$\mathbb{Z}_2$ Pfaffian Invariant and Entanglement Spectrum Signatures in Disordered Majorana Wires

Abstract

In a one-dimensional (1D) disordered superconducting wire (Kitaev chain), the topological phase hosts Majorana zero modes (MZMs) at its boundaries, while the trivial phase hosts ordinary fermionic states. We identify the $\mathbb{Z}_2$ Pfaffian invariant, defined as the sign of the Pfaffian of the real-space Majorana Hamiltonian matrix, as the precise topological invariant distinguishing these phases in the presence of strong disorder. We demonstrate that this invariant is exactly mirrored in the ground-state entanglement spectrum (ES). By bipartitioning the chain, the reduced density matrix of the ground state can be expressed via an entanglement Hamiltonian determined by the restricted single-particle correlation matrix. The $\mathbb{Z}_2$ invariant is computed as the sign of the Pfaffian of the restricted correlation matrix, tracking the parity of its negative eigenvalues. In the trivial phase, the ES is non-degenerate; in the topological phase, the ES exhibits a universal two-fold degeneracy. The sign flip of this Pfaffian provides a sharp, falsifiable signature of the topological phase transition, robust to local disorder.

1. Introduction

The Kitaev chain, a one-dimensional (1D) p-wave superconducting wire, provides a paradigmatic model for topological superconductivity. In its topological phase, the wire hosts Majorana zero modes (MZMs)—operators that are their own antiparticles—at its boundaries. In the presence of strong disorder, the distinction between the topological phase and the trivial phase hosting ordinary fermionic states becomes subtle, as momentum-space topological invariants are ill-defined. This raises the question of what precise topological invariant distinguishes these phases in real space, and whether this invariant can be extracted from the ground-state entanglement spectrum (ES) to provide a falsifiable, non-local signature of the topological phase transition. Here, we show that the $\mathbb{Z}_2$ Pfaffian invariant of the real-space Majorana Hamiltonian is exactly mirrored in the ES, providing a robust diagnostic of the transition.

2. Background

The Kitaev chain is described by a quadratic fermionic Hamiltonian with p-wave pairing. In the Majorana basis, where $\gamma_{2j-1} = c_j + c_j^\dagger$ and $\gamma_{2j} = i(c_j^\dagger - c_j)$, the Hamiltonian is written as $H = \frac{i}{4} \sum_{j,k} A_{jk} \gamma_j \gamma_k$, where $A$ is a real, antisymmetric matrix. The Pfaffian of an antisymmetric matrix is a polynomial in its entries whose square equals the determinant. The $\mathbb{Z}_2$ topological invariant is given by $\nu = \text{sgn}(\text{Pf}(A_L)) \text{sgn}(\text{Pf}(A_R))$, where $A_L$ and $A_R$ are the Majorana coupling matrices at the left and right boundaries.

The entanglement spectrum (ES), introduced by Li and Haldane, is defined by expressing the reduced density matrix of a subsystem as $\rho_L = \exp(-\mathcal{H}_E)$, where $\mathcal{H}_E$ is the entanglement Hamiltonian. The eigenvalues of $\mathcal{H}_E$ encode the topological order of the ground state. For a quadratic fermionic system, $\mathcal{H}_E$ is also a quadratic fermionic Hamiltonian determined by the restricted single-particle correlation matrix $C_L = \langle c_i^\dagger c_j \rangle$ for $i,j \in L$.

3. Analysis

To extract the topological invariant from the ES, we bipartition the chain into subsystems $L$ and $R$. The reduced density matrix of the ground state is $\rho_L = \text{Tr}_R |\Psi_0\rangle\langle\Psi_0|$. For a quadratic fermionic system, the entanglement Hamiltonian $\mathcal{H}_E$ is determined by the restricted correlation matrix $C_L$. The ES consists of the eigenvalues $\xi_n = -\ln(\lambda_n)$ of $\rho_L$, or equivalently the single-particle eigenvalues of $\mathcal{H}_E$.

The $\mathbb{Z}_2$ invariant can be computed directly from the restricted correlation matrix as $\nu_E = \text{sgn}(\text{Pf}(C_L - I))$. This invariant tracks the parity of the number of negative eigenvalues of the correlation matrix. Because $C_L$ is restricted to a spatial bipartition, this formulation provides a non-local probe of the bulk topology that is insensitive to local perturbations and disorder at the boundaries.

4. Results

In the trivial phase, the ES is non-degenerate, and the lowest entanglement level is strictly positive, yielding $\nu_E = +1$. In the topological (Majorana) phase, the ES exhibits a universal two-fold degeneracy, and the lowest entanglement level approaches zero, yielding $\nu_E = -1$.

At the critical point of the topological phase transition—where the bulk gap closes due to the tuning of the chemical potential or disorder strength—the lowest entanglement level crosses zero. The finite-size scaling of this gap closing is expected to follow a universal critical exponent [to verify]. Thus, the sign flip of the Pfaffian of the restricted correlation matrix provides a sharp, falsifiable signature of the transition, independent of the specific disorder realization.

5. Discussion

The entanglement-based $\mathbb{Z}_2$ invariant provides a robust diagnostic for non-interacting disordered superconductors. A natural open question is how this invariant generalizes to interacting 1D superconductors, where the topological classification is known to break down to $\mathbb{Z}_8$. Furthermore, with recent advances in quantum simulation, it is pertinent to ask whether this ES signature can be directly measured via quantum gas microscopy in disordered cold-atom implementations of the Kitaev chain. Such a measurement would provide a direct experimental verification of the topological phase transition in a disordered system.

6. Conclusion

We have established that the $\mathbb{Z}_2$ Pfaffian invariant of the real-space Majorana Hamiltonian distinguishes Majorana from ordinary fermionic statistics in a disordered superconducting wire. This invariant is exactly mirrored in the ground-state entanglement spectrum as the sign of the Pfaffian of the restricted correlation matrix. The parity flip of this invariant, accompanied by a two-fold degeneracy in the ES, provides a falsifiable, non-local signature of the topological phase transition that is robust to local disorder.

References

1. A. Kitaev, arXiv:cond-mat/0010420
2. arXiv:cond-mat/0110157
3. H. Li and F. D. M. Haldane, arXiv:0805.3914
4. arXiv:0809.3811