Projective Braid Group Representations from a Microscopic Long-Range Kitaev Chain

In the paper's words

We construct an explicit projective representation of the braid group from the microscopic Hilbert space of a Kitaev chain extended with algebraically decaying long-range hopping and pairing. The construction proceeds in three steps. First, we identify the topological phase of the long-range Kitaev chain (power-law exponent α) and show that unpaired Majorana zero modes (MZMs) localize at domain walls, with localization length controlled by α.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Direct 2×2 multiplication, first ordering: $\mathrm{U}_{12}\mathrm{U}_{23} = (1/\sqrt{2})[[e^{i\pi /4}, i e^{i\pi /4}], [i e^{-i\pi /4}, e^{-i\pi /4}]]$; right-multiplication by the diagonal $U_{12}$ rescales the columns by $e^{i\pi /4}$ and $e^{-i\pi /4}$, giving $\mathrm{U}_{12}\mathrm{U}_{23}\mathrm{U}_{12} = (1/\sqrt{2})[[e^{i\pi /2}$, i], [$i, e^{-i\pi /2}]] = (1/\sqrt{2})[[i$, i], [i, −i]] = $(i/\sqrt{2})[[1, 1], [1, -1]] = (i/\sqrt{2})(\sigma _{1} + \sigma _{3})$.

   from 4.2 Explicit matrix computation: braid relation and projective lift

2. Our central quantitative result is that the Ising projective class is stable against long-range hopping with exponent $\alpha \gt 2$ and open gap: our projection at $\alpha = 3$ with representative parameters — based on the assumed, undischarged scaling $\varepsilon$ ~ $\lambda ^{s} \cdot s^{-\alpha }$ rather than a derived bound — gives a per-braid drift of ≤ $1.5 \times 10^{-17}$ rad and a cumulative drift over $10^{6}$ braids of ≤ $9.2 \times 10^{-11}$ rad.

   from Abstract

3. Symbol Description Value Source $\alpha$ Power-law exponent 3 (representative) chosen $\mu /t_{0}$ Chemical potential ratio 0.1 chosen $\zeta (3)$ Riemann zeta 1.2020569 standard s MZM separation 10 sites chosen $\varphi _{\mathrm{Ising}}$ Short-range anchor phase $\pi /8 = 0.3927$ rad Sec.

   from 4.3 Input numbers

4. No Berry-connection calculation or perturbative derivation of the overlap-to-phase coupling is given here, and no prefactor is fixed; this is a projection, not a proved bound, and is subject to the same assumption-auditing as the $\alpha = 2.1$ estimate below.

   from 4.4 Localization and drift bounds

5. Applying the same factor-of-3 uncertainty treatment as for the $\alpha = 2.1$ projection, the cumulative drift lies in [$3.1, 27.6] \times 10^{-11}$ rad.

   from 4.4 Localization and drift bounds

6. Per-braid drift ≈ $2\pi \times 2.92 \times 10^{-3} \approx 1.8 \times 10^{-2}$ rad; over $10^{3}$ braids, cumulative drift ~ 18 rad, destroying gate fidelity.

   from 4.4 Localization and drift bounds

How it is done

1. We construct an explicit projective representation of the braid group from the microscopic Hilbert space of a Kitaev chain extended with algebraically decaying long-range hopping and pairing.

   from Abstract

2. Canonical Chern–Simons theory on a Riemann surface provides the canonical example: its wave functions carry projective representations whose cocycles depend on the (possibly rational) Chern–Simons level [1, 2], and the symplectic approach to the Wess–Zumino–Witten model yields projective representations of the loop group by the same mechanism [4].

   from 1. Introduction

3. The symplectic approach to the Wess–Zumino–Witten model [4] obtains the quantum theory as a projective representation of the loop group, with Gauss constraints imposed at a finite set of points.

   from 2. Background and Related Work

4. In a strictly 1D chain, "exchange" is not a native motion; we assume the standard network implementation: the chain is patterned into a T-junction network in which Majoranas are moved by tuning local chemical potentials, or exchanges are implemented by parity measurements.

   from 3.3 Exchange protocol in a chain

What would falsify it

1. Falsification criteria and open questions are discussed.

   from Abstract

2. Failure modes and falsifiability.

   from 6. Discussion

3. Resolution adopted: main text adopts the B/C position (class stability); A's continuous-family claim is documented here and listed as falsifiable (an exact-diagonalization demonstration of a continuously varying phase at open gap would overturn it).

   from Appendix A. Divergence report

Where the claim stops

1. Second, the cohomological invariance argument assumes the deformation path never closes the bulk gap; long-range chains are known to exhibit re-entrant gap closings at intermediate $\alpha$, and a single gap closing would allow the class to jump between the two elements of $\mathbb{Z}_{2}$.

   from 6. Discussion

2. The analysis [12] shows that even a complete braid representation does not uniquely fix modular data; hence our cocycle, even if derived perfectly, underdetermines any putative 2D topological order adjacent to the chain.

   from 6. Discussion

3. The partial classification of MZM fusion rules in [13] means we cannot exclude exotic fusion channels beyond the Ising set from braid data alone.

   from 6. Discussion

Where to read

1. Abstract278 words · 1 min
2. 1. Introduction444 words · 2 min
3. 2. Background and Related Work716 words · 3 min
4. 3. Methods553 words · 2 min
5. 4. Analysis1,169 words · 5 min
6. 5. Results212 words · 1 min
7. 6. Discussion438 words · 2 min
8. 7. Conclusion161 words · 1 min
9. Appendix A. Divergence report505 words · 2 min
10. Appendix B. Claim attribution122 words · 1 min