Stability of entanglement-spectrum crossing in quench dynamics of one dimensional gapped free-fermion systems

TL;DR

This paper investigates the stability of entanglement-spectrum crossings in one-dimensional gapped free-fermion systems after a quench, revealing that some crossings are unstable with finite dispersion while others are protected by fermion parity.

Contribution

It demonstrates that entanglement-spectrum crossings in certain topological classes are unstable under finite energy dispersion, and identifies fermion parity as a protecting symmetry.

Findings

Entanglement-spectrum crossings in classes AIII, DIII, CII are unstable with energy dispersion.

Crossings in classes BDI and D remain stable due to fermion parity conservation.

The quench process in classes BDI and D acts as a $ ext{Z}_2$ fermion parity pump.

Abstract

In a recent work by Gong and Ueda (arXiv:1710.05289), the classification of (1+1)-dimensional quench dynamics for the ten Altland-Zirnbauer classes is achieved, and entanglement-spectrum crossings of the time-dependent states for the topological classes (AIII, DIII, CII, BDI, and D) are discovered as a consequence of the bulk-edge correspondence. We note that, their classification scheme relies on the limit that the energy spectrum of the post-quench Hamiltonian is flat, because any finite band dispersion leads to the break down of time-reversal and chiral symmetries for the parent Hamiltonian (which are used for the classification). We show that, because of the reduction of symmetry by finite energy dispersion, the gapless entanglement-spectrum crossing in the flat-band limit in classes AIII, DIII, and CII is unstable, and could be gapped without closing the bulk gap. The entanglement-spectrum crossing in classes BDI and D is still stable against energy dispersion. We show that the quench process for classes BDI and D can be understood as a $\mathbb{Z}_2$ fermion parity pump, and the entanglement-spectrum crossing for this case is protected by the conservation of fermion parity.