6.3.9 Set of symmetry generators has vector space structure\nsye3v0-2.6.3.10 Commutator (Lie bracket) of generators is linear combination of others: [Qa, Qb] = ifabcQc\nsye3v0-2.6.3.11 fabc are structure constants\nsye3v0-2.6.3.12 Lie algebra: vector space of generators + commutation relations\nsye3v0-2.6.3.13 If group is Abelian, structure constants are zero\nsye3v0-2.6.3.14 Lie algebra does not uniquely define Lie group (discrete transformations not captured)\nsye3v0-2.6.3.15 Lie algebra important for Nambu-Goldstone modes structure (Section 3.2)\nsye3v0-2.6.3.16 Exercise 1.7 (SU(2) Lie Algebra)\nsye3v0-2.6.4 Representation theory\nsye3v0-2.6.4.1 Groups/algebras are abstract mathematical concepts\nsye3v0-2.6.4.2 Representation: set of matrices with same structure (multiplication/addition) as group/algebra elements\nsye3v0-2.6.4.3 Example: cyclic group Cn, rotation of n-sided polygon\nsye3v0-2.6.4.4 Representation as 2x2 real matrices (SO(2))\nsye3v0-2.6.4.5 Representation as 1x1 complex matrices (U(1))\nsye3v0-2.6.4.6 Abstract notion represented in different ways depending on description\nsye3v0-2.6.4.7 Elements satisfy same properties as abstract group elements\nsye3v0-2.6.4.8 Example: SU(2)-invariant 2-vector field ψ=(ψ1, ψ2), Lagrangian L\nsye3v0-2.6.4.9 SU(2) representation in terms of 2x2 matrices (Pauli matrices σa)\nsye3v0-2.6.4.10 Symmetry transformations ψm(x) → Σa,n e-iαa(σa)mn ψn(x)\nsye3v0-2.6.4.11 Distinction between representations of symmetry group and operations on Hilbert space\nsye3v0-2.6.4.12 Lie algebra generators σa do not correspond directly to conserved Noether charges Qa\nsye3v0-2.6.4.13 Qa expressed in terms of fields ψn\nsye3v0-2.6.4.14 Qa = π(-iσaψ) + (iψ†σa)π†\nsye3v0-2.6.4.15 Qa related to representations σa, but not same\nsye3v0-2.6.4.16 Exercise 1.8 (Representations of U(1) and Z)\nsye3v0-2.6.4.17 Faithful representation: distinct group elements represented by distinct matrices\nsye3v0-3.0 Symmetry breaking\nsye3v0-3.1 Spontaneous symmetry breaking (SSB)\nsye3v0-3.1.1 Symmetric system (H, L, or action invariant under unitary transformation) typically has symmetric equilibrium configuration\nsye3v0-3.1.2 Rarely see truly symmetric objects in everyday world\nsye3v0-3.1.3 Explained by theory of spontaneous symmetry breaking\nsye3v0-3.1.4 Definition: stable state (ground state or thermal equilibrium) not symmetric under theory's symmetry\nsye3v0-3.1.5 Consequences of SSB\nsye3v0-3.1.5.1 Order parameter\nsye3v0-3.1.5.2 Tower of states\nsye3v0-3.1.5.3 Effectively restricted configuration space\nsye3v0-3.1.5.4 Singular limits\nsye3v0-3.1.6 Detailed examples: classical physics, harmonic solid, antiferromagnet\nsye3v0-3.1.7 Recurring themes: thermodynamic limit, tower of states, stability\nsye3v0-3.2 Basic notions of SSB\nsye3v0-3.2.1 State |ψ spontaneously breaks symmetry if not left invariant by U\nsye3v0-3.2.2 Multitude of related states with same energy (U|ψ has same energy as |ψ because [U, H]=0)\nsye3v0-3.2.3 Set of inequivalent degenerate states by performing all U on |ψ\nsye3v0-3.2.4 Example: rock localised in space, H translationally invariant, moving rock yields distinct state with same energy\nsye3v0-3.2.5 Order parameter operator O\nsye3v0-3.2.5.1 Eigenstates are inequivalent states in the set\nsye3v0-3.2.5.2 Eigenvalues are different and non-zero for each set\nsye3v0-3.2.5.3 Zero expectation value for symmetric states\nsye3v0-3.2.5.4 Generally does not commute with H (exceptions Section 3.2)\nsye3v0-3.2.5.5 Example: solid object, broken states are eigenstates of position operator X (order parameter), H commutes with momentum P, not X\nsye3v0-3.2.6 Conundrum: states not eigenstates of H, not in thermal equilibrium\nsye3v0-3.2.7 Explained by singularity of thermodynamic limit\nsye3v0-3.2.7.1 Thermodynamic limit: N→∞, V→∞, N/V fixed\nsye3v0-3.2.7.2 Intensive quantities (density, temperature) unchanged\nsye3v0-3.2.7.3 Extensive quantities (N, entropy) grow to infinity\nsye3v0-3.2.7.4 [O, H] expectation value vanishes in thermodynamic limit\nsye3v0-3.2.7.5 Broken states orthogonal in limit, degenerate with symmetric exact eigenstates\nsye3v0-3.2.7.6 In limit, broken states are eigenstates of H, may occur in thermal equilibrium\nsye3v0-3.2.7.7 Limit is singular: qualitatively different for infinite vs finite volume\nsye3v0-3.2.7.8 Real systems have large but finite N, V\nsye3v0-3.2.8 How real, finite-sized objects observed in broken configurations?\nsye3v0-3.2.8.1 Spectra of symmetric Hamiltonians have common properties\nsye3v0-3.2.8.2 H separated into centre-of-mass (k=0) and internal (finite k) parts\nsye3v0-3.2.8.3 Parts commute\nsye3v0-3.2.8.4 Breaking global symmetry only needs collective part\nsye3v0-3.2.8.5 Example: solid object, collective H describes centre-of-mass position/motion\nsye3v0-3.2.8.6 Collective H for free particle of mass mN\nsye3v0-3.2.8.7 Lowest energy levels spaced by ~1/N\nsye3v0-3.2.8.8 Low-energy eigenstates of collective H make up tower of states\nsye3v0-3.2.8.9 States in tower are highly collective and non-local\nsye3v0-3.2.8.10 Cannot be written as product states\nsye3v0-3.2.8.11 For solid, tower consists of eigenstates of total momentum, collectively delocalised\nsye3v0-3.2.8.12 Energy eigenstates in tower increasingly unstable towards local interactions as size increases\nsye3v0-3.2.8.13 Instability prevents them from being realised in everyday world\nsye3v0-3.2.8.14 Weakest asymmetric interaction suffices to destabilise\nsye3v0-3.2.8.15 Stable states are local, may be product states\nsye3v0-3.2.8.16 For rock, localised eigenstates of position operator\nsye3v0-3.2.8.17 Superpositions of states in the tower\nsye3v0-3.2.8.18 Not generally energy eigenstates\nsye3v0-3.2.8.19 Energy uncertainty very small for large systems (spacing ~1/N)\nsye3v0-3.2.8.20 Stable states not orthogonal, overlap drops as e⁻N\nsye3v0-3.2.8.21 Probability of tunnelling to another state exponentially suppressed\nsye3v0-3.2.8.22 Stable states are not symmetric, degenerate in energy expectation value (close to exact ground state)\nsye3v0-3.2.9 Spontaneous symmetry breaking for finite-sized objects\nsye3v0-3.2.9.1 System may exist in stable state not exact eigenstate of symmetric collective H\nsye3v0-3.2.9.2 Symmetric H cannot account for single state selection\nsye3v0-3.2.9.3 External perturbation explicitly breaks symmetry, favours one stable state\nsye3v0-3.2.9.4 Large symmetric system exceedingly sensitive to disturbances\nsye3v0-3.2.9.5 Perturbation ~1/N suffices to single out state\nsye3v0-3.2.9.6 Spontaneous aspect: arbitrarily small perturbation determines fate of large systems\nsye3v0-3.2.9.7 Combined ground state of symmetric collective H + small perturbation is stable symmetry-broken state\nsye3v0-3.2.9.8 In thermodynamic limit, extrapolates to eigenstate of order parameter\nsye3v0-3.2.9.9 In strict absence of perturbations, symmetric ground state is true ground state for any size\nsye3v0-3.2.9.10 State in thermodynamic limit changes qualitatively if infinitesimal perturbation added/removed (singular nature manifestation)\nsye3v0-3.2.10 Because overlap exponentially suppressed, treat system as single symmetry-broken ground state for practical purposes\nsye3v0-3.2.11 Entire dynamics takes place in restricted part of Hilbert space\nsye3v0-3.2.12 Configuration space effectively restricted to small subspace\nsye3v0-3.2.13 For physics of broken phase, consider effective H within subspace\nsye3v0-3.2.14 Important instance of ergodicity breaking (part of phase space not accessible)\nsye3v0-3.2.15 Occurs in disordered systems (glasses Section A.1)\nsye3v0-3.3 Singular limits\nsye3v0-3.3.1 Possibility of SSB closely related to singular nature of thermodynamic limit\nsye3v0-3.3.2 Singular limits occur throughout physics and daily life, not common in standard curricula\nsye3v0-3.3.3 Example: classical perfect cylinder/pencil balanced on table\nsye3v0-3.3.4 Perfectly balanced (θ=0) symmetric under rotations normal to table\nsye3v0-3.3.5 Sharpening tip (b→0)\nsye3v0-3.3.6 Balancing sharp pencil impossible in real life, falls over in single direction (spontaneously breaks symmetry)\nsye3v0-3.3.7 Mathematically: lim b→0 lim θ→0 y > 0, lim θ→0 lim b→0 y = 0 (limits do not commute)\nsye3v0-3.3.8 Failure of limits to commute is necessary and distinctive signal of singular limit\nsye3v0-3.3.9 Implies non-analytic feature in some function\nsye3v0-3.3.10 Example: y = arctan(zx), limits x→0 and z→∞ do not commute\nsye3v0-3.3.11 In limit z→∞, y(x) becomes step function, value at x=0 depends on approach direction\nsye3v0-3.3.12 Value changes qualitatively under infinitesimally small changes around zero\nsye3v0-3.3.13 In pencil example, neither limit b→0 nor θ→0 realised in practice\nsye3v0-3.3.14 Meaning of singular limits: for sufficiently sharp pencil, arbitrarily difficult to balance, always tips over\nsye3v0-3.3.15 Particularities about pencil SSB\nsye3v0-3.3.15.1 Which symmetry broken? Global rotational symmetry relative to table (reference)\nsye3v0-3.3.15.2 Not global gauge freedom of rotating universe\nsye3v0-3.3.15.3 Same issue as Section 1.5.3 (distinguishing gauge freedom from symmetry)\nsye3v0-3.3.15.4 Symmetric state unstable, broken states stable (consistent with Section 2.1)\nsye3v0-3.3.15.5 In classical physics, stable broken states are ground states even for finite systems\nsye3v0-3.3.15.6 Hamiltonian cannot account for direction of fall\nsye3v0-3.3.15.7 External perturbation favouring θ necessary\nsye3v0-3.3.15.8 Spontaneous aspect: perturbation arbitrarily small for sufficiently sharp pencil\nsye3v0-3.3.16 Exercise 2.1 (Classical magnet)\nsye3v0-3.3.16.1 Classical magnet: many microscopic bar magnets coupled by nearest-neighbour interactions\nsye3v0-3.3.16.2 Internal energy E = Σx,δ -|J|Sx · Sx+δ\nsye3v0-3.3.16.3 Energy minimised when all magnets point in same direction (ferromagnet)\nsye3v0-3.3.16.4 Invariant under simultaneous rotation of all magnets (global symmetry)\nsye3v0-3.3.16.5 States of maximum magnetisation degenerate regardless of direction\nsye3v0-3.3.16.6 Thermal expectation value MT = Σstates e-E/kBT M / Σstates e-E/kBT\nsye3v0-3.3.16.7 Expectation value of total magnetisation is zero for any temperature, even T=0\nsye3v0-3.3.16.8 Classical magnets in thermal equilibrium cannot have well-defined north/south pole\nsye3v0-3.3.16.9 Resolution: spontaneous breakdown of symmetry by adding small symmetry-breaking magnetic field h\nsye3v0-3.3.16.10 E' = Σx,δ -|J|Sx · Sx+δ - hn̂ · Sx\nsye3v0-3.3.16.11 Expectation value for magnetisation at T=0 will be Nsn̂ (N=number of magnets, s=|Sx|)\nsye3v0-3.3.16.12 Non-commuting limits: lim N→∞ lim h→0 MT/N = 0, lim h→0 lim N→∞ MT/N = sn̂\nsye3v0-3.3.16.13 Real magnets do not thermalise as described by Eq. (2.4)\nsye3v0-3.3.16.14 States with different magnetisation not accessible on ordinary time scales\nsye3v0-3.3.16.15 Simultaneous rotation of all magnets exceedingly unlikely for large magnets\nsye3v0-3.3.16.16 Large part of configuration space effectively inaccessible\nsye3v0-3.4 Symmetry breaking in the thermodynamic limit\nsye3v0-3.4.1 Spontaneous breakdown signalled by singular thermodynamic limit\nsye3v0-3.4.2 Practical implication: symmetric states unstable, broken states stable in large, finite systems\nsye3v0-3.4.3 Many aspects easier to describe in thermodynamic limit (stable/unstable states degenerate)\nsye3v0-3.4.4 Standard approach: classify broken-symmetry states entirely within thermodynamic limit\nsye3v0-3.4.5 Classification of broken-symmetry states\nsye3v0-3.4.5.1 Consider Hamiltonians/states with translational invariance (periodic arrangement of unit cells)\nsye3v0-3.4.5.2 Examples: crystal, antiferromagnetic N´eel state\nsye3v0-3.4.5.3 Translational invariance allows Fourier transformation (prerequisite for NG modes)\nsye3v0-3.4.5.4 Broken state as product state |ψ = ⊗i |ψi (ψi same local function)\nsye3v0-3.4.5.5 Translational invariance sufficient for distinct broken states to be orthogonal in thermodynamic limit\nsye3v0-3.4.5.6 Overlap <ψ'|ψ> = Πx <ψ'(x)|ψ(x)> = <ψ'(x)|ψ(x)>N\nsye3v0-3.4.5.7 If |ψ> and |ψ'> distinct, local overlap <ψ'(x)|ψ(x)> < 1, full inner product vanishes as N→∞\nsye3v0-3.4.5.8 If states not distinct, differ only by phase |ψ'(x)> = eiϕ|ψ(x)>, inner product is one (up to phase)\nsye3v0-3.4.5.9 Any two symmetry-breaking states in thermodynamic limit are equivalent or orthogonal\nsye3v0-3.4.5.10 Apply group theory (Section 1.6.1) to classify distinct states\nsye3v0-3.4.5.11 Symmetry transformations make up group G\nsye3v0-3.4.5.12 g |ψ = eiϕ |ψ for unbroken symmetry transformations g\nsye3v0-3.4.5.13 Set of unbroken transformations forms subgroup H ⊂ G (residual symmetry group)\nsye3v0-3.4.5.14 Exercise 2.5 (Subgroup of unbroken transformations)\nsye3v0-3.4.5.15 g1 |ψ equivalent to g2 |ψ if g1 = g2h for some h∈H\nsye3v0-3.4.5.16 If g1 = g2h, g1|ψ> and g2|ψ> are distinct/orthogonal broken states\nsye3v0-3.4.5.17 Inequivalent broken states classified by cosets gH of quotient set G/H\nsye3v0-3.4.5.18 G is group of all symmetry transformations, H is subgroup of unbroken transformations\nsye3v0-3.4.5.19 Classification in terms of cosets applies only to thermodynamic limit\nsye3v0-3.4.5.20 Finite size: distinct states need not be orthogonal\nsye3v0-3.4.5.21 Finite object breaking continuous rotational symmetry: infinite states labelled by directions, correspond to infinite cosets\nsye3v0-3.4.5.22 Example: global phase-rotational symmetry G=U(1), broken to trivial group H=e\nsye3v0-3.4.5.23 Quotient set G/H = U(1), states labelled by phase factors eiϕ\nsye3v0-3.4.5.24 Broken symmetry characterising superfluid, different phase values distinguished by Josephson current (Section 2.5.5)\nsye3v0-3.4.5.25 If all symmetry transformations in group broken, states labelled by elements of full group\nsye3v0-3.4.5.26 Example: spin-rotation symmetry G=SU(2) broken to H=U(1) (rotations around single axis)\nsye3v0-3.4.5.27 Quotient set G/H = SU(2)/U(1) ≈ S² (set of points on sphere)\nsye3v0-3.4.5.28 Points indicate directions of residual rotation axis/sublattice magnetisation\nsye3v0-3.4.5.29 S² classifies states, does not have group structure\nsye3v0-3.4.5.30 For continuous groups, classification also in terms of generators Q\nsye3v0-3.4.5.31 Unbroken generators: Q of which state is eigenstate\nsye3v0-3.4.5.32 Broken generators: do not leave state invariant\nsye3v0-3.4.5.33 Continuous group broken to continuous/discrete subgroup or trivial group\nsye3v0-3.4.5.34 Dimension of quotient set G/H equals number of broken generators\nsye3v0-3.4.5.35 Algebraic relations between broken/unbroken generators important for NG modes (Chapter 3)\nsye3v0-3.5 The order parameter\nsye3v0-3.5.1 Operator whose expectation value distinguishes inequivalent broken states\nsye3v0-3.5.2 Ideally: zero expectation value in symmetric state, unique non-zero value for each set of equivalent broken states\nsye3v0-3.5.3 Word "order parameter" often used for any quantity non-zero in broken phase, zero in symmetric phase\nsye3v0-3.5.4 Example: amplitude of gap function in superconductivity (Section 7.2)\nsye3v0-3.5.5 Narrow definition used here instrumental for Goldstone theorem (Chapter 3)\nsye3v0-3.5.6 Definition 2.1: U = eiαQ symmetry transformation, state |ψ breaks symmetry if ∃ operator Φ such that ψ|[Q, Φ]|ψ = 0\nsye3v0-3.5.7 If no such Φ exists, state is symmetric\nsye3v0-3.5.8 Consistent with earlier definition (eigenstates of U)\nsye3v0-3.5.9 Operator Φ is interpolating field (field Φ(x) acting locally)\nsye3v0-3.5.10 Order parameter operator O(x) related to broken symmetry Q\nsye3v0-3.5.11 Expectation value O(x) = ψ|O(x)|ψ is local order parameter\nsye3v0-3.5.12 O(x) = [Q, Φ(x)]\nsye3v0-3.5.13 O(x) is left-hand side of Eq. (2.30)\nsye3v0-3.5.14 O(x) automatically zero if |ψ symmetric, non-zero if |ψ breaks symmetry\nsye3v0-3.5.15 To distinguish inequivalent states, O operator has eigenvalues mapping one-to-one onto G/H\nsye3v0-3.5.16 O(x) different for distinct states, equal for states related by residual symmetry\nsye3v0-3.5.17 O(x) inherits structure of quotient space\nsye3v0-3.5.18 States close in G have small difference in O(x) values\nsye3v0-3.5.19 Eq. (2.30) does not uniquely determine O and Φ\nsye3v0-3.5.20 Multiplying Φ by constant or taking Hermitian conjugate yields alternative definitions\nsye3v0-3.5.21 Convenient choice for O suggested by physics of system\nsye3v0-3.5.22 Example: Heisenberg antiferromagnet (SU(2) broken to U(1))\nsye3v0-3.5.23 Inequivalent states: antiferromagnetic configurations with sublattice magnetisation in different directions (S²)\nsye3v0-3.5.24 For state with magnetisation along z-axis, broken generators Sˣ, Sʸ, unbroken Sᶻ\nsye3v0-3.5.25 Define staggered magnetisation Na_i = (±1)iSa_i (i=position index, a=spin direction)\nsye3v0-3.5.26 To describe breaking of Sˣ, choose Ny as interpolating field\nsye3v0-3.5.27 Order parameter operator [Sˣ_i, Ny_j] = iδijNz_i = iΣi Nz_i\nsye3v0-3.5.28 Staggered magnetisation identified as order parameter\nsye3v0-3.5.29 Example: Schrödinger field theory (global U(1) phase rotation)\nsye3v0-3.5.30 Choose ψ(x) itself as interpolating field\nsye3v0-3.5.31 Associated order parameter [Q, ψ(x)] = -ψ(x)\nsye3v0-3.5.32 Order parameter operator is field ψ(x) itself\nsye3v0-3.5.33 Eigenstates of ψ(x) are coherent states e∫dDx φ(x)ψ*(x)|vac\nsye3v0-3.5.34 Coherent state is eigenstate of annihilation operator ψ(x) with eigenvalue φ(x)\nsye3v0-3.5.35 |φ(x)|² is expectation value of number of field quanta/particles at x\nsye3v0-3.5.36 Translational invariance: φ independent of x\nsye3v0-3.5.37 Coherent state is superposition of infinitely many states with different numbers of quanta\nsye3v0-3.5.38 Eigenstate of number/density operator ψ*(x)ψ(x) is superposition of coherent states with same |φ(x)| but different phase\nsye3v0-3.5.39 Phase and modulus of φ are conjugate variables\nsye3v0-3.5.40 SSB state of Schrödinger field is coherent state (indeterminate particle number, precise phase)\nsye3v0-3.5.41 Consistent with Q ∝ ∫ψ*ψ associated with conservation of particle number\nsye3v0-3.5.42 Distinct states characterised by phase of φ\nsye3v0-3.5.43 Rotations with exp(iαQ) lead to other order parameter operators e iαψ(x)\nsye3v0-3.5.44 Correspond to broken states with different phase values (coset U(1)/1 ≈ U(1))\nsye3v0-3.5.45 Formation of state with indeterminate particle number and precise phase is good interpretation for Bose-Einstein condensates (superfluids, superconductors)\nsye3v0-3.5.46 In these states, zero energy to add/remove particle in condensate\nsye3v0-3.6 The classical state\nsye3v0-3.6.1 Given O definition, tempted to believe broken states are eigenstates of O\nsye3v0-3.6.2 Justifies O as expectation value, agrees with expectation for perfectly ordered state\nsye3v0-3.6.3 In translationally invariant systems, eigenstates of local O(x) are tensor products of local eigenstates |ψ = ⊗x |ψ(x)\nsye3v0-3.6.4 Correspond directly to states of classical Hamiltonian (operators replaced by expectation values)\nsye3v0-3.6.5 Call these eigenstates of order parameter operator classical states\nsye3v0-3.6.6 Real quantum broken states typically not classical states\nsye3v0-3.6.7 Symmetry-breaking perturbation may dominate ground state shape for large systems, but not only contribution\nsye3v0-3.6.8 Remaining symmetric part of H contributes, takes state away from classical ideal\nsye3v0-3.6.9 True quantum broken states have O expectation values close to classical state\nsye3v0-3.6.10 Quantum states thought of as arising in perturbation theory around classical states\nsye3v0-3.6.11 Differences are quantum corrections to classical state\nsye3v0-3.6.12 Corrections: part at zero wave number (tower of states Section 2.6), part at non-zero momentum (Chapter 4)\nsye3v0-3.7 Long-range order\nsye3v0-3.7.1 For uniform ground state, O identifies broken/unbroken symmetries, distinguishes inequivalent states\nsye3v0-3.7.2 Practical calculations: often don't know exact ground state, systems not perfectly uniform\nsye3v0-3.7.3 Alternative way to quantify SSB: two-point correlation function C(x, x') = ψ| O†(x)O(x')|ψ\nsye3v0-3.7.4 If |ψ uniform and eigenstate of O, C(x,x') same for any x,x', equals |O|²\nsye3v0-3.7.5 Advantage: can be used in less clear-cut cases\nsye3v0-3.7.6 Behaviour of C(x,x') as |x-x'|→∞ distinguishes classes\nsye3v0-3.7.7 C(x,x') ∝ constant: long-range ordered\nsye3v0-3.7.8 C(x,x') ∝ e-|x-x'|/l: disordered (l=correlation length)\nsye3v0-3.7.9 Long-range ordered systems: spatial average of O non-zero, l diverges\nsye3v0-3.7.10 Presence of long-range order associated with breaking of symmetry\nsye3v0-3.7.11 C(x,x') signals propensity to break symmetry even for finite systems with symmetric ground state\nsye3v0-3.7.12 O expectation value zero, C(x,x') shows correlations for long separations\nsye3v0-3.7.13 Example: two-spin singlet state |↑↓ - |↓↑> (Heisenberg antiferromagnet)\nsye3v0-3.7.14 Singlet state no preferred staggered magnetisation direction, spins anti-parallel\nsye3v0-3.7.15 C(x,x') easy to 'measure' numerically, widely used to establish order/SSB\nsye3v0-3.7.16 Separation into long-range ordered/disordered not exhaustive\nsye3v0-3.7.17 Special case: low dimensions, C(x,x') ∝ |x-x'|c (algebraic/quasi long-range order Section 6.3)\nsye3v0-3.7.18 Term "off-diagonal long-range order" (ODLRO) introduced by Penrose/Onsager (1950s) for superfluidity\nsye3v0-3.7.19 Concept historical, obsolete with modern O definition\nsye3v0-3.7.20 ODLRO definition: in terms of N-particle wave function Ψ(x1,...,xN)\nsye3v0-3.7.21 ρ(x1,...,xN, y1,...,yN) = Ψ*(x1,...,xN)Ψ(y1,...,yN)\nsye3v0-3.7.22 Coinciding coordinates yi=xi: ρ(xi,xi) is density matrix\nsye3v0-3.7.23 Two-particle reduced density matrix ρD,2(x1,x2) by integrating over N-2 coordinates\nsye3v0-3.7.24 Uniform space: ρD,2 depends on x1-x2\nsye3v0-3.7.25 Diagonal long-range order (DLRO): ρD,2(x1-x2) periodic in x1-x2 (solids, harmonic crystal Section 2.3)\nsye3v0-3.7.26 Called diagonal because considers diagonal elements of ρ(xi,yi) with yi=xi\nsye3v0-3.7.27 Alternative two-point function ρO,2(x,y) by integrating over N-1 coordinates\nsye3v0-3.7.28 ρO,2 almost identical to C(x,x') if O(x) is field operator Ψ(x)\nsye3v0-3.7.29 ODLRO: ρO,2(x,y) does not vanish as x-y→∞ (superfluids)\nsye3v0-3.7.30 Called off-diagonal because involves off-diagonal elements of ρ(xi,yi)\nsye3v0-3.7.31 Applying Eq. (2.34) to crystals: C(x,x') approaches periodic function, not constant\nsye3v0-3.7.32 Both DLRO and ODLRO part of larger family classified by C(x,x') behaviour\nsye3v0-3.7.33 Nothing special about order occurring in crystals vs superfluids\nsye3v0-3.7.34 ODLRO originally introduced to capture long-range ordering of internal U(1) phase degree of freedom in N-body wave function\nsye3v0-3.8 The Josephson effect\nsye3v0-3.8.1 Symmetry must always be defined with respect to some reference (Section 1.1)\nsye3v0-3.8.2 Example: crystal breaking translational symmetry defined with respect to outside observer\nsye3v0-3.8.3 To observe breakdown, observer needs reference frame in broken-symmetry state\nsye3v0-3.8.4 Cannot measure crystal position relative to uniform fluid\nsye3v0-3.8.5 Cannot measure magnetisation direction with non-magnetised plastic\nsye3v0-3.8.6 What about U(1) phase-rotation symmetry in superfluids?\nsye3v0-3.8.7 Question manifested in spontaneous tunnelling current predicted by Josephson (1962)\nsye3v0-3.8.8 Occurs between two separated pieces of superconducting material\nsye3v0-3.8.9 Origin in broken U(1) symmetry\nsye3v0-3.8.10 Discussed for neutral superfluids rather than superconductors\nsye3v0-3.8.11 Superfluid state with broken U(1) phase-rotation symmetry, complex order parameter ψ=|ψ|eiϕ\nsye3v0-3.8.12 Characterised by ability to host supercurrents (flow without viscosity)\nsye3v0-3.8.13 Supercurrent identical to conserved Noether current associated with broken U(1) symmetry: jn = i((∂nψ*)ψ - ψ*(∂nψ))\nsye3v0-3.8.14 Order parameter field ψ is expectation value of field operator ("macroscopic wave function")\nsye3v0-3.8.15 ψ(x) satisfies equations of motion with spatial derivatives, forced to be continuous\nsye3v0-3.8.16 Does not vanish abruptly at boundary, falls off exponentially into vacuum\nsye3v0-3.8.17 Two superfluid samples separated by small gap, order parameter fields overlap in gap\nsye3v0-3.8.18 Implies possibility for field quanta to tunnel (essence of Josephson effect)\nsye3v0-3.8.19 Junction of width w between superfluids with constant order parameters ψ1, ψ2\nsye3v0-3.8.20 Order parameter field in junction ψ(x) = Ae-x/ξ + Bex/ξ (ξ=decay length)\nsye3v0-3.8.21 Samples semi-infinite in x, extend indefinitely in y,z\nsye3v0-3.8.22 Boundary conditions: ψ(-w/2)=ψ1, ψ(w/2)=ψ2\nsye3v0-3.8.23 Coefficients A, B derived from boundary conditions\nsye3v0-3.8.24 Current density in junction jx = 2i(B*A - A*B)/ξ, independent of x\nsye3v0-3.8.25 Substituting A, B yields jx = 2|ψ1||ψ2|/ξsinh(w/ξ) sin(ϕ2-ϕ1)\nsye3v0-3.8.26 Current per unit area jx proportional to sine of phase difference between order parameters\nsye3v0-3.8.27 Flow of supercurrent without chemical potential difference is interesting physical observation\nsye3v0-3.8.28 Gains fundamental interpretation in SSB context\nsye3v0-3.8.29 Phase of one sample determined relative to second by measuring Josephson current\nsye3v0-3.8.30 Analogous to crystal position determined relative to other object\nsye3v0-3.8.31 Discovery of Josephson effect deciding factor in settling debate on SSB in superconductors\nsye3v0-3.8.32 Generalised Josephson effects applicable to materials with any SSB\nsye3v0-3.8.33 Unambiguous general way of measuring order parameter of sample relative to reference broken state\nsye3v0-3.8.34 Exercise 2.6 (Josephson effect)\nsye3v0-3.8.34.1 Generalised Josephson current derived from simple model (Feynman)\nsye3v0-3.8.34.2 Global order parameter operators ΨL(t), ΨR(t)\nsye3v0-3.8.34.3 Hamiltonian H = HL + HR + HK (HK coupling)\nsye3v0-3.8.34.4 For superconductors, HK = K(ψ*RψL + ψ*LψR)\nsye3v0-3.8.34.5 Derive IJ = ∂t(ψ*LψL) using Heisenberg equations of motion\nsye3v0-3.8.34.6 For ferromagnets, HK = KML · MR\nsye3v0-3.8.34.7 Derive "spin Josephson current" ∂tML\nsye3v0-3.9 Nambu-Goldstone modes\nsye3v0-3.9.1 Every symmetry of H/L corresponds to conserved quantity\nsye3v0-3.9.2 Holds regardless of whether state respects symmetry\nsye3v0-3.9.3 Example: classical ball (broken translational symmetry) and electron (symmetric plane-wave) have conserved total momentum\nsye3v0-3.9.4 Relation between conserved global quantity and SSB elucidated in Section 2.6 (tower of states)\nsye3v0-3.9.5 Collective k=0 part of spectrum consists of eigenstates of conserved global quantity\nsye3v0-3.9.6 Both individual eigenstates and SSB superposition conserve global quantity\nsye3v0-3.9.7 Noether's theorem implications beyond global aspects\nsye3v0-3.9.8 For continuous global symmetry, guarantees existence of locally conserved current ∂νjν = 0\nsye3v0-3.9.9 Holds regardless of whether state respects symmetry\nsye3v0-3.9.10 Local conservation law tied to generic property of spectrum of systems with SSB continuous symmetry\nsye3v0-3.9.11 Impacts excitations at non-zero wave number\nsye3v0-3.9.12 Guarantees appearance of gapless modes (Nambu-Goldstone/NG modes)\nsye3v0-3.9.13 In particle physics/relativistic QFT, called Goldstone bosons (massless)\nsye3v0-3.9.14 Difference is nomenclature\nsye3v0-3.9.15 Nature of NG modes: consider temporal component of Noether current jt(x) related to broken symmetry generator Q\nsye3v0-3.9.16 NG mode |π(k) viewed as plane-wave superposition of local excitations created by acting with jt(x) on broken state |ψ>\nsye3v0-3.9.17 |π(k,t) ∝ ∫dDx eik·x jt(x,t)|ψ>\nsye3v0-3.9.18 Goldstone's theorem shows these states are gapless (energy goes to zero as k→0)\nsye3v0-3.9.19 Low-energy excitations correspond to creating local Noether charge density\nsye3v0-3.9.20 Continuity equation guarantees dispersion over time\nsye3v0-3.9.21 Low-energy disturbances carried away like waves\nsye3v0-3.9.22 Systems with SSB endowed with rigidity\nsye3v0-3.10 Goldstone's theorem\nsye3v0-3.10.1 Theorem 3.1 (Goldstone's theorem): Global, continuous symmetry spontaneously broken (no long-ranged interactions, discrete translational symmetry intact) -> mode in spectrum whose energy vanishes as wave number approaches zero\nsye3v0-3.10.2 Includes many assumptions\nsye3v0-3.10.3 If assumptions don't hold, NG mode ceases to exist or is not gapless\nsye3v0-3.10.4 If symmetry explicitly broken (e.g., external field µ), NG mode exists but with gap µ at k→0\nsye3v0-3.10.5 If broken symmetry discrete, no NG mode\nsye3v0-3.10.6 If symmetry with gauge freedom (long-ranged interaction), NG mode couples to gauge field, develops gap (Anderson-Higgs mechanism Section 7.3)\nsye3v0-3.10.7 Original theorem required Lorentz invariance, non-relativistic versions derived later (Section 3.2)\nsye3v0-3.10.8 Requirement of translational invariance in broken state same as for Noether's theorem proof\nsye3v0-3.10.9 Need translational invariance on coarse-grained level\nsye3v0-3.10.10 Momentum is good quantum number, modes have definite momentum\nsye3v0-3.10.11 Define complete set of eigenstates of H, |n, k, labelled by momentum k and energy En(k), n=other quantum numbers\nsye3v0-3.10.12 States orthogonal: <n', k'|n, k> = (2π)Dδnnδ(k-k')\nsye3v0-3.10.13 Resolution of identity I = Σn ∫dDk/(2π)D |n, k><n, k|\nsye3v0-3.10.14 Insert into definition of broken state Eq. (2.30) in terms of interpolating field Φ\nsye3v0-3.10.15 ψ|[Q, Φ]|ψ = Σn ∫dDk/(2π)D <ψ|Q(t)|n, k><n, k|Φ|ψ> - c.c. = 0\nsye3v0-3.10.16 Write Q as integral of jt: ∫ΩdDx Σn ∫dDk/(2π)D <ψ|jt(x,t)|n, k><n, k|Φ|ψ> - c.c. = 0\nsye3v0-3.10.17 Goldstone's theorem addresses modes as k→0, not states at k=0 (tower of states)\nsye3v0-3.10.18 Related to behaviour of Noether charge density integrated over large, finite space part\nsye3v0-3.10.19 Primary assumption: integration volume Ω large but finite\nsye3v0-3.10.20 Interpolating field Φ is local, contributions from outside Ω vanish (causality in relativistic, no long-ranged interactions in non-relativistic/effective)\nsye3v0-3.10.21 Use translational invariance: jt(x,t) translated in time/space using shift operators\nsye3v0-3.10.22 <ψ|jt(x,t)|n, k> = e i(En t - k·x) <ψ|jt(0,0)|n, k>\nsye3v0-3.10.23 ψ is zero-momentum state\nsye3v0-3.10.24 Define (2π)DδΩ(k) = ∫ΩdDx exp(ik·x) (strongly peaked function)\nsye3v0-3.10.25 <ψ|[Q, Φ]|ψ> = Σn ∫dDk/(2π)D δΩ(k) e iEnt <ψ|jt(0,0)|n, k><n, k|Φ|ψ> - c.c. = 0\nsye3v0-3.10.26 Broken state |ψ> implies O non-zero, implies ∃ state |n, k> such that integrand non-zero for large Ω, k→0\nsye3v0-3.10.27 First part of theorem: must exist state near zero momentum excited by jt(0,0) and Φ\nsye3v0-3.10.28 If Φ time-independent, Q time-independent, [Q, Φ] time-independent in thermodynamic limit\nsye3v0-3.10.29 ∂t[Q, Φ] = Σn ∫dDk/(2π)D δΩ(k) iEn e iEnt <ψ|jt(0,0)|n, k><n, k|Φ|ψ> - c.c. = 0\nsye3v0-3.10.30 Implies En(k) → 0 as k→0 for NG mode\nsye3v0-3.10.31 Completes proof: system with SSB has excitation whose energy vanishes as wave number approaches zero\nsye3v0-3.10.32 Goldstone's theorem constructive: indicates how to find modes (act with jt or Φ on broken state)\nsye3v0-3.11 Counting of NG modes\nsye3v0-3.11.1 Derivation suggests one NG mode per broken symmetry generator (not always true)\nsye3v0-3.11.2 Heisenberg ferromagnet: 1 NG mode, 2 spin-rotation symmetries broken\nsye3v0-3.11.3 Energy of NG modes not always linear in momentum (En ∝ k)\nsye3v0-3.11.4 Relativistic systems: Lorentz symmetry dictates time/space derivatives equal footing, En ∝ k\nsye3v0-3.11.5 Non-relativistic systems: Heisenberg ferromagnet NG mode quadratic in momentum\nsye3v0-3.11.6 Goldstone's theorem states at least one NG mode, not dispersion relation other than gapless\nsye3v0-3.11.7 How many NG modes? What replaces intuitive rule? Cleared up recently\nsye3v0-3.11.8 Cannot be found in standard textbooks, accessible in original literature [36-43] or review [44]\nsye3v0-3.11.9 NG modes excited by generator of broken symmetry or interpolating field\nsye3v0-3.11.10 Special case: interpolating field Φ is also generator of broken symmetry\nsye3v0-3.11.11 Example: Heisenberg ferromagnet, S² obtains non-zero expectation value\nsye3v0-3.11.12 Commutator of broken generators Sˣ, Sʸ proportional to S²\nsye3v0-3.11.13 Broken generators act as interpolating fields for each other\nsye3v0-3.11.14 Eq. (3.4) shows they excite same NG mode [38]\nsye3v0-3.11.15 Generally: take two symmetry generators Qa,b = ∫x jt a,b(x)\nsye3v0-3.11.16 Commutator expectation value: <ψ|[Qa, jt b(x)]|ψ> = Σc ifabc <ψ|jt c(x)|ψ>\nsye3v0-3.11.17 If commutator has non-zero expectation value in broken state, they excite same NG mode\nsye3v0-3.11.18 Call such modes type-B, 'ordinary' NG modes type-A [41]\nsye3v0-3.11.19 From Eq. (3.6), type-B modes cannot arise for Abelian symmetry groups (generators commute)\nsye3v0-3.11.20 To systematically count modes: construct Watanabe-Brauner matrix Mab = -i<ψ|[Qa, jt b(x)]|ψ> [40]\nsye3v0-3.11.21 a, b label broken symmetry generators, |ψ is SSB state\nsye3v0-3.11.22 Matrix elements do not depend on position x (translational invariance)\nsye3v0-3.11.23 Numbers nA and nB of type-A and type-B modes given by: nA = dim G/H - rank M, nB = 1/2 rank M [40]\nsye3v0-3.11.24 Two independent proofs [41,42] show type-A modes linear dispersion, type-B quadratic dispersion (in almost all cases)\nsye3v0-3.11.25 Understood using low-energy effective Lagrangian method [39, 41, 43, 45]\nsye3v0-3.11.26 Leff written in terms of fields πa(x) taking values in G/H\nsye3v0-3.11.27 Number of fields = number of broken symmetry generators (not necessarily independent)\nsye3v0-3.11.28 Gapless modes in Leff spectrum correspond to NG modes\nsye3v0-3.11.29 Lowest order terms in Leff: Leff = mab(πa∂tπb - πb∂tπa) + ḡab∂tπa∂tπb - gab∇πa · ∇πb [41]\nsye3v0-3.11.30 mab, ḡab, gab are coefficients constrained by symmetry\nsye3v0-3.11.31 No terms linear in gradients (isotropic space)\nsye3v0-3.11.32 First term mab(πa∂tπb - πb∂tπa) breaks Lorentz invariance, non-zero only in non-relativistic systems\nsye3v0-3.11.33 Watanabe/Murayama show mab given by elements of Mab [41]\nsye3v0-3.11.34 Exercise 3.1 (Number of type-B NG modes)\nsye3v0-3.11.34.1 Part of proof in Ref. [41]: Mab real, antisymmetric\nsye3v0-3.11.34.2 Orthogonal transformation O such that M̃ = OMOT takes block diagonal form with 2x2 antisymmetric blocks Mi\nsye3v0-3.11.34.3 Eigenvalues of M are purely imaginary\nsye3v0-3.11.34.4 Non-zero eigenvalues come in conjugate pairs iλi, -iλi\nsye3v0-3.11.34.5 rank M is even, so nB is integer\nsye3v0-3.11.34.6 For each 2x2 submatrix ei, find unitary wi such that wieiw†i = Mi\nsye3v0-3.11.34.7 M unitarily equivalent to M̃ by WU\nsye3v0-3.11.34.8 M, M̃ real implies orthogonally equivalent\nsye3v0-3.11.35 From effective Lagrangian, find equations of motion and dispersion relations\nsye3v0-3.11.36 Systems where mab=0 (relativistic systems): Leff describes modes with linear dispersions ω²∝k²\nsye3v0-3.11.37 Systems where mab≠0: mab terms dominate at low energies, dispersion quadratic ω∝k²\nsye3v0-3.11.38 πa∂tπa terms are total derivatives, vanish in action\nsye3v0-3.11.39 mab must be antisymmetric\nsye3v0-3.11.40 First term in Eq. (3.9) non-zero only if two fields coupled\nsye3v0-3.11.41 Reduction in gapless modes (two generators excite same mode) and quadratic dispersion go hand-in-hand\nsye3v0-3.11.42 Coefficients gab can be zero\nsye3v0-3.11.43 Higher-order terms taken into account\nsye3v0-3.11.44 Type-A modes can have quadratic dispersion ω²∝k⁴ (Tkachenko modes in vortex lattices [43])\nsye3v0-3.12 Examples of NG modes\nsye3v0-3.12.1 Selection of practical examples\nsye3v0-3.12.2 Superfluid: complex scalar field theory, field operator is order parameter\nsye3v0-3.12.3 Action invariant under phase rotations, Noether's theorem links to particle number conservation\nsye3v0-3.12.4 U(1) phase-symmetry spontaneously broken, particle number indeterminate\nsye3v0-3.12.5 One broken symmetry generator, one NG mode\nsye3v0-3.12.6 Excited by finite-wave-number rotations of phase variable\nsye3v0-3.12.7 NG mode is type-A, linear dispersion in momentum\nsye3v0-3.12.8 Supercurrent (particle current without viscosity) is direct manifestation\nsye3v0-3.12.9 Crystal: D spatial dimensions, breaks D translations and D(D-1)/2 rotations\nsye3v0-3.12.10 Translation group Abelian, associated NG modes are type-A, linear dispersions\nsye3v0-3.12.11 Called phonons or sound waves, one for each direction of space\nsye3v0-3.12.12 Rigidity due to breaking translational symmetry is shear rigidity\nsye3v0-3.12.13 Broken rotational symmetries do not lead to additional NG modes [47-49]\nsye3v0-3.12.14 Rotations and translations not independent symmetry operations\nsye3v0-3.12.15 NG fields excited by broken translations/rotations not independent, contain redundant degrees of freedom\nsye3v0-3.12.16 Broken rotations do not lead to independent NG modes\nsye3v0-3.12.17 Intuitively: exciting rotational NG mode by torque excites transverse sound modes\nsye3v0-3.12.18 Lorentz boosts also spontaneously broken in crystal, do not lead to independent NG modes [50]\nsye3v0-3.12.19 Antiferromagnet: Heisenberg antiferromagnet Section 2.4, breaks two out of three spin-rotational symmetries (Sˣ, Sʸ)\nsye3v0-3.12.20 Commutator in Watanabe-Brauner matrix is magnetisation Sᶻ (vanishes)\nsye3v0-3.12.21 NG modes excited by broken symmetry generator are independent\nsye3v0-3.12.22 Two type-A NG modes with linear dispersions (spin waves)\nsye3v0-3.12.23 Viewed as plane waves of precessions for spins on each sublattice\nsye3v0-3.12.24 Ferromagnet: Heisenberg ferromagnet Exercise 2.7, breaks same spin-rotation symmetries\nsye3v0-3.12.25 Magnetisation Sᶻ is order parameter, expectation value non-zero\nsye3v0-3.12.26 Watanabe-Brauner matrix non-zero, modes excited by two broken generators not independent\nsye3v0-3.12.27 One type-B NG mode with quadratic dispersion\nsye3v0-3.12.28 Canted antiferromagnet: Adding term favours orthogonal alignment, spins uniformly canted\nsye3v0-3.12.29 State with total uniform magnetisation and staggered magnetisation\nsye3v0-3.12.30 Breaks all three spin-rotation symmetries\nsye3v0-3.12.31 Uniform magnetisation excites one type-A NG mode\nsye3v0-3.12.32 Remaining two broken generators excite one type-B NG mode\nsye3v0-3.12.33 Exercise 3.2 (Chiral symmetry breaking)\nsye3v0-3.12.33.1 Complex scalar doublet Φ=(φ1 φ2)T, Lagrangian L\nsye3v0-3.12.33.2 L invariant under Φ → LΦ, L∈SU(2)\nsye3v0-3.12.33.3 L further invariant under φ1→r2φ1+r1φ*2, φ2→-r1φ*1+r2φ2 with r*1r1+r*2r2=1\nsye3v0-3.12.33.4 Write Φ as U(2) matrix Φ˘, collect r1, r2 in SU(2) matrix R\nsye3v0-3.12.33.5 L invariant under Φ˘ → LΦ˘R† (chiral symmetry)\nsye3v0-3.12.33.6 L also invariant under global U(1) phase rotations Φ → eiαΦ\nsye3v0-3.12.33.7 Full symmetry group SU(2)L × SU(2)R × U(1)\nsye3v0-3.12.33.8 SU(2)L and SU(2)R generated by QLa, QRa\nsye3v0-3.12.33.9 Define vector QV a = QLa + QRa and axial QA a = QLa - QRa\nsye3v0-3.12.33.10 Satisfy algebra relations: [QV a, QV b] = iabcQV c, [QA a, QA b] = iabcQV c, [QV a, QA b] = iabcQA c\nsye3v0-3.12.33.11 QV a generate subgroup, QA a do not\nsye3v0-3.12.33.12 If r<0, u>0, potential minimum at Φ = (0 v)T or Φ˘ = diag(v,v)\nsye3v0-3.12.33.13 Vector transformations (L=R) leave <Φ˘> invariant\nsye3v0-3.12.33.14 Axial transformations (L=R†) do not leave <Φ˘> invariant\nsye3v0-3.12.33.15 Symmetry spontaneously broken by Φ˘ from SU(2)L × SU(2)R to diagonal subgroup SU(2)L+R\nsye3v0-3.12.33.16 Broken generators QA a (three)\nsye3v0-3.12.33.17 Expect three type-A NG modes\nsye3v0-3.12.33.18 Lagrangian expressed in terms of Φ˘ describes Higgs field in Standard Model\nsye3v0-3.12.33.19 NG bosons not massless in Standard Model (explained Section 7.3)\nsye3v0-3.12.33.20 SU(3) × SU(3) → SU(3) chiral symmetry breaking in quantum chromodynamics (QCD) as quark masses go to zero\nsye3v0-3.13 NG-like excitations\nsye3v0-3.13.1 Systems harbouring excitations related to SSB/NG modes but not satisfying all assumptions\nsye3v0-3.13.2 Gapped NG mode: SSB system exposed to external field explicitly breaking symmetry\nsye3v0-3.13.3 Energy gap proportional to external field\nsye3v0-3.13.4 Example: spin waves in ferromagnet exposed to magnetic field parallel to magnetisation\nsye3v0-3.13.5 Interpreted as model without explicit SSB, modified time-dependent symmetry generators\nsye3v0-3.13.6 Pseudo NG mode: symmetry broken explicitly due to weak coupling to other fields\nsye3v0-3.13.7 Bosonic particle with energy gap\nsye3v0-3.13.8 Example: lightest eight pseudoscalar mesons in Standard Model (approximate SU(3) × SU(3) chiral symmetry broken)\nsye3v0-3.13.9 Quasi NG mode: ground state symmetry group larger than H itself, symmetry broken spontaneously\nsye3v0-3.13.10 NG-like excitation emerges\nsye3v0-3.13.11 Called pseudo NG boson confusingly [55]\nsye3v0-3.13.12 Occurs in particle physics (charged pions) and condensed matter (helium-3 superfluids, spinor Bose-Einstein condensates)\nsye3v0-3.13.13 Goldstino: broken generators Qa are creation operators for NG modes\nsye3v0-3.13.14 Usually NG modes are bosons (obey commutation relations)\nsye3v0-3.13.15 Sometimes vector/tensor structure assigned\nsye3v0-3.13.16 If generators satisfy anti-commutation relations, NG modes are fermions\nsye3v0-3.13.17 Example: supersymmetry breaking (Poincaré algebra extension with fermionic generators)\nsye3v0-3.13.18 Associated fermionic NG modes called Goldstinos\nsye3v0-3.14 Gapped partner modes\nsye3v0-3.14.1 Type-B NG modes excited by two distinct broken symmetry generators\nsye3v0-3.14.2 Commutator has non-zero expectation value\nsye3v0-3.14.3 Effective Lagrangian Eq. (3.9) signalled by two fields π1, π2 not independent\nsye3v0-3.14.4 Generally two modes, second is gapped [56-59]\nsye3v0-3.14.5 Leff = 2M(π1∂tπ2 - π2∂tπ1) + 1/c²(∂tπ1)² + 1/c²(∂tπ2)² - 1/2(∇π1)² - 1/2(∇π2)² [60]\nsye3v0-3.14.6 Dispersion relations: ω± = √c²k² + M²c⁴ ± Mc² [60]\nsye3v0-3.14.7 One gapless NG mode ω- = k²/2M + ..., one gapped partner mode ω+ = 2Mc² + k²/2M + ...\nsye3v0-3.14.8 Coefficients 2M and c interpreted as effective mass and velocity\nsye3v0-3.14.9 Limit M→0: two modes decouple, degenerate linear dispersion ω±=ck (two type-A NG modes)\nsye3v0-3.14.10 Limit c→∞: gap goes to infinity, single gapless mode (corresponds to ferromagnet)\nsye3v0-3.14.11 Ferromagnet: only terms with single time derivatives in Eq. (3.18), no gapped mode in spectrum\nsye3v0-3.14.12 Physically: NG modes excited by lowering maximally polarised spins\nsye3v0-3.14.13 In Sˣ, Sʸ, only S⁻ part excites mode\nsye3v0-3.14.14 Acting with S⁺ on maximally polarised ferromagnet annihilates state (not physical excitation)\nsye3v0-3.14.15 Action of two broken symmetry generators on SSB state entirely equivalent in this limit\nsye3v0-3.14.16 More generally, non-zero M and c indicate presence of both type-B NG mode and gapped partner\nsye3v0-3.14.17 Existence of gapped mode seems to invalidate Goldstone's theorem proof (Section 3.1)\nsye3v0-3.14.18 Proof argued broken generator must be gapless for order parameter to be time-independent\nsye3v0-3.14.19 No general proof, but checked in cases: terms in Eq. (3.5) contain product of two matrix elements\nsye3v0-3.14.20 <ψ|jt(0,0)|n, k><n, k|Φ|ψ>\nsye3v0-3.14.21 In verified cases, jt(0,0) excites gapped partner with non-zero En\nsye3v0-3.14.22 <ψ|Φ|n, k> proportional to energy of accompanying gapless mode (vanishes for k→0)\nsye3v0-3.14.23 Existence of gapped partner does not contradict time-independence of order parameter\nsye3v0-3.14.24 Exercise 3.3 (Heisenberg Ferrimagnet)\nsye3v0-3.14.24.1 Ferrimagnet: Heisenberg H on square lattice, positive J, spins on A/B sublattices have different sizes SA, SB\nsye3v0-3.14.24.2 Calculate average magnetisation per unit cell for N´eel-like state: (SA-SB)\nsye3v0-3.14.24.3 State breaks spin-rotation symmetries Sˣ, Sʸ\nsye3v0-3.14.24.4 Calculate matrix elements of Watanabe-Brauner matrix (Mab)\nsye3v0-3.14.24.5 Mab = (-i<ψ|[Qa, jt b(x)]|ψ>) for Qa,b = Sˣtot, Sʸtot\nsye3v0-3.14.24.6 Mxy = (0 i(SA-SB); -i(SA-SB) 0)\nsye3v0-3.14.24.7 Local magnetisation Sᶻi is order parameter\nsye3v0-3.14.24.8 Second order parameter: staggered magnetisation Nz_i = (-1)iSᶻi\nsye3v0-3.14.24.9 Calculate expectation value per unit cell for staggered magnetisation: (SA+SB)\nsye3v0-3.14.24.10 Breaking spin-rotation symmetry described by two distinct order parameters\nsye3v0-3.14.24.11 One commutes with H, one does not\nsye3v0-3.14.24.12 Not sufficient to find O not commuting with H to claim type-A system\nsye3v0-3.14.24.13 Ferrimagnet has one quadratically dispersing NG mode, one gapped partner mode\nsye3v0-3.14.24.14 Gap scales with spin size difference ∆ ∝ |SA-SB|\nsye3v0-3.14.15 The tower of states for systems with type-B NG modes\nsye3v0-3.14.15.1 NG modes (k>0) cousins of collective excitations (k=0) in tower of states\nsye3v0-3.14.15.2 Relation between internal/collective modes persists in type-A/type-B distinction\nsye3v0-3.14.15.3 Heisenberg antiferromagnet, harmonic crystal: unique ground states, tower of low-energy states, unstable (general for type-A NG modes)\nsye3v0-3.14.15.4 Ferromagnet: macroscopically degenerate ground state, no tower of states (general for systems with type-B NG mode and gapped partner)\nsye3v0-3.14.15.5 Number of exact ground states infinite in thermodynamic limit, order N for finite systems\nsye3v0-3.14.15.6 Systems with type-B NG mode and gapped partner (M, c non-zero): ground state degeneracy, no tower of states\nsye3v0-3.14.15.7 Relation to collective modes seen in Lieb lattice antiferromagnet example [61]\nsye3v0-3.14.15.8 Lieb lattice: twice as many A sites as B sites\nsye3v0-3.14.15.9 Classical ground state: N´eel-type, non-zero magnetisation Sᶻ = S(NA-NB) = 1/3 SN\nsye3v0-3.14.15.10 Magnetisation finite implies type-B NG modes\nsye3v0-3.14.15.11 Exact ground states have finite overlap with Lieb-Mattis model (k=0 part of H)\nsye3v0-3.14.15.12 Ground states have total spin Stot = S(NA-NB), degeneracy 2Stot+1 (order N)\nsye3v0-3.14.15.13 Energy to excite remaining k=0 states is order O(J)\nsye3v0-3.14.15.14 Usual antiferromagnet (type-A): k=0 excitations energy vanishes O(J/N)\nsye3v0-3.14.15.15 No tower of states for Lieb lattice antiferromagnet\nsye3v0-3.14.15.16 Relation between NG mode types and spectrum of collective excitations summarised in Table 3.1\nsye3v0-3.14.15.17 Table 3.1: NG mode dispersion, ground state degeneracy, tower of states\nsye3v0-4.0 Quantum corrections and thermal fluctuations\nsye3v0-4.1 Eigenstates of local order parameter operator (classical states) not generally eigenstates of symmetric H\nsye3v0-4.2 Except for systems with conserved O and type-B NG modes (ferromagnets, ferrimagnets)\nsye3v0-4.3 Symmetric H have unique, symmetric ground states\nsye3v0-4.4 Classical states resemble states realised in nature due to stability (Section 2.7)\nsye3v0-4.5 Classical states not exactly states in quantum materials\nsye3v0-4.6 Due to subtle effect of excitations with finite wave number reducing perfect local order\nsye3v0-4.7 Difference between classically expected and actually encountered state at T=0 is quantum corrections\nsye3v0-4.8 At non-zero T, supplemented by thermal fluctuations (further suppress local order)\nsye3v0-4.9 Distinction: thermal fluctuations vs quantum corrections\nsye3v0-4.10 Quantum corrections often called "quantum fluctuations"\nsye3v0-4.11 Reasoning: effect of quantum corrections similar to thermal fluctuations\nsye3v0-4.12 May lead to quantum phase transition at T=0 (analogous to thermal phase transition Section A.4)\nsye3v0-4.13 Terminology misleading: thermal fluctuations describe actual random fluctuations in thermal state\nsye3v0-4.14 At T=0, nothing fluctuates\nsye3v0-4.15 Ground state unique from maximally ordered phase to just before quantum phase transition\nsye3v0-4.16 Expectation value of local O strongly decreasing\nsye3v0-4.17 For given parameters, nothing about ground state evolves/fluctuates in time\nsye3v0-4.18 Prefer term quantum corrections for difference between quantum ground state and classical state\nsye3v0-4.19 Study Heisenberg antiferromagnet example (Section 4.1) to introduce techniques and show significant magnitude of corrections\nsye3v0-4.20 In most ordered systems in 3D, quantum corrections are tiny\nsye3v0-4.21 Example: chair/table well-described by classical state\nsye3v0-4.22 In low dimensions, quantum corrections/thermal fluctuations generically large, prevent ordered state/SSB of continuous symmetries\nsye3v0-4.23 Heuristic derivation of Mermin-Wagner-Hohenberg-Coleman theorem (Section 4.2) explains dimensionality effect and link between corrections/fluctuations\nsye3v0-4.24 Linear spin-wave theory\nsye3v0-4.24.1 No known exact expression for broken state of Heisenberg antiferromagnet in D>1\nsye3v0-4.24.2 Describe by starting from classical state (N´eel state) and looking for deviations\nsye3v0-4.24.3 Done in spin-wave theory (reformulation of H in terms of boson operators)\nsye3v0-4.24.4 Bosonic excitation lowers O expectation value\nsye3v0-4.24.5 Linear spin-wave theory: bosonic H additionally linearised, ground state identified by diagonalisation\nsye3v0-4.24.6 Exact ground state of approximate H serves as approximate ground state of exact H\nsye3v0-4.24.7 Linear spin-wave theory considers first non-trivial order in O expansion\nsye3v0-4.24.8 Heisenberg spin-S antiferromagnet on bipartite lattice in d dimensions (Eq. 2.17)\nsye3v0-4.24.9 Coupling constant J positive (anti-align)\nsye3v0-4.24.10 Lattice vectors δ run over z nearest neighbours\nsye3v0-4.24.11 Factor 1/2 avoids double counting\nsye3v0-4.24.12 Lattice bipartite A/B sublattices\nsye3v0-4.24.13 Spins on A positive magnetisation, B negative in N´eel order\nsye3v0-4.24.14 Introduce rotated spin operators Nj (Eq. 4.2) to track local magnetisation direction\nsye3v0-4.24.15 Coordinate system for spins on B rotated by π around x-axis relative to A\nsye3v0-4.24.16 Nj are proper spin-S operators, obey SU(2) algebra\nsye3v0-4.24.17 Heisenberg H in terms of rotated spins (Eq. 4.3)\nsye3v0-4.24.18 On bipartite lattices, A sites neighbours on B, vice versa\nsye3v0-4.24.19 N±j = Nx j ± iNy j are raising/lowering operators for transformed spins\nsye3v0-4.24.20 Classical N´eel state is eigenstate of Nz tot with maximal eigenvalue\nsye3v0-4.24.21 Not eigenstate of H (due to N⁺N⁻ terms)\nsye3v0-4.24.22 Local excitations by applying spin-lowering operator N⁻j\nsye3v0-4.24.23 Lower local staggered magnetisation by quantised amount\nsye3v0-4.24.24 Similar to ladder operators of harmonic oscillator/boson creation operators\nsye3v0-4.24.25 Express transformed spin operators in terms of boson operators aj, a†j (Holstein-Primakoff bosons)\nsye3v0-4.24.26 N⁺j = √2S(1-nj/2S) aj, N⁻j = √2S a†j(1-nj/2S), Nz j = S - a†jaj (Eq. 4.4-4.6)\nsye3v0-4.24.27 Boson operators obey canonical commutation relations [ai, a†j] = δij, nj = a†jaj\nsye3v0-4.24.28 Square root defined by power series expansion\nsye3v0-4.24.29 Definition of Nj respects SU(2) algebra\nsye3v0-4.24.30 Bosonic vacuum aj|N´eel> = 0 corresponds to N´eel state\nsye3v0-4.24.31 States with non-zero bosons correspond to states with non-maximal N´eel order\nsye3v0-4.24.32 Holstein-Primakoff transformation is exact, but square roots prevent simple diagonalisation\nsye3v0-4.24.33 Linear approximation of square roots (keep first terms)\nsye3v0-4.24.34 N⁺j ≈ √2Saj, N⁻j ≈ √2Sa†j, Nz j unaltered\nsye3v0-4.24.35 Approximation good for large S, low boson excitation numbers\nsye3v0-4.24.36 Resulting approximate ground state/excitations quite accurate even for spin-1/2\nsye3v0-4.24.37 Linear approximation for H (Eq. 4.7)\nsye3v0-4.24.38 N=total sites, z=coordination number\nsye3v0-4.24.39 Diagonalise approximate H using Fourier transformed operators ak = 1/√N Σj eik·jaj\nsye3v0-4.24.40 Momentum-space bosons obey canonical commutation relations [ak, a†k'] = δkk'\nsye3v0-4.24.41 Terms in H become sums over k (Eq. 4.8-4.10)\nsye3v0-4.24.42 Introduce γk ≡ 1/z Σδ eik·δ\nsye3v0-4.24.43 Square lattice: γk real\nsye3v0-4.24.44 H in momentum space (Eq. 4.11)\nsye3v0-4.24.45 Products of two creation/annihilation operators hinder simple ground state identification\nsye3v0-4.24.46 Indicates N´eel state/bosonic vacuum not eigenstate of H\nsye3v0-4.24.47 Find exact ground state of linearised H using Bogoliubov transformation\nsye3v0-4.24.48 Introduce second set of boson operators bk, b†k (Eq. 4.12)\nsye3v0-4.24.49 ak = cosh uk bk + sinh uk b†-k, a†k = cosh uk b†k + sinh uk b-k\nsye3v0-4.24.50 uk unknown real function of k, uk = u-k\nsye3v0-4.24.51 New operators obey canonical commutation relations [bk, b†k'] = δkk'\nsye3v0-4.24.52 H in terms of new bosons (Eq. 4.13)\nsye3v0-4.24.53 H diagonal if terms in final line vanish\nsye3v0-4.24.54 Choose uk such that γk cosh 2uk + sinh 2uk = 0\nsye3v0-4.24.55 Using cosh²x - sinh²x = 1, sinh 2uk = -γk/√1-γk², cosh 2uk = 1/√1-γk² (Eq. 4.14)\nsye3v0-4.24.56 uk ill-defined at k=0 (γk=0=1)\nsye3v0-4.24.57 Collective k=0 part corresponds to tower of states\nsye3v0-4.24.58 Diagonalisation forces treating collective excitations separately\nsye3v0-4.24.59 Collective excitations correspond to quantum corrections/NG modes\nsye3v0-4.24.60 Writing approximate H in terms of Bogoliubov transformed excitations, diagonalising, omitting k=0 part\nsye3v0-4.24.61 Linear spin-wave Hamiltonian (Eq. 4.15)\nsye3v0-4.24.62 H consists of three parts: classical energy, quantum corrections, NG modes\nsye3v0-4.24.63 Classical energy: energy expectation value of classical N´eel state\nsye3v0-4.24.64 Quantum corrections: negative, lowers energy below classical state, difference in ground state energy\nsye3v0-4.24.65 NG modes term: contains excitations propagating with non-zero k, positive energy\nsye3v0-4.24.66 NG modes absent in ground state (vacuum for b bosons)\nsye3v0-4.24.67 Ground state energy evaluated numerically in continuum limit (Table 4.1)\nsye3v0-4.24.68 For spin-1/2 antiferromagnets, quantum corrections to energy substantial\nsye3v0-4.24.69 O expectation value affected by quantum corrections\nsye3v0-4.24.70 Calculate <Nz/N> using Eq. (4.6) in terms of Holstein-Primakoff bosons\nsye3v0-4.24.71 <Nz/N> = S - 1/N Σj <0|a†jaj|0>\nsye3v0-4.24.72 Ground state |0> is vacuum of bk, not ak\nsye3v0-4.24.73 Using Bogoliubov transformation (Eq. 4.12), <0|a†kak|0> = sinh²uk\nsye3v0-4.24.74 <Nz/N> = S - 1/N Σk sinh²uk (Eq. 4.18)\nsye3v0-4.24.75 Continuum limit: integral diverges in one dimension\nsye3v0-4.24.76 Quantum corrections suppress order parameter in 1D\nsye3v0-4.24.77 General phenomenon (Section 4.2)\nsye3v0-4.24.78 Integral evaluated numerically (Table 4.2)\nsye3v0-4.24.79 Quantum corrections take O substantially away from classical value, especially for low-spin\nsye3v0-4.24.80 Sizes of corrections exceptional for Heisenberg antiferromagnet\nsye3v0-4.24.81 Most ordered systems in 3D, corrections tiny\nsye3v0-4.24.82 Linear spin-wave approximation gives unexpectedly good results\nsye3v0-4.24.83 Best estimates for ground state energy within few percent [62]\nsye3v0-4.24.84 Exercise 4.1 (XY-model quantum corrections)\nsye3v0-4.24.84.1 XY-model: interactions between rotors in plane, global U(1) rotational symmetry\nsye3v0-4.24.84.2 Written in terms of spin operators HXY (Eq. 4.19)\nsye3v0-4.24.84.3 Calculate quantum corrections in d dimensions using linear spin-wave theory [63]\nsye3v0-4.24.84.4 Assume J<0 for simplicity\nsye3v0-4.24.84.5 Trick: take different reference frame, H˜XY = JΣij (SˣiSˣj + SᶻiSᶻj)\nsye3v0-4.24.84.6 Write Sˣj in terms of S±j\nsye3v0-4.24.84.7 Write H˜XY in terms of Holstein-Primakoff bosons (Eq. 4.20)\nsye3v0-4.24.84.8 Perform Fourier transform, then Bogoliubov transformation (Eq. 4.12)\nsye3v0-4.24.84.9 Choose uk to diagonalise H\nsye3v0-4.24.84.10 Numerically evaluate ground state energy density E/N and order parameter density Sᶻ/N\nsye3v0-4.24.84.11 Results for O: S-0.0609 (2D), S-0.0225 (3D)\nsye3v0-4.24.84.12 Corrections considerable, but not as large as antiferromagnet\nsye3v0-4.25 Mermin-Wagner-Hohenberg-Coleman theorem\nsye3v0-4.25.1 Lowering of O expectation value in broken state (Heisenberg antiferromagnet) is general property of SSB systems\nsye3v0-4.25.2 Low dimensions (D=1 for antiferromagnet): quantum corrections may preclude non-zero O\nsye3v0-4.25.3 Similar at elevated temperatures: thermal fluctuations may prevent SSB in D≤2\nsye3v0-4.25.4 Thermal limit to ordering: Mermin-Wagner-Hohenberg theorem\nsye3v0-4.25.5 Zero-temperature absence of SSB in 1D: Coleman theorem\nsye3v0-4.25.6 Calculation showing divergence of quantum corrections in Heisenberg antiferromagnet not neatly generalised\nsye3v0-4.25.7 Consider effective Lagrangian Eq. (3.18) (coarse-grained description of SSB system with NG modes)\nsye3v0-4.25.8 Considering type-A or type-B NG modes has significant implications for how corrections/fluctuations affect SSB\nsye3v0-4.25.9 Highest spatial dimension where corrections/fluctuations prevent long-range order is lower critical dimension\nsye3v0-4.25.10 Work out lower critical dimensions for various NG modes types at T=0 and T>0\nsye3v0-4.25.11 Analysis requires imaginary-time path integrals (Matsubara summation)\nsye3v0-4.25.12 Results summarised in Table 4.3\nsye3v0-4.26 The variance of the order parameter\nsye3v0-4.26.1 SSB and associated emergence of long-range order described by local O with non-zero expectation value\nsye3v0-4.26.2 For order to survive corrections/fluctuations, variance of O smaller than expectation value\nsye3v0-4.26.3 Variance of local O: <O(x,t)²> - <O(x,t)>² = lim δO(x,t)δO(x',t') as x'→x, t'→t\nsye3v0-4.26.4 Expand O around expectation value: O = <O> + δO\nsye3v0-4.26.5 Average of Gaussian fluctuations δO is zero\nsye3v0-4.26.6 At low T/energies, gapless NG modes πa(x,t) dominate fluctuations\nsye3v0-4.26.7 lim δOδO = lim Σa πa(x,t)πa(x',t') + ...\nsye3v0-4.26.8 Precise expression in terms of NG modes in [43]\nsye3v0-4.26.9 Approximate form suffices to understand lower critical dimensions\nsye3v0-4.26.10 πa(x,t)πa(x',t') coincides with real-space propagator of NG mode\nsye3v0-4.26.11 Interested in long-ranged ordered systems with translational invariance\nsye3v0-4.26.12 Convenient to calculate propagator in momentum space (Fourier transform)\nsye3v0-4.26.13 G(0) = lim ∫dω/2π ∫dDk/(2π)D Σa πa(-k,-ω)πa(k,ω) as x'→x, t'→t\nsye3v0-4.26.14 G(0) = ∫dω/2π ∫dDk/(2π)D Σa πa(-k,-ω)πa(k,ω)\nsye3v0-4.26.15 Obtain thermal average from ground state expectation value\nsye3v0-4.26.16 Apply analytic continuation to imaginary time (t→-iτ)\nsye3v0-4.26.17 Introduce bosonic Matsubara frequencies ω→iωn (ωn = 2πn/β, β=1/kBT) [64-66]\nsye3v0-4.26.18 G(0) = 1/β Σn ∫dDk/(2π)D Σa πa(-k,-iωn)πa(k,iωn) (Eq. 4.24)\nsye3v0-4.27 Matsubara summation\nsye3v0-4.27.1 Technique to evaluate sum over Matsubara frequencies\nsye3v0-4.27.2 Change sum to contour integral over poles of g(z)\nsye3v0-4.27.3 G(0) = 1/β Σn g(iωn)\nsye3v0-4.27.4 Replace g(iωn) with g(z) (analytic continuation)\nsye3v0-4.27.5 Use Cauchy residue theorem: contour integral = 2πi Σn Res f(zn)\nsye3v0-4.27.6 Trick: function F(z) = (eβz-1)⁻¹ has poles at zn = 2iπn/β (coincide with iωn)\nsye3v0-4.27.7 Resz=zn (g(z)F(z)) = 1/β g(zn) for simple poles\nsye3v0-4.27.8 Write Matsubara summation as contour integral (Eq. 4.27)\nsye3v0-4.27.9 Contours Cn tightly enclose poles at zn\nsye3v0-4.27.10 g(z) may also have poles\nsye3v0-4.27.11 Use fact that contour integrals can be reshaped if no poles crossed\nsye3v0-4.27.12 Circular contour integral with infinite radius vanishes\nsye3v0-4.27.13 Equate contour integral over imaginary axis poles to contour integral over real axis poles\nsye3v0-4.27.14 Matsubara summation written as sum over poles zj of g(z) (Eq. 4.28)\nsye3v0-4.27.15 Easier to evaluate than original sum\nsye3v0-4.28 General NG mode propagator\nsye3v0-4.28.1 Effective Lagrangian Eq. (3.18) with dispersions ω± = √c²k² + M²c⁴ ± Mc² (Eq. 3.19)\nsye3v0-4.28.2 Propagator G(0) (Eq. 4.29)\nsye3v0-4.28.3 Poles at zj = ω+ and zj = -ω- (first term), zj = -ω+ and zj = ω- (second term)\nsye3v0-4.28.4 Evaluating Eq. (4.28) yields G(0) (Eq. 4.30)\nsye3v0-4.28.5 nB(ε) = (eβε-1)⁻¹ is Bose-Einstein distribution\nsye3v0-4.29 Type-A NG modes\nsye3v0-4.29.1 Apply Matsubara summation technique to NG mode propagator\nsye3v0-4.29.2 Start from effective Lagrangian Eq. (3.9) simplified for single type-A mode (Eq. 4.31)\nsye3v0-4.29.3 Leff = 1/c² ∂tπ∂tπ - ∇π · ∇π\nsye3v0-4.29.4 Use Fourier transforms (Eq. 4.32)\nsye3v0-4.29.5 Leff = π(-k,-iωn) (1/c² (iωn)² - k²) π(k,iωn)\nsye3v0-4.29.6 Propagator G(0) found by inverting quadratic part (Eq. 4.33)\nsye3v0-4.29.7 G(0) = c² / ((iωn)² - c²k²)\nsye3v0-4.29.8 g(iωn) = c² / ((iωn)² - c²k²)\nsye3v0-4.29.9 Sum over bosonic Matsubara frequencies\nsye3v0-4.29.10 Apply procedure Eq. (4.27), (4.28)\nsye3v0-4.29.11 Poles of g(z) at z± = ±ck\nsye3v0-4.29.12 G(0) = ∫dDk Σj=± Resz=zj (c²/(eβz-1)(z-z+)(z-z-))\nsye3v0-4.29.13 G(0) = ∫dDk (c²/(eβz+-1)(z+-z-) + c²/(eβz--1)(z--z+))\nsye3v0-4.29.14 G(0) = ∫dDk c/2k (1/(eβck-1) - 1/(e-βck-1)) (Eq. 4.34)\nsye3v0-4.29.15 G(0) = (∫dΩ) ∫dk kD-2 c (nB(ck) + 1/2)\nsye3v0-4.29.16 nB(ε) = (eβε-1)⁻¹ is Bose-Einstein distribution\nsye3v0-4.29.17 Zero temperature (β→∞): Bose factor vanishes\nsye3v0-4.29.18 Momentum integral ∫dk kD-2 c (1/2) diverges in D≤1\nsye3v0-4.29.19 Infrared divergence\nsye3v0-4.29.20 Variance of O unbounded in D≤1 at T=0\nsye3v0-4.29.21 Quantum corrections preclude SSB/long-range order in D≤1 at T=0 (Coleman theorem)\nsye3v0-4.29.22 Non-zero temperature: expand nB for small βck (low k)\nsye3v0-4.29.23 nB(x) = 1/x + 1/2 + O(x) (Eq. 4.35)\nsye3v0-4.29.24 Substitute into G(0) expression\nsye3v0-4.29.25 Momentum integral over function ∝ T kD-3\nsye3v0-4.29.26 Thermal fluctuations have larger effect than quantum corrections\nsye3v0-4.29.27 Variance of O diverges in D≤2 for any non-zero T (Mermin-Wagner-Hohenberg theorem)\nsye3v0-4.30 Type-B NG modes\nsye3v0-4.30.1 Result for lower critical dimensions different from type-A\nsye3v0-4.30.2 Arise when terms with single time derivative in Leff\nsye3v0-4.30.3 Type-B systems with/without gapped partner modes behave same for lower critical dimensions\nsye3v0-4.30.4 Consider simplest type-B system without gapped partner (Eq. 4.36)\nsye3v0-4.30.5 Leff = 2m (π1∂tπ2 - π2∂tπ1) - ∇πa · ∇πa\nsye3v0-4.30.6 Necessarily two NG fields π1, π2 coupled by time derivatives\nsye3v0-4.30.7 Use Fourier transforms, write Leff in matrix form (Eq. 4.37)\nsye3v0-4.30.8 Sum of propagators for two NG fields found by inverting quadratic part and taking trace\nsye3v0-4.30.9 g(iωn) = 2k² / ((2miωn)² - k⁴) = 1/2m (1/(iωn - k²/2m) - 1/(iωn + k²/2m)) (Eq. 4.38)\nsye3v0-4.30.10 Use combined g(iωn) in Matsubara summation (Eq. 4.27, 4.28)\nsye3v0-4.30.11 Poles of g(z) at z± = ±k²/2m\nsye3v0-4.30.12 G(0) = ∫dDk Σi=± Resz=zi (g(z)/(eβz-1))\nsye3v0-4.30.13 G(0) = ∫dDk 1/2m (1/(eβk²/2m-1) - 1/(e-βk²/2m-1)) (Eq. 4.39)\nsye3v0-4.30.14 G(0) = ∫dDk 1/2m coth(1/2 β k²/2m)\nsye3v0-4.30.15 Zero temperature (β→∞): coth → 1\nsye3v0-4.30.16 Momentum integral ∫dDk 1/2m (1) diverges in D≤0\nsye3v0-4.30.17 Finite in any spatial dimension D≥1\nsye3v0-4.30.18 Introduce upper (ultraviolet) cutoff for atomic lattice discreteness\nsye3v0-4.30.19 Variance finite at T=0 in any spatial dimension\nsye3v0-4.30.20 SSB/long-range order may occur at T=0 in any dimension for systems with only type-B NG modes\nsye3v0-4.30.21 Non-zero temperatures: dominant contribution from low k\nsye3v0-4.30.22 Expand integrand for small k: coth(x) = 1/x + ...\nsye3v0-4.30.23 Thermal population of type-B modes induces variance\nsye3v0-4.30.24 G(0) = ∫(2π)D k²/(βk²) + ... = (∫dΩ) ∫dk kD-3 2T/kB (Eq. 4.40)\nsye3v0-4.30.25 Integral diverges in D≤2\nsye3v0-4.30.26 Thermal population of type-B modes at non-zero T yields same lower critical dimension as thermal population of type-A modes\nsye3v0-4.30.27 Same result for type-B systems with gapped partner modes (from Eq. 4.30)\nsye3v0-4.30.28 Final classification of lower critical dimensions summarised in Table 4.3\nsye3v0-4.30.29 Table 4.3: NG mode dispersion, ground state degeneracy, tower of states, quantum corrections, lower critical dimension (T=0, T>0)\nsye3v0-4.30.30 Method applied to Tkachenko modes (type-A, ω∝k²): D_lc=2 (T=0), D_lc=4 (T>0)\nsye3v0-4.30.31 Systems with such modes cannot order at non-zero T even in 3D [68]\nsye3v0-5.0 Phase transitions\nsye3v0-5.1 Properties of equilibrium phases with SSB discussed so far\nsye3v0-5.2 How are such phases created? How is symmetry broken in practice?\nsye3v0-5.3 At infinite T, thermal density matrix is identity, invariant under all symmetry transformations\nsye3v0-5.4 Long-range ordered state at low T requires some symmetries broken at critical temperature Tc\nsye3v0-5.5 Phase transition from symmetric high-T state to SSB low-T state\nsye3v0-5.6 Study of phase transitions is major field [20, 69, 70]\nsye3v0-5.7 Brief overview of central theoretical concepts relevant to SSB\nsye3v0-5.8 Classification of phase transitions\nsye3v0-5.8.1 Ordered SSB phase has non-zero O expectation value\nsye3v0-5.8.2 Symmetric/disordered phase has zero O\nsye3v0-5.8.3 Distinguish types by how O goes to zero at transition\nsye3v0-5.8.4 Discontinuous/first-order phase transitions\nsye3v0-5.8.4.1 O jumps discontinuously from zero to non-zero at Tc\nsye3v0-5.8.4.2 Sudden change in entropy, requires latent heat\nsye3v0-5.8.4.3 Often show hysteresis\nsye3v0-5.8.4.4 Transitions between states with same broken symmetries (gas-to-liquid) almost always first-order\nsye3v0-5.8.4.5 Transitions involving breakdown of symmetry can be discontinuous (liquid-to-solid)\nsye3v0-5.8.5 Continuous/second-order phase transitions\nsye3v0-5.8.5.1 O increases continuously from zero as Tc traversed\nsye3v0-5.8.5.2 Entropy changes continuously\nsye3v0-5.8.5.3 Correlation length and related energy scales diverge at Tc\nsye3v0-5.8.5.4 At Tc, systems become scale-invariant\nsye3v0-5.8.5.5 Many SSB phase transitions are second-order (superfluidity, (anti)ferromagnetism, liquid crystals)\nsye3v0-5.8.6 Terminology from Ehrenfest classification (obsolete)\nsye3v0-5.8.7 n-th order if n-th derivative of free energy discontinuous at Tc\nsye3v0-5.8.8 Heat capacity in many systems diverges, not just discontinuous [20]\nsye3v0-5.8.9 Prefer classification based on O behaviour\nsye3v0-5.8.10 Generalised to quantum phase transitions (at T=0 vs other parameter)\nsye3v0-5.8.11 Summarise O behaviour vs parameters using phase diagram\nsye3v0-5.8.12 Phase diagram: plot of externally controllable parameters (T, pressure)\nsye3v0-5.8.13 Symmetric/SSB phases indicated, transitions denoted by lines\nsye3v0-5.8.14 Critical line: line of second-order transitions\nsye3v0-5.8.15 First-order line can end in critical point\nsye3v0-5.8.16 Critical point: transition exactly at this point is continuous\nsye3v0-5.8.17 Example: phase diagram of helium-4 (Fig. 5.1)\nsye3v0-5.8.18 Liquid-solid transition (first-order, SSB)\nsye3v0-5.8.19 Liquid-gas transition (first-order, no SSB), ends in critical point (scale invariance, critical opalescence)\nsye3v0-5.8.20 Liquid-superfluid transition (second-order, SSB)\nsye3v0-5.9 Landau theory\nsye3v0-5.9.1 Modern view of phase transitions, also known as Landau theory\nsye3v0-5.9.2 Originated with Van der Waals equation of state (gas-liquid transition)\nsye3v0-5.9.3 Below Tc, pressure decreases when volume decreased (unlike gas)\nsye3v0-5.9.4 No symmetry broken in gas-liquid transition\nsye3v0-5.9.5 Liquid has preferred density (order parameter o)\nsye3v0-5.9.6 o is not order parameter in sense of Section 2.5.2 (no symmetries broken)\nsye3v0-5.9.7 Role of preferred density same as SSB order parameters\nsye3v0-5.9.8 Landau functional\nsye3v0-5.9.8.1 Free energy written as functional of order parameter o(T)\nsye3v0-5.9.8.2 Continuous phase transition: o(T) small close to Tc\nsye3v0-5.9.8.3 Taylor expansion for small o(T)\nsye3v0-5.9.8.4 Types of terms depend on nature of O and symmetries\nsye3v0-5.9.8.5 Simplest case: real-valued scalar field o, free energy FL[o, T] (Eq. 5.1)\nsye3v0-5.9.8.6 Assumed invariance under o → -o (odd powers absent)\nsye3v0-5.9.8.7 Actual value of o(T) found by minimizing FL\nsye3v0-5.9.8.8 FL must be bounded from below (highest-order term positive prefactor)\nsye3v0-5.9.8.9 Assume u(T)>0\nsye3v0-5.9.8.10 If r(T)>0, minimum at o(T)=0 (symmetric state)\nsye3v0-5.9.8.11 If r(T)<0, minimum at non-zero o(T) = ±√|r(T)|/u(T) (SSB state)\nsye3v0-5.9.8.12 Phase transition: T variation causes r(T) to change sign (u(T) positive)\nsye3v0-5.9.8.13 Tc defined where r(T) goes through zero\nsye3v0-5.9.8.14 o(T) smoothly changes from zero to non-zero (continuous phase transition)\nsye3v0-5.9.8.15 FL plotted for different r values (Fig. 5.2)\nsye3v0-5.9.8.16 Close to Tc, r(T) Taylor expanded: r(T) ≈ r0 (T-Tc)/Tc = r0t (reduced temperature t)\nsye3v0-5.9.8.17 Assume u constant near Tc\nsye3v0-5.9.8.18 Minimizing FL gives o(T) ∝ (T-Tc)¹/² (Eq. 5.4)\nsye3v0-5.9.8.19 Scale invariance at continuous transition guarantees o(T) ∼ (T-Tc)β (β=critical exponent)\nsye3v0-5.9.8.20 Similar critical exponents for specific heat (α), susceptibility (γ)\nsye3v0-5.9.8.21 Values of critical exponents depend on form of FL (symmetry)\nsye3v0-5.9.9 First-order phase transitions\nsye3v0-5.9.9.1 No guarantee expansion in powers of O makes sense close to discontinuous transition\nsye3v0-5.9.9.2 Landau theory describes these transitions\nsye3v0-5.9.9.3 Consider u(T)<0, expansion to sixth order (Eq. 5.5)\nsye3v0-5.9.9.4 Assume w(T)>0 (FL bounded below)\nsye3v0-5.9.9.5 Consider r(T)=w(T)=1\nsye3v0-5.9.9.6 Discontinuous jump in minimum location as u decreases (Fig. 5.3)\nsye3v0-5.9.9.7 O jumps discontinuously from zero to non-zero\nsye3v0-5.9.9.8 Phase transition described by these parameters is first order\nsye3v0-5.9.9.9 At discontinuous transitions, no scale invariance, no critical exponents\nsye3v0-5.10 Symmetry breaking in Landau theory\nsye3v0-5.10.1 Landau theory describes continuous/discontinuous transitions in terms of O\nsye3v0-5.10.2 O value changes from zero to non-zero at Tc\nsye3v0-5.10.3 Relation to SSB: shape of free energies (Figs. 5.2, 5.3)\nsye3v0-5.10.4 At lowest T, two minima with equal energies, related by transformation o→-o\nsye3v0-5.10.5 Free energy has discrete Z2 symmetry\nsye3v0-5.10.6 System realises specific ground state (positive or negative O value), symmetry spontaneously broken\nsye3v0-5.10.7 Mexican hat potential\nsye3v0-5.10.7.1 Example of continuous SSB in Landau theory\nsye3v0-5.10.7.2 Complex scalar field ψ(x) with continuous U(1) phase rotation symmetry\nsye3v0-5.10.7.3 Ordering transition described by FL functional (Eq. 5.6)\nsye3v0-5.10.7.4 FL invariant under phase rotations (depends on |ψ|²)\nsye3v0-5.10.7.5 Assume u constant, r(T) linear function changing sign at t=0\nsye3v0-5.10.7.6 High T (r>0): single minimum at ψ=0\nsye3v0-5.10.7.7 Low T (r<0): minimum at non-zero amplitude |ψ|=√|r|/u\nsye3v0-5.10.7.8 Amplitude of field is relevant Landau order parameter\nsye3v0-5.10.7.9 Similar to Section 2.5.2 (field operator ψ obtains expectation value ψ=0 in ordered state)\nsye3v0-5.10.7.10 Typical low-T free energy shape (Fig. 5.4) is "Mexican hat potential"\nsye3v0-5.10.7.11 Prototype example in SSB discussions\nsye3v0-5.10.7.12 Continuum of states with amplitude √|r|/u and arbitrary phase minimise FL\nsye3v0-5.10.7.13 FL invariant under U(1) rotations, only single points on circle of minima are stable\nsye3v0-5.10.7.14 Circle isomorphic to quotient space U(1)/1 ≈ U(1) classifying broken states\nsye3v0-5.10.7.15 Traversing transition: one state on circle (value for phase) spontaneously chosen\nsye3v0-5.10.7.16 Minimum-energy state with spontaneously chosen O\nsye3v0-5.10.7.17 Excitations changing phase but not amplitude cost no potential energy\nsye3v0-5.10.7.18 Potential/FL along circle is flat ("flat direction" in field space [71])\nsye3v0-5.10.7.19 Nambu-Goldstone mode associated with U(1) symmetry breaking is long wavelength modulation of phase of ψ\nsye3v0-5.10.7.20 Excitation where O oscillates along flat direction in O parameter space\nsye3v0-5.10.7.21 Result generalised to more complicated SSB instances\nsye3v0-5.10.7.22 Excitations oscillating in flat directions correspond to NG modes\nsye3v0-5.10.8 From Hamiltonian to Landau functional\nsye3v0-5.10.8.1 FL constructed as expansion in terms of O\nsye3v0-5.10.8.2 Symmetry dictates form of O and allowed terms\nsye3v0-5.10.8.3 Coefficients typically determined by fitting to experimental data\nsye3v0-5.10.8.4 Seems phenomenological, disconnected from microscopic theory\nsye3v0-5.10.8.5 Often possible to derive FL from microscopic theory (H or L)\nsye3v0-5.10.8.6 Find terms in FL and coefficients starting from microscopic description\nsye3v0-5.10.8.7 Illustrate with Heisenberg antiferromagnet on D-dimensional hypercube (Eq. 2.17)\nsye3v0-5.10.8.8 H = JΣi,δ Si · Si+δ\nsye3v0-5.10.8.9 i runs over N sites, δ over z=2d nearest neighbours\nsye3v0-5.10.8.10 Good O operator: staggered magnetisation Oi = (-1)iSi\nsye3v0-5.10.8.11 Vector of O parameters to emphasise no preferred axis in symmetric system\nsye3v0-5.10.8.12 Expectation value m = (-1)iSi independent of position i in disordered/translationally invariant state\nsye3v0-5.10.8.13 First step: rewrite operators in terms of average value (mean-field value) m plus deviations δSi\nsye3v0-5.10.8.14 Si = (-1)im + δSi (Eq. 5.8)\nsye3v0-5.10.8.15 H becomes (Eq. 5.9)\nsye3v0-5.10.8.16 H = -JNz|m|² - 2Jz Σi (-1)im · δSi + J Σiδ δSi · δSi+δ (Eq. 5.10)\nsye3v0-5.10.8.17 Rigorous approximation: if system ordered, <δSi> small\nsye3v0-5.10.8.18 Neglect terms quadratic/higher in deviations\nsye3v0-5.10.8.19 Resulting mean-field Hamiltonian H[m] = -JNz|m|² - 2Jz Σi (-1)im · δSi\nsye3v0-5.10.8.20 H[m] = +JNz|m|² - 2Jz Σi (-1)im · Si (Eq. 5.11) (reintroduced original spins, sign change)\nsye3v0-5.10.8.21 Mean-field H depends on |m|, value unknown\nsye3v0-5.10.8.22 Linear in spin operators, can be solved exactly\nsye3v0-5.10.8.23 Compute partition function Z = Tr e-βH, free energy F = -1/β log Z\nsye3v0-5.10.8.24 F as function of m\nsye3v0-5.10.8.25 Since H proportional to m · Sj, energy eigenstates chosen to coincide with Sᶻj (z-axis parallel to m)\nsye3v0-5.10.8.26 Partition function computed by summing over Sᶻj eigenvalues\nsye3v0-5.10.8.27 For spin-1/2: Z = e-βJNz|m|² (2 cosh Jzβ|m|)N (Eq. 5.12)\nsye3v0-5.10.8.28 F = JNz|m|² - N/β log (2 cosh Jzβ|m|) (Eq. 5.13)\nsye3v0-5.10.8.29 Expression for F in terms of mean-field O\nsye3v0-5.10.8.30 Compare to Landau functional, perform Taylor expansion of F for small m\nsye3v0-5.10.8.31 F/N = -log 2/β + Jz(1 - Jzβ/2)|m|² + 1/12 (Jz)⁴β³|m|⁴ + ... (Eq. 5.14)\nsye3v0-5.10.8.32 Exactly shape of Landau free energy Eq. (5.1)\nsye3v0-5.10.8.33 High T (low β): coefficients of quadratic/quartic terms positive, F minimised for |m|=0 (symmetric)\nsye3v0-5.10.8.34 As T lowered, quadratic term coefficient decreases, goes through zero at Jzβ=2, becomes negative\nsye3v0-5.10.8.35 At that point, F minimised by non-zero |m| (SSB, long-range antiferromagnetic order)\nsye3v0-5.10.8.36 Expansion yields description of antiferromagnetic phase transition\nsye3v0-5.10.8.37 Analysis not internally consistent: started defining m as ground state expectation value, ended claiming to find m by minimizing F\nsye3v0-5.10.8.38 Minimizing mean-field F is exactly same as computing mean-field expectation value of O\nsye3v0-5.10.8.39 Exercise 5.1 (Mean-field order parameter)\nsye3v0-5.10.8.40 Expectation value of O found self-consistently in mean-field theory\nsye3v0-5.10.8.41 Consistency condition allows deriving mean-field equations\nsye3v0-5.10.8.42 Show thermal expectation value of magnetisation |m| = |(-1)iSi| is 1/2 tanh Jzβ|m| in mean-field H (Eq. 5.11)\nsye3v0-5.10.8.43 Use exact expression for F (Eq. 5.13) to compute m by minimizing F\nsye3v0-5.10.8.44 Expectation value of O w.r.t mean-field H is same as equilibrium value w.r.t Landau free energy\nsye3v0-5.10.8.45 Triumph: connection by Gor'kov between microscopic BCS theory and phenomenological Ginzburg-Landau theory\nsye3v0-5.11 Spatial fluctuations\nsye3v0-5.11.1 Expansions assumed local O has same value everywhere\nsye3v0-5.11.2 Appropriate for equilibrium state of translationally invariant system\nsye3v0-5.11.3 Landau theory more generally: consider spatially varying O\nsye3v0-5.11.4 Used to study role of fluctuations near phase transition\nsye3v0-5.11.5 FL expanded in powers of O and spatial derivatives (assumed small)\nsye3v0-5.11.6 Ginzburg-Landau theory: Landau functional including spatial derivatives\nsye3v0-5.11.7 Minimal extension for second-order transition (Eq. 5.1) is FGL[o(x), T] (Eq. 5.15)\nsye3v0-5.11.8 FGL depends on O field o(x) with different values at different positions\nsye3v0-5.11.9 If o(x) dimensionless, c² must have units energy density * length²\nsye3v0-5.11.10 Ginzburg-Landau theory tells typical size of fluctuations in O field\nsye3v0-5.11.11 Trick: add small local perturbation µ(x) to potential\nsye3v0-5.11.12 Fµ[o, T] = FGL[o, T] - ∫dDx' µ(x')o(x') (Eq. 5.16)\nsye3v0-5.11.13 Perturbation is delta function µ(x') = µ0δ(x'-x)\nsye3v0-5.11.14 Equilibrium configuration minimises Fµ\nsye3v0-5.11.15 O is position-dependent function, minimum found by setting functional derivative δF/δo(x) = 0\nsye3v0-5.11.16 Result: -c²∇²o(x) + ro(x) + u[o(x)]³ = µ0δ(x) (Eq. 5.17)\nsye3v0-5.11.17 Perturbation small: assume deviations δo(x) from uniform average ō are small o(x) = ō + δo(x)\nsye3v0-5.11.18 Discard O((δo)²) terms\nsye3v0-5.11.19 Yields: -c²∇²δo(x) + rō + rδo(x) + uō³ + 3uō²δo(x) = µ0δ(x) (Eq. 5.18)\nsye3v0-5.11.20 Small local perturbation does not affect average ō\nsye3v0-5.11.21 ō equals value from uniform Landau theory (Eq. 5.2)\nsye3v0-5.11.22 T > Tc: ō=0, equation -c²∇²δo(x) + rδo(x) = µ0δ(x)\nsye3v0-5.11.23 T < Tc: ō=√|r|/u, equation -c²∇²δo(x) - 2rδo(x) = µ0δ(x) (Eq. 5.19)\nsye3v0-5.11.24 Ordinary differential equations for δo(x)\nsye3v0-5.11.25 In 3D, solution δo(x) = µ0/4πc² e-|x|/ξ / |x| (Eq. 5.20)\nsye3v0-5.11.26 δo(x) falls off exponentially with characteristic length ξ (coherence length)\nsye3v0-5.11.27 ξ indicates scale over which O fluctuations persist\nsye3v0-5.11.28 Sometimes called healing length\nsye3v0-5.11.29 Typical size of spontaneously generated thermal fluctuations\nsye3v0-5.11.30 Length scales >ξ: O field well approximated by average value\nsye3v0-5.11.31 Order/SSB determine system look at large scales\nsye3v0-5.11.32 Length scales <ξ: local configuration dominated by perturbations\nsye3v0-5.11.33 Average O hard to distinguish among microscopic fluctuations\nsye3v0-5.11.34 Long-range order emerges from underlying local physics on scales >ξ\nsye3v0-5.11.35 Coherence length ≠ correlation length (Section 2.33)\nsye3v0-5.11.36 Correlation length: likelihood of two distant regions behaving same way\nsye3v0-5.11.37 Long-range ordered system: correlation length infinitely long, coherence length finite\nsye3v0-5.11.38 Parameter r depends on T\nsye3v0-5.11.39 r goes from positive to negative in Landau description of second-order transition\nsye3v0-5.11.40 As T approaches Tc, r→0, coherence length diverges\nsye3v0-5.11.41 Divergence is general feature of second-order transitions (all relevant scales diverge)\nsye3v0-5.11.42 Discussed more in Section 5.5\nsye3v0-5.11.43 Divergence of fluctuations as phase transition approached problem for Ginzburg-Landau expansion\nsye3v0-5.11.44 Based on assumption variations in O field are small\nsye3v0-5.11.45 As T tuned towards Tc, assumption breaks down\nsye3v0-5.11.46 Value tG=0 where Ginzburg-Landau theory no longer applicable\nsye3v0-5.11.47 Ginzburg temperature tG can be determined within theory\nsye3v0-5.11.48 Consider correlation function o(x)o(0)\nsye3v0-5.11.49 Long-range ordered: O does not vary much, C(x,0) close to ō²\nsye3v0-5.11.50 Variance of O should be small compared to ō²\nsye3v0-5.11.51 ∫d³x (o(x)o(0) - ō²) << ∫d³x ō² (Eq. 5.21)\nsye3v0-5.11.52 Right-hand side easily evaluated: integrates over constant ō²\nsye3v0-5.11.53 Left-hand side: use fluctuation-dissipation theorem\nsye3v0-5.11.54 Relates thermal average of fluctuations to derivatives of free energy in presence of perturbations [20, 73]\nsye3v0-5.11.55 In Ginzburg-Landau theory, relation is property of Fµ (Eq. 5.16)\nsye3v0-5.11.56 Thermal average <A> = Tr A e-βF/Z\nsye3v0-5.11.57 <o(x)o(0)> - ō² = kBT/4πc² e-|x|/ξ / |x| in 3D (Eq. 5.23)\nsye3v0-5.11.58 Insert into Ginzburg criterion (Eq. 5.21)\nsye3v0-5.11.59 In 3D, integrate ∫d³x e-|x|/ξ / |x| from 0 to ξ\nsye3v0-5.11.60 Result: kBT/c² (1-2/e)ξ² << 4/3 πξ³ō² (Eq. 5.24)\nsye3v0-5.11.61 Close to Tc, T≈Tc, r changes sign, r≈r0t\nsye3v0-5.11.62 ξ = c/√|r|, ō² = |r|/u\nsye3v0-5.11.63 Substitute into inequality\nsye3v0-5.11.64 Entire fraction on left diverges as 1/√t as Tc approached\nsye3v0-5.11.65 Region of temperatures around Tc where Ginzburg-Landau theory breaks down\nsye3v0-5.11.66 Ginzburg temperature tG: reduced temperature where fraction = 1\nsye3v0-5.11.67 √tG = √2(3-6/e)kBTcu / 4π√r0c³ (Eq. 5.25)\nsye3v0-5.11.68 tG not exact quantitative bound, indicates order of magnitude of region\nsye3v0-5.11.69 Discarded higher order fluctuation terms in Eq. (5.18)\nsye3v0-5.11.70 Implies Ginzburg-Landau theory is mean-field theory\nsye3v0-5.11.71 Expression for tG is for 3D\nsye3v0-5.11.72 In D≥4, suppression of local order due to fluctuations does not diverge\nsye3v0-5.11.73 Mean-field results robust up to Tc for D≥4\nsye3v0-5.11.74 Spatial dimension below which thermal fluctuations invalidate mean-field: upper critical dimension D_uc\nsye3v0-5.11.75 For Ginzburg-Landau theory Eq. 5.15, D_uc=4\nsye3v0-5.11.76 Contrasted with lower critical dimension (Section 4.2)\nsye3v0-5.11.77 For Ginzburg-Landau theory Eq. 5.15, D_lc=2\nsye3v0-5.11.78 At non-zero T, only in 3D can phase transition exist where local fluctuations destroy long-range ordered phase\nsye3v0-5.12 Universality\nsye3v0-5.12.1 Landau theory based on expansion of FL in powers of local O\nsye3v0-5.12.2 Nature of O and allowed terms determined by symmetries of phases\nsye3v0-5.12.3 Symmetries determine many observable properties of phase transition\nsye3v0-5.12.4 Example: T dependence of O near transition (mean-field β=1/2)\nsye3v0-5.12.5 Value of critical exponent depends only on symmetry of transition (second-order, real scalar O)\nsye3v0-5.12.6 Profound implication: knowing only symmetries allows deducing observable properties near transition\nsye3v0-5.12.7 Example of universality in physics\nsye3v0-5.12.8 Observable properties near transition universal, shared among different systems with same symmetry properties\nsye3v0-5.12.9 Models with same symmetry properties collected into universality classes\nsye3v0-5.12.10 Ising model in same universality class as liquid-gas transition\nsye3v0-5.12.11 Superfluid transition in same universality class as XY-model (Eq. 4.19)\nsye3v0-5.12.12 Universality guarantees experimental measurement of T dependence of specific heat near transition does not depend on microscopic details\nsye3v0-5.12.13 Precisely at second-order transition/critical point, universality even stronger\nsye3v0-5.12.14 Correlation function C(x,x') depends on coherence length ξ\nsye3v0-5.12.15 At transition, ξ diverges, C(x,x') ∝ 1/|x|\nsye3v0-5.12.16 1/|x| is example of scale invariant function (does not define typical length)\nsye3v0-5.12.17 Looks same at every scale\nsye3v0-5.12.18 Contrasted with functions depending on characteristic length scale x0\nsye3v0-5.12.19 Direct physical consequence: observables look same regardless of scale measured\nsye3v0-5.12.20 Famous example: critical opalescence at liquid-gas critical point\nsye3v0-5.12.21 Transparent water becomes opaque, does so for light at all wavelengths\nsye3v0-5.12.22 Reason: presence of scale-invariant fluctuations covering all length scales\nsye3v0-5.12.23 Scatter light at all wavelengths\nsye3v0-5.12.24 C(x) ∝ 1/|x| is homogeneous function, C(|x|) = bκC(b|x|) (κ=1)\nsye3v0-5.12.25 Fact C homogeneous is key ingredient in theory of renormalisability/renormalisation group\nsye3v0-5.12.26 Describes T region immediately around Tc where ξ diverges, Ginzburg-Landau theory breaks down\nsye3v0-5.12.27 Crudely: identify small length a (lattice constant)\nsye3v0-5.12.28 Coarse-grain/average over excitations/fluctuations at length scales between a and ba (b>1)\nsye3v0-5.12.29 Coarse grained description: effective Landau free energy using rescaled coordinate x' = bx\nsye3v0-5.12.30 Symmetries not affected by coarse-graining\nsye3v0-5.12.31 FL looks same, but parameters different values\nsye3v0-5.12.32 Using Eq. (5.26), ξ' = f(ξ)\nsye3v0-5.12.33 If ξ' < ξ, effect of fluctuations smaller, Tc approached more closely before Ginzburg-Landau theory invalid\nsye3v0-5.12.34 Repeating coarse-graining many times, approach Tc arbitrarily closely\nsye3v0-5.12.35 Procedure known as real-space renormalisation group\nsye3v0-5.12.36 Similar procedures: momentum space, order parameter fields\nsye3v0-5.12.37 Approaches capture effects of thermal fluctuations near phase transitions in realistic settings\nsye3v0-5.12.38 Study of renormalisation group is major field [20, 67, 70, 75]\nsye3v0-6.0 Topological defects\nsye3v0-6.1 Fluctuations of local O from average value play important role in SSB state stability\nsye3v0-6.2 Proliferation at high T causes long-range order to melt (phase transition)\nsye3v0-6.3 Mermin-Wagner-Hohenberg theorem: low dimensions, fluctuations prevent long-range order\nsye3v0-6.4 Fluctuations considered so far: Nambu-Goldstone modes (small, wave-like modulations of O)\nsye3v0-6.5 Other types of fluctuations influence SSB state: topological excitations/defects\nsye3v0-6.6 Meaning of topological and defect\nsye3v0-6.6.1 Most intuitive example: U(1) order state (XY-model Eq. 4.19)\nsye3v0-6.6.2 Degrees of freedom visualised by unit vectors in 2D plane\nsye3v0-6.6.3 Ordered state: all vectors point in same direction\nsye3v0-6.6.4 NG modes: plane wave excitations, direction oscillates as system traversed (Fig. 6.1a)\nsye3v0-6.6.5 Long-wavelength NG excitations: neighbouring vectors never far from parallel, energy cost arbitrarily low\nsye3v0-6.6.6 Other modulation: direction winds around a circle (Fig. 6.1b)\nsye3v0-6.6.7 This vortex fundamentally different from plane wave\nsye3v0-6.6.8 Closed contour encircling vortex core: vectors rotate over integer multiple of 2π\nsye3v0-6.6.9 Integer is winding number\nsye3v0-6.6.10 Winding number does not depend on contour location (as long as it encircles core)\nsye3v0-6.6.11 Property of vortex itself: topological charge/invariant\nsye3v0-6.6.12 Topological because contour can be freely deformed\nsye3v0-6.6.13 Smooth change of vector-field configuration does not alter winding number\nsye3v0-6.6.14 Singularity in order parameter field somewhere within contour (O direction undefined)\nsye3v0-6.6.15 In actual systems, singularity avoided: O occurs in discrete lattice or amplitude goes to zero (superfluid)\nsye3v0-6.6.16 Core of vortex around singularity has radial size of coherence length ξ (Eq. 5.20)\nsye3v0-6.6.17 Topological excitations exist for ordered states in any dimension\nsye3v0-6.6.18 1D chain of vectors (XY ferromagnet): all left point up, all right point down\nsye3v0-6.6.19 Zero-dimensional topological defect: domain wall at origin\nsye3v0-6.6.20 2D system: vortex (point-like defect)\nsye3v0-6.6.21 3D volume: vectors point outwards from centre (hedgehog/monopole configuration)\nsye3v0-6.6.22 Generally: p-dimensional defects in D-dimensional system\nsye3v0-6.6.23 D-1 defects: domain walls\nsye3v0-6.6.24 D-2 defects: vortices\nsye3v0-6.6.25 D-3 defects: monopoles (nomenclature varies)\nsye3v0-6.6.26 Vortices in Fig 6.1 have cores pointlike in 2D, linelike in 3D\nsye3v0-6.6.27 Creating single topological defect involves changing O orientation almost everywhere\nsye3v0-6.6.28 Typically costs lot of energy (scaling with log of volume or faster)\nsye3v0-6.6.29 Extremely unlikely introduced spontaneously by thermal/other fluctuations\nsye3v0-6.6.30 Isolated defects created when forced in from outside\nsye3v0-6.6.31 Example: superfluid in container, start spinning container\nsye3v0-6.6.32 Superfluid irrotational in ground state (O field vanishing vorticity)\nsye3v0-6.6.33 Superfluid remains still, ignoring rotation\nsye3v0-6.6.34 External torque exceeds energy cost of forming single vortex\nsye3v0-6.6.35 Topological defect moves into system, causes O phase to wind throughout superfluid\nsye3v0-6.6.36 Quantised amounts of angular momentum imposed (proportional to winding number)\nsye3v0-6.6.37 Single topological defect very stable once formed (need extensive change to remove)\nsye3v0-6.6.38 Local disturbances cannot alter topological charge\nsye3v0-6.6.39 Topological defects under investigation for quantum computation/information storage\nsye3v0-6.6.40 In contrast to isolated defects, common in topologically neutral combinations\nsye3v0-6.6.41 Vortices: total topological charge of multiple defects found by contour enclosing cores\nsye3v0-6.6.42 If O phase does not wind along contour, defects form neutral configuration (Fig. 6.2)\nsye3v0-6.6.43 Neutral combinations affect O within isolated part, energy cost grows with separation\nsye3v0-6.6.44 Can be created as thermal excitations\nsye3v0-6.7 Topological melting: the D = 1 Ising model\nsye3v0-6.7.1 Importance of topological defects in study of long-range order/SSB\nsye3v0-6.7.2 1D chain with classical Ising spins (point up/down)\nsye3v0-6.7.3 Ferromagnet of Ising spins: two broken states (all up/all down)\nsye3v0-6.7.4 Example of discrete symmetry breaking\nsye3v0-6.7.5 Hamiltonian H = -JΣi σᶻi σᶻi+1 (Eq. 6.1)\nsye3v0-6.7.6 Ground state two-fold degenerate, energy E0 = -JN\nsye3v0-6.7.7 Simplest excitation: single spin flip (Fig. 6.3b), costs energy Ef = 4J\nsye3v0-6.7.8 Flipped spin aligned antiferromagnetically with neighbours\nsye3v0-6.7.9 Ising spins cannot be continuously rotated\nsye3v0-6.7.10 Localised spin-flip is best one can do\nsye3v0-6.7.11 Massless NG mode in continuous symmetry case becomes gapped excitation in discrete case\nsye3v0-6.7.12 Single spin flip reduces total magnetisation by 2\nsye3v0-6.7.13 Not-extensive number of spin flips cannot completely remove magnetisation\nsye3v0-6.7.14 1D Ising ferromagnet seems stable at non-zero T\nsye3v0-6.7.15 Conclusion wrong, neglected topological defects\nsye3v0-6.7.16 Topological defect: domain wall created by splitting chain into two segments (Fig. 6.3c)\nsye3v0-6.7.17 Spins up in one segment, down in other\nsye3v0-6.7.18 Only single pair of neighbouring spins aligned antiferromagnetically\nsye3v0-6.7.19 Energetic cost of domain wall Edw = 2J\nsye3v0-6.7.20 Lower energy than spin flip in 1D case\nsye3v0-6.7.21 Single domain wall involves reorientation of macroscopic number of spins\nsye3v0-6.7.22 Strongly affects average magnetisation\nsye3v0-6.7.23 Include states with single domain wall in low-energy effective model\nsye3v0-6.7.24 Consequences drastic\nsye3v0-6.7.25 Chain with N spins, domain wall at position j, magnetisation M = 2j-N\nsye3v0-6.7.26 N possible configurations for single domain wall, entropy S = ln N\nsye3v0-6.7.27 Free energy of single wall Fone wall = 2J - kBT ln N\nsye3v0-6.7.28 Large systems, finite T: entropic gain outweighs energetic cost\nsye3v0-6.7.29 Introduce domain walls into system\nsye3v0-6.7.30 Thermal expectation value of magnetisation vanishes in thermodynamic limit\nsye3v0-6.7.31 Ferromagnetic order cannot occur at any non-zero T in 1D Ising chain\nsye3v0-6.7.32 Simplest example of topological melting (local order destroyed by proliferation of topological defects)\nsye3v0-6.7.33 Topological melting vs NG modes\nsye3v0-6.8 Berezinskii-Kosterlitz-Thouless phase transition\nsye3v0-6.8.1 Mermin-Wagner-Hohenberg-Coleman theorem (Section 4.2): thermal fluctuations destroy long-range order at T>0 in 2D systems with type-A NG modes\nsye3v0-6.8.2 Rather than exponentially decaying correlations (disordered states)\nsye3v0-6.8.3 Some 2D systems have low-T correlation function decaying as power laws C(x,x') ∝ |x-x'|⁻c (Eq. 6.2)\nsye3v0-6.8.4 Not long-ranged, but qualitatively different from short-ranged\nsye3v0-6.8.5 States with power law correlations exhibit algebraic long-range order\nsye3v0-6.8.6 Low to high T: critical temperature where algebraic order gives way to true disorder\nsye3v0-6.8.7 Power law cannot be analytically continued to exponential function\nsye3v0-6.8.8 Correlation function non-analytic at critical temperature\nsye3v0-6.8.9 True phase transition rather than smooth crossover\nsye3v0-6.8.10 Occurs despite no symmetry truly broken in either phase\nsye3v0-6.8.11 Phase transition described by another form of topological melting: unbinding of pairs of topological defects\nsye3v0-6.8.12 Topologically neutral defect-antidefect pairs (Fig. 6.2) occur as finite-energy excitations\nsye3v0-6.8.13 System at T=0 develops thermal population of such pairs at T>0\nsye3v0-6.8.14 Energy cost of pair scales with separation between cores\nsye3v0-6.8.15 Low T: no defect has sufficient energy to wander far from partner\nsye3v0-6.8.16 Low T phase characterised by thermal population of bound pairs (algebraic long-range order)\nsye3v0-6.8.17 As T raised, pairs more prolific, defects further separated\nsye3v0-6.8.18 At some T, average separation = separation between pairs\nsye3v0-6.8.19 Individual defect no longer associated with partner, single excitations roam freely\nsye3v0-6.8.20 Unbinding of topological defects\nsye3v0-6.8.21 Picture put forward by Berezinskii [76], Kosterlitz and Thouless [77, 78]\nsye3v0-6.8.22 Known as BKT phase transition\nsye3v0-6.8.23 Applied to systems with U(1) or XY-symmetry\nsye3v0-6.8.24 Requirement: stable point-like topological defects formed in 2D systems with SSB at T=0\nsye3v0-6.8.25 Reason pairs unbind at high T: heuristic argument (Kosterlitz/Thouless [78]) inspired by 1D Ising model\nsye3v0-6.8.26 Energy of single isolated vortex ∝ lnL/ξ (L=system size, ξ=coherence length)\nsye3v0-6.8.27 Entropy associated with single defect ≈ kB lnL²/ξ\nsye3v0-6.8.28 Energy and entropy scale same way with size\nsye3v0-6.8.29 Free energy of single isolated defect Fone defect = E - TS ≈ (J - kBT) lnL/ξ (Eq. 6.3)\nsye3v0-6.8.30 Low T: energy cost > entropy gain, isolated defects do not occur\nsye3v0-6.8.31 High T: entropic term outweighs energetic, isolated defects proliferate, destroying order\nsye3v0-6.8.32 Argument shows thermal phase transition unavoidable, but only part of story\nsye3v0-6.8.33 Neglects physics of defect-antidefect pairs screening interactions\nsye3v0-6.8.34 Screening allows pairs to drift further apart, lowers energy cost of additional pairs\nsye3v0-6.8.35 Eventually culminates in proliferation at critical temperature\nsye3v0-6.8.36 BKT transition cannot be described within usual Landau paradigm\nsye3v0-6.8.37 Sometimes called "infinite-order" transition (FL and derivatives continuous)\nsye3v0-6.8.38 Has distinct critical exponents (used to identify experimentally)\nsye3v0-6.8.39 Exponents calculated using appropriate renormalisation group version\nsye3v0-6.8.40 Evidence first found in films of superfluid helium, later in anisotropic magnets, ultracold atomic gases, colloidal discs, thin-film superconductors\nsye3v0-6.8.41 Removal of vortex by rotating vectors out of plane (Fig. 6.4)\nsye3v0-6.8.42 Called "escape in the third dimension"\nsye3v0-6.8.43 Shows vortex not stable topological defect for O represented by 3D vector\nsye3v0-6.9 Classification of topological defects\nsye3v0-6.9.1 Which defects arise determined by broken symmetries defining O\nsye3v0-6.9.2 Details beyond scope, superficial introduction [79]\nsye3v0-6.9.3 O parameter values/directions in broken states correspond to quotient space G/H (Section 2.5.1)\nsye3v0-6.9.4 G=group of all symmetries, H=subgroup of unbroken symmetries\nsye3v0-6.9.5 If O direction varies, different points in real space map to different points in G/H\nsye3v0-6.9.6 System described by map from real space to G/H\nsye3v0-6.9.7 Mathematical structures categorising topologically distinct mappings: homotopy groups\nsye3v0-6.9.8 Example: winding of XY-vectors around vortex (Fig. 6.1)\nsye3v0-6.9.9 System U(1) symmetry broken completely, H=1\nsye3v0-6.9.10 Quotient space G/H = U(1) ≈ S¹ (points on circle)\nsye3v0-6.9.11 Vectors in ordered state point in any direction in 2D plane\nsye3v0-6.9.12 Points on real space contour map to points on S¹\nsye3v0-6.9.13 Go around 1-loop in real space, trace closed path on S¹\nsye3v0-6.9.14 If 1-loop does not encircle singularity, path in G/H covers part of S¹, can be contracted to point\nsye3v0-6.9.15 If single vortex enclosed, G/H traversed completely\nsye3v0-6.9.16 Smooth deformation cannot transform single covering of S¹ into point\nsye3v0-6.9.17 Situation with/without vortex give topologically distinct paths in G/H\nsye3v0-6.9.18 Charge-two vortex (Fig. 6.1c) corresponds to path going around S¹ twice\nsye3v0-6.9.19 Topological index quantifying difference: total number of times S¹ covered (winding number)\nsye3v0-6.9.20 S¹ covered any integer number of times\nsye3v0-6.9.21 First homotopy group π1(U(1)) ≈ Z\nsye3v0-6.9.22 Generally: p-dimensional topological defects in D-dimensional system classified by homotopy group πD-p-1(G/H)\nsye3v0-6.9.23 Contour to detect defects is (D-p-1)-dimensional\nsye3v0-6.9.24 Example: XY-vectors, 1D contour in 2D system characterises 0D point defect\nsye3v0-6.9.25 Domain walls in 1D Ising model: characterised by π0(Z2) = Z2 ("zeroth" homotopy group counts disconnected components)\nsye3v0-6.9.26 Link in chain is domain wall or not\nsye3v0-6.9.27 Vortices in U(1) symmetry: point-like in 2D, line-like in 3D, characterised by π1(U(1)) = Z\nsye3v0-6.9.28 Second homotopy group π2(U(1)) trivial (no monopoles in 3D)\nsye3v0-6.9.29 Ferromagnetic configurations of 3D spins break SU(2) down to U(1)\nsye3v0-6.9.30 O takes values on S² ≈ SU(2)/U(1) (Section 2.5.1)\nsye3v0-6.9.31 Any closed 1-loop on S² can be contracted to point\nsye3v0-6.9.32 π1(S²) = 0 (no stable vortices in Heisenberg ferromagnets)\nsye3v0-6.9.33 If vortex introduced, spins smoothly rotated to perpendicular direction to remove singularity (Fig. 6.4)\nsye3v0-6.9.34 Zero-dimensional defects in 3D Heisenberg ferromagnet: hedgehog/monopole configuration\nsye3v0-6.9.35 Spins point radially outward from origin\nsye3v0-6.9.36 Classified by π2(S²) = Z (integer index counts times 2D surface covers S²)\nsye3v0-6.9.37 Crystalline solids: two types of π1 topological defects (points in 2D/lines in 3D)\nsye3v0-6.9.38 Dislocations: associated with translational symmetry breaking\nsye3v0-6.9.39 Disclinations: associated with rotational symmetry breaking\nsye3v0-6.9.40 Dislocation has vector-valued topological charge (Burgers vector)\nsye3v0-6.9.41 Row of misaligned atomic bonds\nsye3v0-6.9.42 Disclination characterised by angle (wedge of superfluous/deficient material)\nsye3v0-6.9.43 Famously: neutral pairs of dislocations in hexagonal lattices might proliferate\nsye3v0-6.9.44 Led to prediction of new phase: hexatic liquid crystal\nsye3v0-6.9.45 Liquid (translationally symmetric) but possesses hexatic order (rotational symmetry broken to C6) [81-83]\nsye3v0-6.9.46 Transition between crystalline and hexatic liquid-crystalline phases: dislocation-mediated melting (similar to BKT)\nsye3v0-6.9.47 Time-dependent topological excitations\nsye3v0-6.9.48 Defect exists only at one point in space-time: instanton\nsye3v0-6.9.49 Appears in study of Yang-Mills theories, nucleation at first-order transitions\nsye3v0-6.9.50 In 4D space-time, enclosed by 3D contour\nsye3v0-6.9.51 Characterised by third homotopy group π3(G/H)\nsye3v0-6.9.52 Systems with SSB SU(N) symmetry: π3(G/H) can be non-trivial\nsye3v0-6.9.53 Instantons play important role\nsye3v0-6.9.54 Book by Shifman [84] good reference\nsye3v0-6.10 Topological defects at work\nsye3v0-6.10.1 Play role in many physical phenomena\nsye3v0-6.10.2 Brief introduction, not comprehensive\nsye3v0-6.10.3 Duality mapping\nsye3v0-6.10.3.1 Traditional picture: Landau, symmetric disordered state → SSB ordered state\nsye3v0-6.10.3.2 BKT transition: useful to take complementary approach\nsye3v0-6.10.3.3 Proliferation of topological defects leads towards disordered state from ordered phase\nsye3v0-6.10.3.4 Possible to treat topological defects as particles\nsye3v0-6.10.3.5 Transition from low- to high-T phase described as Bose-Einstein condensation of defects\nsye3v0-6.10.3.6 Low/high-T phases both ordered, but differently\nsye3v0-6.10.3.7 Order parameter for defect condensate acts as disorder parameter for original particles, vice versa\nsye3v0-6.10.3.8 Creation of topological defect involves reorientation of particles throughout system\nsye3v0-6.10.3.9 Creation operator for defect extremely non-local in terms of underlying particles' operators\nsye3v0-6.10.3.10 Sometimes: writing original operators in terms of defect operators results in convenient form\nsye3v0-6.10.3.11 Choose to describe physics in terms of original particles or defects\nsye3v0-6.10.3.12 Both pictures give same results, one often easier to apply (low-T vs high-T)\nsye3v0-6.10.3.13 Mathematical map between two descriptions of same system: duality mapping\nsye3v0-6.10.3.14 First example: Kramers-Wannier for 2D Ising model [85]\nsye3v0-6.10.3.15 Writing model in terms of domain walls enabled Onsager to solve exactly [86]\nsye3v0-6.10.3.16 Existence of duality mapping usually allows exploring critical point more easily\nsye3v0-6.10.3.17 Met with success in description of U(1)-symmetry breaking in 2D/3D (boson-vortex duality)\nsye3v0-6.10.3.18 Recently extended to systems with multiple species (fermions) ("web of dualities" [87, 88])\nsye3v0-6.10.3.19 In all cases, phase transition described by duality mapping viewed as unbinding/condensation of defects\nsye3v0-6.10.4 Kibble-Zurek mechanism\nsye3v0-6.10.4.1 Dynamics of phase transitions falls outside scope\nsye3v0-6.10.4.2 Topological defects come into existence by going through continuous phase transition "too quickly"\nsye3v0-6.10.4.3 Near continuous transition, all length/energy scales diverge (Section 5.4)\nsye3v0-6.10.4.4 Characteristic time scales diverge (relaxation time)\nsye3v0-6.10.4.5 Effect of increasing time scales: critical slowing down\nsye3v0-6.10.4.6 Plagues numerical simulations and experiments\nsye3v0-6.10.4.7 Driving system across transition: impossible to retain equilibrium\nsye3v0-6.10.4.8 No matter how slowly cooled, always exceed relaxation rate close to transition\nsye3v0-6.10.4.9 Implication: systems necessarily in highly excited state when entering ordered phase\nsye3v0-6.10.4.10 Relaxing towards equilibrium: long-range order built up gradually\nsye3v0-6.10.4.11 Topological stability of defects present in excited state prevents removal by local relaxation\nsye3v0-6.10.4.12 Result: ordered state with non-zero density of topological defects\nsye3v0-6.10.4.13 Way of creating defects by crossing continuous transition first proposed by Kibble [89]\nsye3v0-6.10.4.14 To explain structure formation in universe after Big Bang\nsye3v0-6.10.4.15 Later refined by Zurek [90]\nsye3v0-6.10.4.16 Derived expected density of defects associated with quench rate and universality class\nsye3v0-6.10.4.17 Now referred to as Kibble-Zurek mechanism\nsye3v0-6.10.5 Topological solitons and skyrmions\nsye3v0-6.10.5.1 Special category of topological objects\nsye3v0-6.10.5.2 Topological solitons (name sometimes applied only to 1D systems)\nsye3v0-6.10.5.3 Topological charge takes quantised values, cannot be altered by smooth deformations\nsye3v0-6.10.5.4 Contrast to usual defects: do not require singularity in O field\nsye3v0-6.10.5.5 Have finite energy, does not scale with system size\nsye3v0-6.10.5.6 Strongly localised near centre of soliton\nsye3v0-6.10.5.7 How created: consider XY ferromagnet on 1D line\nsye3v0-6.10.5.8 Soliton localised near centre, spins far away unaffected\nsye3v0-6.10.5.9 O undisturbed and constant almost everywhere on line\nsye3v0-6.10.5.10 Describe soliton: mathematical transformation maps boundary (or infinity) onto single point\nsye3v0-6.10.5.11 Allowed since O takes same value at these points\nsye3v0-6.10.5.12 1D line: many points from both sides taken to same point, turning line into circle\nsye3v0-6.10.5.13 Mathematical procedure: compactification\nsye3v0-6.10.5.14 Soliton spin configuration in real space corresponds to vortex configuration on compactified space\nsye3v0-6.10.5.15 Vortex core lies in centre of circle\nsye3v0-6.10.5.16 O field along circle smooth and well-defined (Fig. 6.5)\nsye3v0-6.10.5.17 Similar solitons in higher dimensions D, categorised by πD(G/H)\nsye3v0-6.10.5.18 2D plane compactified into 2-sphere (stereographic projection)\nsye3v0-6.10.5.19 S²-valued O (Heisenberg ferromagnet) arranged in hedgehog/monopole configuration\nsye3v0-6.10.5.20 All spins pointing radially outward on sphere\nsye3v0-6.10.5.21 Folding sphere back out into flat plane: resulting spin configuration is skyrmion\nsye3v0-6.10.5.22 Appears in quantum Hall systems, some magnetic materials\nsye3v0-6.10.5.23 In nuclear physics: same configuration called baby skyrmion\nsye3v0-6.10.5.24 Name skyrmion reserved for 3D siblings (introduced by Skyrme [91])\nsye3v0-6.10.5.25 Possible way of creating pointlike objects within smooth 3D vector field\nsye3v0-7.0 Gauge fields\nsye3v0-7.1 Briefly discussed gauge freedom in Section 1.5.2\nsye3v0-7.2 Main focus on systems with global symmetry without gauge fields\nsye3v0-7.3 Gauge freedom presence affects phenomenology of SSB phases/transitions\nsye3v0-7.4 Introduce effects using superconducting state example\nsye3v0-7.4.1 Global phase rotation symmetry broken in presence of local U(1) gauge freedom\nsye3v0-7.4.2 Example makes apparent physics appearing in more complicated non-Abelian types\nsye3v0-7.4.3 Non-Abelian gauge fields briefly discussed in Exercise 7.4\nsye3v0-7.5 Ginzburg-Landau superconductors\nsye3v0-7.5.1 Real-world superconducting materials: metals cooled to low T, undergo phase transition\nsye3v0-7.5.2 Instability of metallic state understood by microscopic BCS theory [92]\nsye3v0-7.5.3 Main ingredient: Cooper instability\nsye3v0-7.5.4 Attractive force between electrons in Fermi liquid leads to bound states of two electrons (Cooper pairs)\nsye3v0-7.5.5 Attractive force from interaction with phonons\nsye3v0-7.5.6 Cooper pair: two fermions, behaves like boson\nsye3v0-7.5.7 Bose-condense at sufficiently low T\nsye3v0-7.5.8 Superconducting state viewed as superfluid of Cooper pairs\nsye3v0-7.5.9 Pairs electrically charged\nsye3v0-7.5.10 Dissipationless flow of superfluid is resistance-free electric supercurrent\nsye3v0-7.5.11 SSB transition in superconductors: Bose-condensation of Cooper pairs\nsye3v0-7.5.12 Discuss broken symmetry, relation to gauge freedom, consequences\nsye3v0-7.5.13 Largely ignore pairs consist of two electrons bound by phonons\nsye3v0-7.5.14 Start from (metallic) normal fluid of charged bosons\nsye3v0-7.5.15 Landau potential Eq. (5.6) describes FL of complex order parameter field ψ(x)\nsye3v0-7.5.16 Ordered state corresponds to neutral superfluid\nsye3v0-7.5.17 Effect of charged pairs seen if allow for density fluctuations\nsye3v0-7.5.18 |ψ(x)|² represents density of bosons\nsye3v0-7.5.19 Add lowest order term in expansion of gradients: ħ²/2m* |∇ψ|² (m*=mass of single boson)\nsye3v0-7.5.20 Density fluctuations of charged pairs create/affected by electromagnetic fields\nsye3v0-7.5.21 Minimal way to introduce coupling: Peierls substitution ∇ψ → (∇-ie*A/ħ)ψ\nsye3v0-7.5.22 A is electromagnetic vector potential, e* is charge of isolated boson\nsye3v0-7.5.23 Cooper pair has two electrons, e*=2e\nsye3v0-7.5.24 Assume scalar potential V=0\nsye3v0-7.5.25 Full FL of Ginzburg-Landau theory for superconductivity (Eq. 7.1)\nsye3v0-7.5.26 Includes potential energy of EM field (µ0 magnetic constant)\nsye3v0-7.5.27 Consider time-independent fields (equilibrium phases)\nsye3v0-7.5.28 FGL explains large part of superconductivity phenomenology\nsye3v0-7.5.29 Includes dissipationless current, Meissner effect, vortex topological defects, Josephson effect\nsye3v0-7.5.30 Intimately related to SSB\nsye3v0-7.5.31 Many excellent textbooks on superconductivity [72, 74, 93, 94]\nsye3v0-7.5.32 Exercise 7.1 (Peierls substitution)\nsye3v0-7.5.32.1 Verify e*A/ħ has units of inverse length\nsye3v0-7.5.32.2 Reason ħ appears: EM fields couple to charged matter via quantum electrodynamics\nsye3v0-7.5.33 FGL invariant under global U(1) symmetry transformation ψ(x) → e-iαψ(x) (Eq. 7.2)\nsye3v0-7.5.34 Global symmetry associated with conserved Noether current\nsye3v0-7.5.35 Current obtained via Noether procedure or applying Peierls substitution to neutral superfluid current (Eq. 1.21)\nsye3v0-7.5.36 Resulting Noether current (Eq. 7.3)\nsye3v0-7.5.37 j = iħ/2m*((∇ψ*)ψ - ψ*(∇ψ)) - e*/m* ψ*ψA\nsye3v0-7.5.38 Write ψ = |ψ|eiϕ: j = ħ/m* |ψ|² (∇ϕ - e*A/ħ)\nsye3v0-7.5.39 Conserved Noether charge transported by this current is number of Cooper pairs\nsye3v0-7.5.40 Charged pairs: current corresponds to actual electric current je = e*j\nsye3v0-7.5.41 Current manifested in ordered phase by NG modes (lifetime infinity in long-wavelength limit)\nsye3v0-7.5.42 Infinitely long-lived current of Cooper pairs: supercurrent\nsye3v0-7.5.43 Coupling to dynamic gauge field suppresses finite-frequency modes (Section 7.3.1)\nsye3v0-7.5.44 Besides global symmetry, FGL invariant under local U(1) gauge transformation\nsye3v0-7.5.45 ψ(x) → e-iα(x)ψ(x), A(x) → A(x) - ħ/e* ∇α(x) (Eq. 7.4)\nsye3v0-7.5.46 Gauge freedom result of introducing superfluous degrees of freedom (longitudinal component of A)\nsye3v0-7.5.47 Longitudinal component does not contribute to observable E and B\nsye3v0-7.5.48 Minimal coupling: phase of ψ(x) also subject to gauge freedom\nsye3v0-7.5.49 In FGL, longitudinal component of A(x) traded for local rotation of ψ(x)\nsye3v0-7.5.50 Gauge transformations are not symmetries (Section 1.5)\nsye3v0-7.5.51 Consistency requirements, not physical manipulation\nsye3v0-7.5.52 Any physical observable derived from FGL must be invariant under Eq. (7.4)\nsye3v0-7.5.53 Gauge invariance can never be broken\nsye3v0-7.5.54 For constant α(x)=α, gauge transformation Eq. (7.4) coincides with global symmetry Eq. (7.2)\nsye3v0-7.5.55 Situation same as Section 1.5.3 (ferromagnet spin rotation vs coordinate rotation)\nsye3v0-7.5.56 Distinction clear when using careful definition of global symmetry w.r.t. external reference\nsye3v0-7.5.57 Gauge-invariant definition for order parameter needed\nsye3v0-7.6 Gauge-invariant order parameter\nsye3v0-7.6.1 Ignoring spatial variations in Cooper pair density, Landau potential same as neutral superfluid (Eq. 5.6)\nsye3v0-7.6.2 For r<0, minimum at ψ≠0\nsye3v0-7.6.3 Global U(1) symmetry broken by choosing phase for minimum energy configuration\nsye3v0-7.6.4 Field variable ψ(x) could be used as O for superfluid\nsye3v0-7.6.5 Local fluctuations make superconductor different from superfluid\nsye3v0-7.6.6 Tempting to introduce ψ as O for superconductor\nsye3v0-7.6.7 Quantity ψ(x) not invariant under gauge transformation Eq. (7.4)\nsye3v0-7.6.8 Any physical quantity must be gauge invariant\nsye3v0-7.6.9 ψ cannot be good O choice\nsye3v0-7.6.10 Three ways of dealing with this, used in practice\nsye3v0-7.6.11 1. Ignore complication, simply use ψ(x) as O\nsye3v0-7.6.12 Not as silly as it sounds\nsye3v0-7.6.13 Many cases: no external EM field, induced fields negligible\nsye3v0-7.6.14 Vector potential ≈ 0, FGL reduces to neutral superfluid FL\nsye3v0-7.6.15 Ignoring EM fields suffices for many physical predictions\nsye3v0-7.6.16 Original BCS theory used O of this form [92]\nsye3v0-7.6.17 Ginzburg-Landau [95] and Josephson [29] papers treated O similarly\nsye3v0-7.6.18 2. Choose particular gauge fix\nsye3v0-7.6.19 Impose additional arbitrary constraints on A and ψ phases to remove gauge freedom\nsye3v0-7.6.20 Given ψ and A, choose gauge transformation ψ→ψ', A→A' satisfying constraints\nsye3v0-7.6.21 Constraints chosen for mathematical convenience/aesthetics\nsye3v0-7.6.22 Two useful choices for superconductivity\nsye3v0-7.6.23 Unitary gauge fix: ψ phase zero everywhere, all degrees of freedom in A\nsye3v0-7.6.24 Coulomb/London gauge fix: ∇ · A = 0 everywhere, longitudinal degree of freedom in ψ phase\nsye3v0-7.6.25 Constraints implemented by gauge transformation, guaranteed not to affect physical observables\nsye3v0-7.6.26 Choosing gauge fix allowed at any step, cannot affect final predictions\nsye3v0-7.6.27 Choosing unitary gauge fix does not mean ψ phase is zero in measurement\nsye3v0-7.6.28 Physical outcome must be gauge invariant\nsye3v0-7.6.29 3. Define gauge-invariant but non-local O\nsye3v0-7.6.30 Not possible to have gauge-invariant local O(x) including only operators in small neighbourhood\nsye3v0-7.6.31 Possible to define non-local O: Dirac [96] proposed ψD(x,t) = ψ(x,t) ei∫d³y Z(y-x)·A(y,t) (Eq. 7.5)\nsye3v0-7.6.32 Z(x) defined by ∇ · Z(x) = e*δ(x)/ħ (proportional to E field from point charge)\nsye3v0-7.6.33 ψD(x) non-local (integrate over space to find value)\nsye3v0-7.6.34 ψD(x) reduces to ψ(x) in Coulomb gauge fix ∇ · A = 0\nsye3v0-7.6.35 Gauge-invariant formulation exists, possible to impose gauge fix and work with ψ(x) as local O\nsye3v0-7.6.36 Remember gauge fix imposed, final predictions must be gauge invariant\nsye3v0-7.6.37 Exercise 7.2 (Dirac order parameter)\nsye3v0-7.6.37.1 Verify Dirac order parameter is invariant under gauge transformation Eq. (7.4)\nsye3v0-7.6.38 Using Dirac O, difference between global symmetry Eq. (7.2) and uniform local gauge freedom Eq. (7.4) clear\nsye3v0-7.6.39 Symmetry broken in superconducting phase: global U(1) phase rotation of ψD(x)\nsye3v0-7.6.40 ψD(x) invariant under any gauge transformation except global α(x)=α\nsye3v0-7.6.41 Reason: local phase of ψ(x) effectively measured in gauge invariant way w.r.t phase at infinity (taken zero)\nsye3v0-7.6.42 Global α(x)=α changes phase everywhere, including infinity\nsye3v0-7.6.43 Analogous to rotating all spins in universe (no measurable effect on ferromagnet)\nsye3v0-7.6.44 Magnetisation orientation measured w.r.t. magnetic field from second magnet\nsye3v0-7.6.45 Crystal position defined w.r.t. reference frame (lab)\nsye3v0-7.6.46 Symmetry spontaneously broken in superconductor: rotation of ψD(x) phase constant throughout material, leaving external reference fixed\nsye3v0-7.6.47 Observable describing relative phase differences: gauge-invariant equal-time correlation function CD(x,x') (Eq. 7.6)\nsye3v0-7.6.48 CD(x,x') = <ψD(x,t)ψ†D(x',t)> = <ψ(x,t)ei∫d³y(Z(y-x)-Z(y-x'))·A(y,t)ψ†(x',t)>\nsye3v0-7.6.49 For separated superconductors, CD proportional to Josephson current (Section 2.5.5)\nsye3v0-7.6.50 Josephson current provides gauge-invariant global O akin to total magnetisation of ferromagnet or centre-of-mass position of crystal\nsye3v0-7.6.51 Local O from which it's built: phase of ψD(x) defined w.r.t. external coordinate system\nsye3v0-7.6.52 Josephson effect used to measure local O value (scanning-tunnelling experiment)\nsye3v0-7.7 The Anderson-Higgs mechanism\nsye3v0-7.7.1 Superconductor spontaneously breaks continuous symmetry\nsye3v0-7.7.2 Reasonably expect it to host NG modes\nsye3v0-7.7.3 Gauge fields mediate long-ranged Coulomb interactions between Cooper pairs\nsye3v0-7.7.4 Goldstone's theorem does not apply with long-ranged interactions\nsye3v0-7.7.5 No guarantee gapless modes exist in ordered state\nsye3v0-7.7.6 Coupling gapless NG mode of neutral superfluid to gapless photon of Maxwell EM makes both massive\nsye3v0-7.7.7 Quickest way to see: rewrite Ginzburg-Landau effective free energy FGL (Eq. 7.1)\nsye3v0-7.7.8 Write field in terms of amplitude |ψ| and phase ϕ: ψ = |ψ|eiϕ\nsye3v0-7.7.9 Define à ≡ A - ħ/e* ∇ϕ\nsye3v0-7.7.10 FGL in terms of |ψ| and à (Eq. 7.7)\nsye3v0-7.7.11 Ã(x) invariant under gauge transformations Eq. (7.4)\nsye3v0-7.7.12 à is physical, observable degree of freedom (like |ψ|)\nsye3v0-7.7.13 à proportional to Noether current Eq. (7.3)\nsye3v0-7.7.14 In superconducting phase, r negative, FGL minimum for non-zero |ψ|\nsye3v0-7.7.15 Second term in Eq. (7.7) e*²/2m* |ψ|² ò interpreted as mass term for Ã\nsye3v0-7.7.16 Check by minimizing FGL for fixed |ψ|\nsye3v0-7.7.17 In ordered phase, both à and |ψ| are massive (gapped dispersion)\nsye3v0-7.7.18 In gauge-invariant description, would-be NG mode ϕ(x) seems to disappear\nsye3v0-7.7.19 Described as "removed" by imposing unitary gauge fix ϕ≡0\nsye3v0-7.7.20 Also said "in unphysical part of Hilbert space" (Coulomb gauge ∇·A=0, ϕ does not couple to observables)\nsye3v0-7.7.21 In neutral superfluid, ϕ(x) is real propagating mode\nsye3v0-7.7.22 Physical degrees of freedom cannot disappear when including interactions\nsye3v0-7.7.23 ϕ excitation has not disappeared, included in newly defined Ã\nsye3v0-7.7.24 Before coupling, free to choose Coulomb gauge ∇·A=0 (eliminates longitudinal component)\nsye3v0-7.7.25 New field à invariant under gauge transformations, none of its components removed\nsye3v0-7.7.26 Degree of freedom carried by ϕ transferred to/represented by longitudinal component of Ã\nsye3v0-7.7.27 Field-theory formulation: massless vector field in 3D carries 2 transverse degrees of freedom\nsye3v0-7.7.28 Massive vector field like à carries 3 degrees of freedom (includes longitudinal component)\nsye3v0-7.7.29 Alternative/more detailed derivation in Hamiltonian constraints formalism (Section 1.5.3) [24]\nsye3v0-7.7.30 Transformation from A and ϕ to à described as "vector field has eaten Goldstone boson"\nsye3v0-7.7.31 Vector field said to "have gotten fat by becoming massive"\nsye3v0-7.7.32 Mechanism for emergence of massive vector field: Anderson-Higgs mechanism [97, 98]\nsye3v0-7.7.33 Also known by other names depending on subfield\nsye3v0-7.7.34 Terminology: field ψ is Higgs field, excitations of |ψ| are Higgs bosons\nsye3v0-7.7.35 Higgs boson has no a priori connection to gauge freedom\nsye3v0-7.7.36 Neutral superfluids, charge density waves, etc. have amplitude modes\nsye3v0-7.7.37 If gauge fields couple to Higgs field ψ, and ψ has vacuum expectation value ψ≠0, Anderson-Higgs mechanism ensures gauge fields massive\nsye3v0-7.7.38 Standard Model: W and Z gauge bosons of electroweak interaction become massive this way (Exercise 7.4)\nsye3v0-7.7.39 Gauge fields becoming massive by coupling to Higgs fields and simultaneous conversion of NG mode into massive component of gauge field is Anderson-Higgs mechanism\nsye3v0-7.7.40 Exercise 7.4 (Non-Abelian gauge fields)\nsye3v0-7.7.40.1 Recall Higgs Lagrangian Eq. (3.11) in terms of U(2)-field Φ˘\nsye3v0-7.7.40.2 L invariant under global Φ˘ → LΦ˘, L∈SU(2)\nsye3v0-7.7.40.3 Not invariant under local transformation L(x)\nsye3v0-7.7.40.4 Try to fix by introducing gauge field\nsye3v0-7.7.40.5 Define Aµ(x) = Σa=1³ Aaµ(x)Ta, Ta are SU(2) Lie algebra generators\nsye3v0-7.7.40.6 Aaµ are real-valued vector fields\nsye3v0-7.7.40.7 Define SU(2)-gauge-covariant derivative Dµ ≡ ∂µI - igAµ\nsye3v0-7.7.40.8 Show DµΦ˘(x) transforms as L(x)(DµΦ˘(x)) under combined transformations (Eq. 7.10)\nsye3v0-7.7.40.9 Field strength of non-Abelian gauge field (Yang-Mills field): Fµν = ∂µAν - ∂νAµ - ig[Aµ, Aν] (Eq. 7.11)\nsye3v0-7.7.40.10 Fµν(x) = Σa=1³ Faµν(x)Ta, Faµν(x) real-valued\nsye3v0-7.7.40.11 Show Fµν transforms as LFµνL⁻¹ under Eq. (7.10)\nsye3v0-7.7.40.12 Lagrangian L = Tr(1/2 (DµΦ˘)†(DµΦ˘) - 1/2 rΦ˘†Φ˘ - 1/4 u(Φ˘†Φ˘)² - 1/4 FµνFµν) (Eq. 7.12)\nsye3v0-7.7.40.13 L invariant under Eq. (7.10)\nsye3v0-7.7.40.14 k>0 components of transformations are gauge freedoms, not symmetries\nsye3v0-7.7.40.15 Denote superfluous degrees of freedom in Aaµ\nsye3v0-7.7.40.16 Global symmetry L(x)=L also transforms gauge field (contrast Abelian case Eq. 7.4)\nsye3v0-7.7.40.17 L also invariant under global Φ˘ → Φ˘R, R∈SU(2) (Exercise 3.2)\nsye3v0-7.7.40.18 When r<0, global part of L together with R broken to diagonal subgroup R=L† [100]\nsye3v0-7.7.40.19 Potential minimum at detΦ˘=0\nsye3v0-7.7.40.20 Show can perform gauge transformation L(x) such that Φ˘=diag(v,v), v∈R (SU(2) equivalent of unitary gauge fix)\nsye3v0-7.7.40.21 Assume detΦ˘ constant (no Higgs boson), in unitary gauge fix ∂µΦ=0\nsye3v0-7.7.40.22 Show mass term 1/2 M²AaµAaµ in L (Eq. 7.12), determine mass M\nsye3v0-7.7.40.23 Substitute <Φ˘>=diag(v,v), use SU(2)-anticommutator {Ta,Tb}=1/2 δabI\nsye3v0-7.7.40.24 M = gv\nsye3v0-7.7.40.25 Anderson-Higgs mechanism for SU(2) gauge-Higgs theory: gauge fields massive, massless NG bosons absent\nsye3v0-7.7.40.26 Standard Model: unified SU(2)×U(1) electroweak force couples to Higgs field, breaks to residual U(1) EM subgroup, photon remains massless\nsye3v0-7.7.40.27 Chiral gauge theory (only L local): issue with chiral anomaly (beyond scope) [71, 101]\nsye3v0-7.7.40.28 Another example: colour superconductivity in quantum chromodynamics (QCD)\nsye3v0-7.7.40.29 Suggested in quark matter at high density (neutron star)\nsye3v0-7.7.40.30 Quarks form Cooper pairs, condense\nsye3v0-7.7.40.31 SU(3) gauge fields ("gluons") become massive\nsye3v0-7.8 Vortices\nsye3v0-7.8.1 In neutral superfluids, broken phase-rotation symmetry allows topological defects (vortex excitations)\nsye3v0-7.8.2 Enter superfluid when external torque applied\nsye3v0-7.8.3 Topological charge = winding number of phase\nsye3v0-7.8.4 Presence of vortex affects O orientation throughout superfluid\nsye3v0-7.8.5 Vortices exert long-range forces on each other\nsye3v0-7.8.6 Energy of single vortex grows with system size\nsye3v0-7.8.7 In charged superfluid/superconductor, combination of phase degree of freedom with EM vector potential changes vortex nature\nsye3v0-7.8.8 If vortex present, O amplitude vanishes at core\nsye3v0-7.8.9 Meissner effect (expels magnetic fields) not operative in core region\nsye3v0-7.8.10 Magnetic field penetrates core\nsye3v0-7.8.11 Resulting field profile for vortex excitation (Fig. 7.1)\nsye3v0-7.8.12 How much magnetic flux penetrates? Consider relation between supercurrent and vector potential (Eq. 7.8)\nsye3v0-7.8.13 Integrate both sides along closed contour C encircling core\nsye3v0-7.8.14 Contour far from core, magnetic field screened by Meissner effect, left side zero\nsye3v0-7.8.15 ∫C dx · ∇ϕ = ∫C dx · A = ∫S dS · B (Eq. 7.13) (S=area enclosed by C)\nsye3v0-7.8.16 Left side: ∫∇ϕ = 2πn (winding number)\nsye3v0-7.8.17 Right side: total flux through S\nsye3v0-7.8.18 Magnetic flux penetrating superconductor through vortices is quantised in units of Φ0 = h/e* (flux quantum)\nsye3v0-7.8.19 Decay of magnetic field radially outward from core understood intuitively\nsye3v0-7.8.20 Winding of ϕ does not depend on contour size\nsye3v0-7.8.21 Close to core, |ψ| low, phase winding leads to circular electric supercurrent (Eq. 7.3)\nsye3v0-7.8.22 Supercurrent opposes externally applied magnetic field, leading to decay\nsye3v0-7.8.23 Same physics as surface of superconductor, length scale is London penetration depth λL\nsye3v0-7.8.24 Generated magnetic field cancels supercurrent, also decays within λL\nsye3v0-7.8.25 Stark contrast to neutral superfluid: O field outside area of radius λL unaffected by defect\nsye3v0-7.8.26 Well-separated vortices in superconductor have no interaction\nsye3v0-7.8.27 Energy of single vortex does not depend on system size\nsye3v0-7.8.28 For 3D superconductor with λL >> ξ, energy of straight vortex line of length lz: Evortex ≈ lz (nΦ0/λL)² ln√λL/2ξ (Eq. 7.14) [74]\nsye3v0-7.8.29 n is winding number\nsye3v0-7.8.30 Energy scales as ln λL, not ln L\nsye3v0-7.8.31 O affected only up to distances <λL\nsye3v0-7.8.32 Total vortex free energy: field-independent cost Evortex + magnetic energy gained by allowing field to pass\nsye3v0-7.8.33 Magnetic energy gained -H · B (H=external field, B=Φ0lz induced field)\nsye3v0-7.8.34 Above critical field Hc1 = Evortex/Φ0lz, energetically favourable to let field in through vortex lines\nsye3v0-7.8.35 Resulting state: Abrikosov vortex lattice\nsye3v0-7.8.36 Distinctive feature of type-II superconductors with λL > √2ξ\nsye3v0-7.8.37 Increasing field beyond Hc1: vortices more closely packed, cores overlap, superconductivity destroyed\nsye3v0-7.8.38 Total flux distributed over vortices, each carrying single flux quantum\nsye3v0-7.8.39 Cores cover entire superconductor when Φ0/H ∝ ξ²\nsye3v0-7.8.40 Superconductivity breaks down for fields >Hc2 ∝ Φ0/ξ² > Hc1\nsye3v0-7.8.41 Hc1, Hc2 are lower/upper critical fields in type-II superconductors\nsye3v0-7.8.42 If λL < √2ξ, Evortex estimate breaks down\nsye3v0-7.8.43 More careful analysis: energetically favourable to create defects with highest vorticity\nsye3v0-7.8.44 Abrikosov lattice not formed, bulk remains in Meissner state\nsye3v0-7.8.45 External field increased until destroys superconducting order: single critical field Hc\nsye3v0-7.8.46 Type of behavior: type-I superconductivity\nsye3v0-7.9 Charged BKT phase transition\nsye3v0-7.9.1 Coupling of phase degree of freedom to EM field alters characteristics of phase transition vs neutral superfluid\nsye3v0-7.9.2 Situation in 2D particularly interesting\nsye3v0-7.9.3 Brings together several topics\nsye3v0-7.9.4 Physics described in terms of heuristic arguments, comparing vortices\nsye3v0-7.9.5 Neutral systems with U(1) symmetry (superfluids) undergo BKT phase transition in 2D (Section 6.3)\nsye3v0-7.9.6 Comes from two ingredients\nsye3v0-7.9.7 1. Mermin-Wagner-Hohenberg-Coleman theorem (Section 4.2): thermal fluctuations destroy long-range order at T>0 in 2D (type-A NG modes)\nsye3v0-7.9.8 Due to divergence of thermal corrections to O\nsye3v0-7.9.9 Algebraic long-range order possible (bound vortex-antivortex pairs)\nsye3v0-7.9.10 2. Energy cost of pairs balanced by entropy (both scale logarithmically)\nsye3v0-7.9.11 Leads to critical T where pairs unbind\nsye3v0-7.9.12 In charged superfluid, both arguments adjusted\nsye3v0-7.9.13 1. Anderson-Higgs mechanism (Section 7.3) renders all excitations massive\nsye3v0-7.9.14 No gapless NG modes\nsye3v0-7.9.15 No divergence as k→0 in O corrections (Eq. 4.22)\nsye3v0-7.9.16 At first sight, no obstruction to truly long-range ordered 2D superconductors at T>0\nsye3v0-7.9.17 2. Vortex excitations of neutral superfluid altered by EM field\nsye3v0-7.9.18 Section 7.4: winding of O counteracted by penetrating magnetic field\nsye3v0-7.9.19 Energy of one vortex finite, does not scale with system size (order of ln λL/ξ)\nsye3v0-7.9.20 Total vortex free energy: balance between entropy and energy\nsye3v0-7.9.21 Entropy of vortex counts possible locations for core S∝lnL²/ξ² (valid for superconductors)\nsye3v0-7.9.22 Energy of superconducting vortices no longer scales with system size\nsye3v0-7.9.23 Large systems (L>>λL): entropic gain outweighs energetic cost even at lowest T\nsye3v0-7.9.24 BKT transition T where pairs unbind pushed all way to zero, should destroy superconducting order at any T\nsye3v0-7.9.25 In reality, live in 3D world, even thinnest superconductors couple to 3D EM fields\nsye3v0-7.9.26 Vortices interact over large distances through EM fields in vacuum surrounding film\nsye3v0-7.9.27 Described by effective penetration depth of magnetically mediated interactions λL,2D = λL²/w (w=thickness) [102]\nsye3v0-7.9.28 If w small compared to λL, effective penetration depth very large\nsye3v0-7.9.29 λL inversely proportional to e*, large penetration depth equivalent to weak coupling\nsye3v0-7.9.30 Truly 2D superconductor embedded in 3D vacuum behaves like neutral superfluid\nsye3v0-7.9.31 Gap of would-be NG modes becomes very small\nsye3v0-7.9.32 Thermal fluctuations prevent formation of true long-range order\nsye3v0-7.9.33 Simultaneously, energy cost of vortices grows, depends on system size, pushing BKT transition T up from zero\nsye3v0-7.9.34 Explains experimental observation of BKT transitions in thin type-II superconductors and Josephson junction arrays [102]\nsye3v0-7.10 Order of the superconducting phase transition\nsye3v0-7.10.1 Even in 3D, true long-range order exists\nsye3v0-7.10.2 Superconducting phase transition affected by coupling between phase and EM vector potential\nsye3v0-7.10.3 Both fields can fluctuate\nsye3v0-7.10.4 Fluctuations alter critical behaviour from mean-field expectation (style of Sections 5.4, 5.5)\nsye3v0-7.10.5 One way to investigate: integrate out gauge field\nsye3v0-7.10.6 Partition function sum over all configurations\nsye3v0-7.10.7 Possible to perform partial sum over gauge fields keeping O field fixed\nsye3v0-7.10.8 Natural in path integral formulation of QFT\nsye3v0-7.10.9 Result: new 'effective' theory for O field, gauge field does not appear explicitly\nsye3v0-7.10.10 May include different terms, changed coefficients\nsye3v0-7.10.11 Shortcut: follow Ref. [103], start from FGL (Eq. 7.1) in 3D\nsye3v0-7.10.12 Expand minimal coupling term |(∇-ie*A/ħ)ψ|²\nsye3v0-7.10.13 Replace A and powers by expectation values (calculated with fixed ψ)\nsye3v0-7.10.14 No static EM fields in superconducting state, terms linear in A must vanish\nsye3v0-7.10.15 Fluctuations A · A do not vanish\nsye3v0-7.10.16 Make replacement ∫d³x e*²/2m* |ψ|²A² → e*²/2m* |ψ|² ∫d³x A² (Eq. 7.15)\nsye3v0-7.10.17 Expectation value for fluctuations computed from London equation (7.9)\nsye3v0-7.10.18 Two-point correlation function for A in Coulomb gauge, momentum space: <Ai(k)Aj(-k)> = µ0 (δij - kikj/k²) / (k² + 1/λL²) (Eq. 7.16)\nsye3v0-7.10.19 Use Eq. (7.16) to carry out integral in final line of Eq. (7.15)\nsye3v0-7.10.20 Result: e*²µ0/4πm* |ψ|²Λ - e*³µ0³/²/8m*³/² |ψ|³ (Eq. 7.17) (Λ=cutoff momentum scale)\nsye3v0-7.10.21 Final expression replaces coupling term ∝ |ψ|²A² in Ginzburg-Landau theory\nsye3v0-7.10.22 Modified, effective theory formulated in terms of O field\nsye3v0-7.10.23 First term ∝ |ψ|² renormalises r value\nsye3v0-7.10.24 Second term introduces new term ∝ |ψ|³\nsye3v0-7.10.25 Causes minimum of FL to develop at non-zero |ψ| (similar to Fig. 5.3)\nsye3v0-7.10.26 Phase transition in presence of fluctuations must be first-order (discontinuous)\nsye3v0-7.10.27 Called fluctuation-induced first-order phase transition\nsye3v0-7.10.28 In field theory, way of arriving at discontinuous transition known as Coleman-Weinberg mechanism\nsye3v0-7.10.29 Experiments on many superconductors show critical properties close to mean-field predictions (e.g., β=1/2)\nsye3v0-7.10.30 One reason: interval where fluctuations important (quantified by tG) is very small\nsye3v0-7.10.31 tG (Eq. 5.25) can be sizeable for strongly type-II superconductors (λL >> ξ)\nsye3v0-7.10.32 Same conditions allow emergence of stable vortices (Section 7.4)\nsye3v0-7.10.33 Using duality mapping (Section 6.5.1), found presence of topological defects alters effective theory in critical region\nsye3v0-7.10.34 Causes superconducting phase transition in 3D to stay second-order as long as λL/ξ << 1 [104]\nsye3v0-7.10.35 Prediction confirmed in numerical simulations\nsye3v0-8.0 Other aspects of spontaneous symmetry breaking\nsye3v0-8.1 Glasses\nsye3v0-8.1.1 Rigidity of collective states governed by SSB\nsye3v0-8.1.2 Apparent in crystalline solids (gapless NG modes/phonons assure rigidity)\nsye3v0-8.1.3 Most solids in everyday life are not crystalline (glasses)\nsye3v0-8.1.4 Glass: solid when it comes to short-time response\nsye3v0-8.1.5 Macroscopic ways indistinguishable from crystalline solids (rigid, carry phonons, break, break translational symmetry)\nsye3v0-8.1.6 Microscopic level: glasses are disordered\nsye3v0-8.1.7 Example: SiO2 molecules in windowpane sit at random positions, no long-range order in conventional correlation function\nsye3v0-8.1.8 Differ from polycrystalline systems (crystalline order at intermediate length scale)\nsye3v0-8.1.9 Detailed theoretical description of glass phase elusive\nsye3v0-8.1.10 Major open question: is glass phase genuine phase of matter or slow, viscous liquid?\nsye3v0-8.1.11 Related question: true thermodynamic glass transition separating high-T liquid from low-T stable glass phase\nsye3v0-8.1.12 Difficult to answer: most glasses made by supercooling liquid to avoid crystallisation\nsye3v0-8.1.13 Viscosity increases exponentially upon further cooling (Section A.3)\nsye3v0-8.1.14 Theoretical simulations: "glasses" often used for systems with quenched disorder\nsye3v0-8.1.15 Described by Hamiltonians with random parameter values (probability distribution)\nsye3v0-8.1.16 Example: Sherrington-Kirkpatrick model (Ising spin with random interaction with every other spin)\nsye3v0-8.1.17 Low-T rigid phase without long-range order (akin to glass)\nsye3v0-8.1.18 States observed do not spontaneously break translational symmetry (quenched disorder explicitly broke it)\nsye3v0-8.1.19 Within ensemble of Hamiltonians, quenched disorder glasses break symmetry between different disorder realisations (replica symmetry breaking)\nsye3v0-8.1.20 Good books on quenched disorder glasses [105-107]\nsye3v0-8.1.21 Detailed introduction of glasses without quenched disorder [108, 109]\nsye3v0-8.1.22 Second class of solids without long-range order: quasicrystals\nsye3v0-8.1.23 No periodicity in atomic positions\nsye3v0-8.1.24 Fourier transform of atomic positions has perfectly sharp peaks (infinitely many, fractal pattern)\nsye3v0-8.1.25 Discovered by unusual but sharp electron diffraction pattern\nsye3v0-8.1.26 From SSB perspective: quasicrystal completely breaks translational symmetry (unlike normal crystals)\nsye3v0-8.1.27 Can break rotational symmetry down to discrete subgroup\nsye3v0-8.1.28 Remaining rotational symmetries can be types not allowed in crystalline matter (five-fold, ten-fold)\nsye3v0-8.2 Many-body entanglement\nsye3v0-8.2.1 Recent development in quantum physics\nsye3v0-8.2.2 Topic introduced by reviewing two-particle entanglement\nsye3v0-8.2.3 Good discussions in textbooks [1, 110, 111]\nsye3v0-8.2.4 System with two spin-1/2 degrees of freedom\nsye3v0-8.2.5 Full Hilbert space spanned by four states\nsye3v0-8.2.6 Particularly interesting: singlet state |0> = 1/√2 (|↓1 ⊗ |↑2> - |↑1 ⊗ |↓2>) (A.1)\nsye3v0-8.2.7 State is entangled, not easy to quantify how\nsye3v0-8.2.8 One way: consider properties of reduced density matrix\nsye3v0-8.2.9 Arbitrarily split full Hilbert space into two parts\nsye3v0-8.2.10 Reduced density matrix on spin 1 obtained by 'tracing' full density matrix over spin 2 degrees of freedom\nsye3v0-8.2.11 Pure state Eq. (A.1), full density matrix ρ = |0><0|\nsye3v0-8.2.12 Trace operation carried out explicitly: ρ1 = Tr2 ρ (A.2)\nsye3v0-8.2.13 ρ1 = 1/2 (|↑1><↑1| + |↓1><↓1|)\nsye3v0-8.2.14 ρ1 no longer represents pure state (spins 1 and 2 entangled)\nsye3v0-8.2.15 Quantify entanglement: Von Neumann proposed entanglement entropy SvN = -Trρ1 log ρ1 (A.3)\nsye3v0-8.2.16 Singlet state: SvN = log 2\nsye3v0-8.2.17 Recent years: entanglement studied in systems with many degrees of freedom (many-spin system)\nsye3v0-8.2.18 Still talk about bipartite entanglement\nsye3v0-8.2.19 Arbitrarily split system into A and B, compute reduced density matrix on one subsystem, use SvN\nsye3v0-8.2.20 Typically interesting: how SvN scales with size of subsystem\nsye3v0-8.2.21 Special cases\nsye3v0-8.2.22 'Product state' (N´eel antiferromagnet): no entanglement between any two spins, SvN=0\nsye3v0-8.2.23 Spins entangled with few close spins: SvN scales with size of boundary of region A ('area law' entanglement)\nsye3v0-8.2.24 All ground states of locally interacting systems obey area law\nsye3v0-8.2.25 Extreme case: each spin entangled with every other spin\nsye3v0-8.2.26 SvN scales with volume of subsystem A ('volume law')\nsye3v0-8.2.27 Typically applies to sufficiently highly excited eigenstates\nsye3v0-8.2.28 Topologically ordered states (Section A.5) considered most entangled (discussion beyond scope [112])\nsye3v0-8.2.29 Possible connection between entanglement entropy and classical statistical entropy at any T (active research field)\nsye3v0-8.2.30 Relation to SSB: Section 2.7 showed exact ground states of symmetric H displaying long-range correlations indicate SSB\nsye3v0-8.2.31 Implies unstable ground states always highly entangled\nsye3v0-8.2.32 Stable SSB states unentangled product states satisfying cluster decomposition\nsye3v0-8.2.33 Careful before equating absence of bipartite entanglement with stability\nsye3v0-8.2.34 Example: unstable ground state of Ising model (Eq. 2.50) is maximally entangled (GHZ state)\nsye3v0-8.2.35 SvN = log 2 regardless of system division\nsye3v0-8.2.36 Entanglement entropy by itself not sufficient to distinguish stable/unstable ground states\nsye3v0-8.2.37 Search for alternative measures ongoing\nsye3v0-8.2.38 SvN captures subtle properties related to SSB\nsye3v0-8.2.39 Type A SSB systems with symmetric finite size ground state: SvN has logarithmic corrections to area law\nsye3v0-8.2.40 Coefficient in front of logarithm counts number of Nambu-Goldstone modes NNG [113, 114]\nsye3v0-8.2.41 SvN(LD) = aLD-1 + NNG(D-1)/2 log L + ... (A.4)\nsye3v0-8.2.42 So-called entanglement Hamiltonian H ≡ -log ρA has same structure as tower of states Section 2.6\nsye3v0-8.2.43 Two examples of how info about SSB state/excitations hidden inside entanglement structure of symmetric ground state\nsye3v0-8.3 Dynamics of spontaneous symmetry breaking\nsye3v0-8.3.1 Focus on equilibrium properties, neglected how transition comes about/evolves\nsye3v0-8.3.2 Mentioned Kibble-Zurek mechanism (Section 6.5.2) describing defect formation during second-order transition\nsye3v0-8.3.3 Several other related processes connected to dynamical phase transitions\nsye3v0-8.3.4 Example: paramagnetic-to-ferromagnetic transition (second-order)\nsye3v0-8.3.5 Typically results in ferromagnetic state with multiple magnetic domains\nsye3v0-8.3.6 Domains with more-or-less random orientations\nsye3v0-8.3.7 Even if no topological defects, torque exerted by domains\nsye3v0-8.3.8 System tends to avoid large single domain formation (sizeable magnetostatic energy cost)\nsye3v0-8.3.9 First-order phase transitions: related phenomenon due to energy barrier between symmetric/asymmetric phases\nsye3v0-8.3.10 Prevents system relaxing to SSB configuration from symmetric metastable state\nsye3v0-8.3.11 Example: slowly cooling pure liquid, remains liquid below Tc (supercooling)\nsye3v0-8.3.12 Perturbed by motion/T difference/imperfection: local potential energy causes crossing barrier\nsye3v0-8.3.13 Liquid crystallises locally, enables neighbouring regions to crystallise\nsye3v0-8.3.14 Solidification spreads quickly (nucleation event)\nsye3v0-8.3.15 Scale invariance associated with critical point (Section 5.5) pertains to dynamical phenomena\nsye3v0-8.3.16 Leads to emergence of dynamical critical exponents for quantities like damping/relaxation [115, 116]\nsye3v0-8.4 Quantum phase transitions\nsye3v0-8.4.1 Discussion of phase transitions in Chapter 5 covered ordinary, thermal transitions\nsye3v0-8.4.2 Change from one phase to another result of change in T\nsye3v0-8.4.3 Transition from ordered to disordered phase (low to high T) due to thermal fluctuations\nsye3v0-8.4.4 Possible to study change at fixed T as other parameter varied (density, field, pressure, current)\nsye3v0-8.4.5 Readily accessible tuning parameters\nsye3v0-8.4.6 Transitions viewed as thermal transitions by tuning to critical value of alternative parameter\nsye3v0-8.4.7 Obvious when considering phase diagram with T on one axis, alternative parameter on other\nsye3v0-8.4.8 Crossing transition line can be reached in purely thermal transition\nsye3v0-8.4.9 Not true for transitions at precisely zero T\nsye3v0-8.4.10 System undergoes quantum phase transition between two stable phases as parameter p tuned through quantum critical point pc\nsye3v0-8.4.11 At T=0, system in ground state at any p value (Section 2.6)\nsye3v0-8.4.12 Thermal excitations/fluctuations do not play role\nsye3v0-8.4.13 Sometimes said due to quantum fluctuations (misnomer Section 4)\nsye3v0-8.4.14 System in ground state of H throughout transition, nothing fluctuates/changes in time\nsye3v0-8.4.15 Actual situation: far from pc, quantum system close to classical state (Section 2.5.3)\nsye3v0-8.4.16 Quantum corrections become more important as p approaches pc\nsye3v0-8.4.17 Completely overwhelm classical correlations at quantum critical point\nsye3v0-8.4.18 Far across transition, system approaches different classical state\nsye3v0-8.4.19 Just like thermal transitions, quantum critical point characterised by scale invariance/universality\nsye3v0-8.4.20 Universality class of quantum critical point linked to that of thermal critical point in one dimension higher\nsye3v0-8.4.21 Understood by recalling thermal/quantum corrections calculation (Section 4.2)\nsye3v0-8.4.22 At T=0, Matsubara frequencies taken into account, effectively additional dimension\nsye3v0-8.4.23 At non-zero T, finite number of Matsubara frequencies interpreted as thermal corrections\nsye3v0-8.4.24 Exactly zero T beyond reach of experiment\nsye3v0-8.4.25 Discussion of quantum phase transitions seems academic\nsye3v0-8.4.26 Existence of quantum critical point has large influence on physics near pc even at non-zero T (hundreds of Kelvins)\nsye3v0-8.4.27 Referred to as quantum criticality\nsye3v0-8.4.28 Strongest manifestations of non-classical physics in condensed matter\nsye3v0-8.4.29 Good resource: Sachdev's textbook [117]\nsye3v0-8.4.30 Many phase transitions in (particle) field theory at zero density/T\nsye3v0-8.4.31 Sort of quantum phase transition\nsye3v0-8.4.32 Lorentz invariant: time dimension treated on equal footing with spatial\nsye3v0-8.4.33 No quantum critical region in phase diagram (finite T not accessible by construction)\nsye3v0-8.5 Topological order\nsye3v0-8.5.1 Discovery of integer quantum Hall effect made clear symmetry/SSB alone do not exhaust interesting phases\nsye3v0-8.5.2 Quantum Hall effect in 2D electron gases in perpendicular magnetic field\nsye3v0-8.5.3 Field forces electrons into orbital motion (Landau levels)\nsye3v0-8.5.4 Longitudinal conductivity completely suppressed for values of field where electrons fill Landau levels\nsye3v0-8.5.5 Going from 1 filled level to 2 filled levels: transition between different quantum Hall phases\nsye3v0-8.5.6 Characterised by different (quantised) values of transverse/Hall conductivity\nsye3v0-8.5.7 None of quantum Hall states have non-trivial SSB\nsye3v0-8.5.8 From symmetry perspective, indistinguishable from ordinary gas/liquid\nsye3v0-8.5.9 No local order parameter in any\nsye3v0-8.5.10 Instead of O, states labelled by quantised topological invariant (value of Hall conductivity)\nsye3v0-8.5.11 Not directly related to topological defects of Section 6\nsye3v0-8.5.12 Invariants topological: calculated by integral over whole system, unaffected by local deformations\nsye3v0-8.5.13 Physics of quantum Hall states generalised to topological insulators/superconductors\nsye3v0-8.5.14 Also characterised by topological invariants\nsye3v0-8.5.15 Always remain disordered in sense of SSB\nsye3v0-8.5.16 Often occur in crystalline materials, lattice symmetries play intricate role\nsye3v0-8.5.17 Invariants topological: insensitive to local perturbations\nsye3v0-8.5.18 Adding weak interactions typically does not destroy topology (as long as symmetries not broken)\nsye3v0-8.5.19 Special feature: necessary existence of gapless mode at interface between two materials with different topological invariants\nsye3v0-8.5.20 Known as symmetry-protected edge modes\nsye3v0-8.5.21 Cannot be avoided as long as interface does not break necessary symmetries\nsye3v0-8.5.22 Returning to quantum Hall effect: states with zero longitudinal/quantised Hall conductance emerge at rational filling fractions (fractional quantum Hall effect)\nsye3v0-8.5.23 Interactions between excitations strong/essential in establishing state\nsye3v0-8.5.24 To distinguish fractional quantum Hall states/other interacting liquids: Wen pioneered notion of topological order [119]\nsye3v0-8.5.25 Unlike integer QH/topological insulators, topologically ordered materials have non-zero ground state degeneracy\nsye3v0-8.5.26 Distinct ground states labelled by topological invariant\nsye3v0-8.5.27 Occur in strongly interacting systems\nsye3v0-8.5.28 Qualitatively changes other features\nsye3v0-8.5.29 Long-range entangled (Section A.2) vs short-range entangled (topological insulators)\nsye3v0-8.5.30 Typically have fractionalised excitations (quasiparticles in fractional QH states, charge is rational fraction of elementary charge)\nsye3v0-8.5.31 Reviews of topological order [119, 120, 121]\nsye3v0-8.5.32 Superconductors where condensed bosons composite (charge fraction of constituent) considered topologically ordered [122]\nsye3v0-8.6 Time crystals\nsye3v0-8.6.1 Crystalline solid: medium where translational symmetry broken to discrete infinite subgroup\nsye3v0-8.6.2 Wonder if systems exist breaking time translation symmetry to discrete infinite subgroup (time crystal)\nsye3v0-8.6.3 Wilczek proposed idea [123]\nsye3v0-8.6.4 Pointed out spontaneous breaking of time translation symmetry fundamentally impossible in equilibrium [124, 125]\nsye3v0-8.6.5 No-go theorems did not exclude possibility out of equilibrium\nsye3v0-8.6.6 Oscillating state with period T: time translation symmetry discrete from outset\nsye3v0-8.6.7 Described using Floquet theory\nsye3v0-8.6.8 Periodic evolution can have further spontaneous breaking of time translations\nsye3v0-8.6.9 Leads to recurring dynamics with longer period nT (n integer)\nsye3v0-8.6.10 Such systems called discrete time crystals\nsye3v0-8.6.11 To avoid trivialities, emergence of longer period accompanied by rigidity\nsye3v0-8.6.12 Even when driving not at preferred frequency, time-ordered state emerges\nsye3v0-8.6.13 Systems displaying discrete time-crystalline order reported in experiments\nsye3v0-8.6.14 Trapped ions driven by periodic laser pulses [126]\nsye3v0-8.6.15 Diamond spin impurities driven by microwave radiation [127]\nsye3v0-8.6.16 Recent review [128]\nsye3v0-8.7 Higher-form symmetry\nsye3v0-8.7.1 Noether charge defined as Q = ∫V dDx j0(x) (integral of density over space)\nsye3v0-8.7.2 Operator is global symmetry generator for ordinary local fields (point particles)\nsye3v0-8.7.3 Point particles charged under this generator\nsye3v0-8.7.4 Notion of generalised global symmetry/higher-form symmetry [129]\nsye3v0-8.7.5 Generalises concept: considers spatially extended charged objects (lines, surfaces)\nsye3v0-8.7.6 Symmetry generators defined as integrals over lower-dimensional space that extended object intersects\nsye3v0-8.7.7 Illustrated using ordinary Maxwell electromagnetism in empty 3+1D spacetime\nsye3v0-8.7.8 U(1) gauge freedom defined (Eq. 1.38)\nsye3v0-8.7.9 Define line operators along contour C: WC = ei∫C dxµAµ (Wilson loop), HC = ei∫C dxµÃµ ('t Hooft loop) (A.5, A.6)\nsye3v0-8.7.10 Maxwell field strength tensor Fµν used to define dual photon field õ\nsye3v0-8.7.11 Electric/magnetic symmetry generators act on these line operators\nsye3v0-8.7.12 QE = ∫Σ dStm Ftm = ∫Σ dStm Em, QM = ∫Σ dSmn Fmn = ∫Σ dSmn mnkBk (A.7)\nsye3v0-8.7.13 Σ is surface perpendicular to C\nsye3v0-8.7.14 In language of differential forms, these are 1-form symmetries\nsye3v0-8.7.15 Symmetry transformation acting on line operators results in 1-form (vector) valued phase\nsye3v0-8.7.16 Similarly, p-form symmetries defined on (D-p)-dimensional subspace, act on p-dimensional objects\nsye3v0-8.7.17 Higher-form symmetries can be broken\nsye3v0-8.7.18 Line operators act as (non-local) order parameters/generalised correlation functions\nsye3v0-8.7.19 Expectation value of line operator depends on integration contour C\nsye3v0-8.7.20 If satisfies area law WC ∝ e-|S| (S=area enclosed), symmetry unbroken\nsye3v0-8.7.21 If satisfies perimeter law WC ∝ e-|C| (|C|=length of C), symmetry broken\nsye3v0-8.7.22 Generalises to higher forms\nsye3v0-8.7.23 Magnetic symmetry QM spontaneously broken in Maxwell vacuum ('t Hooft line HC satisfies perimeter law)\nsye3v0-8.7.24 Associated NG mode is photon Aµ itself\nsye3v0-8.7.25 Higher-form symmetry restored in superconductor (photon becomes gapped by Anderson-Higgs mechanism Section 7.3)\nsye3v0-8.7.26 Dual description of superconducting phase transition\nsye3v0-8.7.27 Understanding: in vacuum, magnetic fields free/unconfined (cost little energy), analogous to bosons in BEC\nsye3v0-8.7.28 In superconductor, magnetic flux confined into vortex lines, expensive to create, analogous to gapped bosonic excitations\nsye3v0-8.7.29 p-form generalisation of Goldstone theorem [130, 131]\nsye3v0-8.7.30 Interesting connection to topology of Sec. A.5\nsye3v0-8.7.31 Higher form discrete symmetry spontaneously broken, system exhibits topological order [132]\nsye3v0-8.8 Decoherence and the measurement problem\nsye3v0-8.8.1 Hallmark of quantum physics: ability to create superpositions of any two states\nsye3v0-8.8.2 Completely isolated system: superpositions never decay\nsye3v0-8.8.3 Superposition of energy eigenstates shows Rabi oscillations\nsye3v0-8.8.4 Reality: no system completely isolated\nsye3v0-8.8.5 Hard to maintain coherent superpositions of macroscopically distinct states for extended time\nsye3v0-8.8.6 Detailed understanding: decoherence program [133-138]\nsye3v0-8.8.7 Important success: identification of set of stable states\nsye3v0-8.8.8 Quantum states described using any Hilbert space basis\nsye3v0-8.8.9 Why everyday objects in eigenstates of position, hardly ever momentum?\nsye3v0-8.8.10 Resolution: coupling of observable system to environment selects preferred basis (pointer basis)\nsye3v0-8.8.11 Pointer basis: eigenstates of full H describing system, environment, interaction\nsye3v0-8.8.12 If energy scale of interaction > energy scale of systems/environment, pointer basis set by interaction H\nsye3v0-8.8.13 Second important observation: process of decoherence itself\nsye3v0-8.8.14 Starting from pure state |ψ>, write density matrix ρ = |ψ><ψ| in pointer basis\nsye3v0-8.8.15 Interaction with environment (large number of uncontrollable degrees of freedom/heat bath)\nsye3v0-8.8.16 Typically causes off-diagonal elements of reduced density matrix (traced over environment) to decay exponentially quickly\nsye3v0-8.8.17 Leaving mixed ensemble of pointer states\nsye3v0-8.8.18 Understanding decoherence important in experiments exploiting quantum superpositions (computations/simulations)\nsye3v0-8.8.19 Good exposition: textbook by Schlosshauer [139]\nsye3v0-8.8.20 Evolution of pure state to mixed state suggestive of what happens during quantum measurement\nsye3v0-8.8.21 Following unitary time evolution, interaction between microscopic quantum object and machine leads to superposition of pointer states\nsye3v0-8.8.22 In single experiment, only single outcome observed\nsye3v0-8.8.23 Decoherence program addresses some aspects of measurement problem\nsye3v0-8.8.24 Explains which outcomes may be observed (defines pointer basis)\nsye3v0-8.8.25 Explains suppression of quantum interference between pointer states (disappearance of off-diagonal elements)\nsye3v0-8.8.26 Does not solve measurement problem\nsye3v0-8.8.27 Cannot explain which of available pointer states will be observed as single outcome\nsye3v0-8.8.28 Not case of ignorance as in classical measurement\nsye3v0-8.8.29 Cannot say anything about outcome of single measurement even with perfect knowledge/control\nsye3v0-8.8.30 Quantum measurement is more fundamental problem\nsye3v0-8.8.31 Under unitary time evolution, quantum dynamics evolves single initial state to unique final state\nsye3v0-8.8.32 Given initial superposition, sometimes observe one outcome, sometimes another\nsye3v0-8.8.33 Description of measurement process must be non-unitary [140, 141]\nsye3v0-8.8.34 Several authors noticed peculiar parallels between measurement problem and SSB [24, 142-146]\nsye3v0-8.8.35 Given symmetric H, any superposition of states in broken manifold equally likely to be realised\nsye3v0-8.8.36 Some states turn out to be stable, others not\nsye3v0-8.8.37 Starting from symmetric state at high T, cooling object through phase transition results in one of many equivalent stable SSB states 'spontaneously' chosen\nsye3v0-8.8.38 Unlike equilibrium case, imperfections don't give singular limit in dynamics of phase transitions\nsye3v0-8.8.39 Evolution of single symmetric state to single ordered state must again be non-unitary [145]\nsye3v0-8.8.40 Based on this observation, suggested that