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

In a recent work by Gong and Ueda [Phys. Rev. Lett. 121, 250601 (2018)], 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 for the (1+1)-dimensional bulk topology as well as the adoption of entanglement-spectrum crossings as a diagnosis of topological edge states focus on the limit that the spectrum of the postquench Hamiltonian is flat. When band dispersion of the postquench Hamiltonian is considered, even though the bulk topology is still stable, the entanglement-spectrum crossing at the edge might be unstable due to the fact that the band dispersion can be regarded as a “symmetry-breaking disorder” in the frequency domain. Such a symmetry-breaking disorder lowers the symmetry for the edge side, and the conventional bulk-edge correspondence could not be applied directly for the quench dynamics. We show that because of the reduction of symmetry by finite energy dispersion the gapless entanglement-spectrum crossing in the flatband 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.

Figure 1

(a) and (c) Many-particle ES $\lambda$ and (b) and (d) single-particle ES $\xi$ of the quenched Kitaev chain (in class D). In (a) and (b), we quench the chemical potential $\mu =1.5\to 0$ (the spectrum of the postquench Hamiltonian is naturally flat for this case); in (c) and (d), we quench the chemical potential $\mu =1.5\to 0.25$ (the spectrum of the postquench Hamiltonian is dispersive). The length of the chain is set as $L=10$. When calculating the ES, the five sites of the right-half chain are traced out. We also explicitly label the parities of the largest two components of the many-particle ES in (a) and (c): red for the odd parity and black for the even parity.

Figure 2

Stability of the ES crossing for two coupled SSH-like chains (in class BDI). (a) A schematic of the model [Eq. (8)] in which the nearest-neighbor ($J_1$ and $J_2$), third-nearest-neighbor ($J_3$), and interchain ($J_c$) hopping amplitudes are indicated. This ladder is a bipartite lattice where the $A$ and $B$ sublattices are denoted in blue and red, respectively. (b) and (c) The open-boundary condition (OBC) results for the evolution of the time-dependent ES for the quench process $(J_1,J_2,J_3,J_c)=(1,0.5,0.6,0.2)E\to (0.5,1,-0.4,0.2)E$. The prequench Hamiltonian $H$ is trivial, and the postquench Hamiltonian $H_1$ is nontrivial with winding number 2. Here, (b) shows the ES calculated with an artificially flattened [18] postquench Hamiltonian ${\tilde{H}}_1$, whereas (c) shows the ES with a conventional (nonflatband) $H_1$. We see that ES crossings in the flatband case (b) are generally not stable against band dispersion of the postquench Hamiltonian and could be gapped (c). The length of the chain is set as $L=50$ (unit cells). We have also calculated the ES using the periodic boundary condition (PBC). It turns out that the results for the PBC for the current case are two almost indistinguishable (but still slightly different) copies of the ones for the OBC.

Figure 3

The entanglement gap $\mathrm{\Delta }\xi$ as a function of band-flatness ratio $\alpha$ for the quenched SSH-like chain model as shown in Fig. 2. Here, $\alpha =0$ corresponds to the flatband limit for which we see that the entanglement gap vanishes. The entanglement gap opens as long as the band dispersion is considered. The results for the PBC almost overlap with the ones for the OBC. (But if we zoom into the region of small $\alpha$, they are slightly different.)

Figure 4

Stability of the ES crossing for a SSH-like model with complex third-nearest-neighbor hopping amplitude $J_3$ (class AIII). (a) A schematic of the SSH-like model [Eq. (10)], which is just one leg of the SSH-like ladder in Fig. 2. (b) and (c) The OBC results for the ES evolution of the SSH-like model for the quench process $(J_1,J_2,J_3)=(1,0.5,0.2i)E\to (0.5,1,0.01i)E$. The prequench Hamiltonian $H$ is trivial, whereas the postquench Hamiltonian $H_1$, which has only chiral symmetry, is nontrivial with winding number 1. The difference between (b) and (c) is that in (b) we use an artificially flattened postquench Hamiltonian ${\tilde{H}}_1$ whereas in (c) we use a conventional $H_1$ with a dispersive spectrum (the same as the corresponding ones in Fig. 2). (d) and (e) The same as (b) and (c) but with the PBC. The system size is set as $L=100$ (unit cells).

Figure 5

The entanglement gap $\mathrm{\Delta }\xi$ as a function of band-flatness ratio $\alpha$ for the quenched SSH-like model as shown in Fig. 4. In the flatband limit ($\alpha =0$), we see that the entanglement gap vanishes; and the entanglement gap opens as long as the band dispersion is considered.