Entanglement spectrum of Su-Schrieffer-Heeger-Hubbard model

We investigate the entanglement spectrum of the ground state of the Su-Schrieffer-Heeger-Hubbard model. The topological phases of the model can be identified by degeneracy of the largest eigenvalues of entanglement spectrum. The study of the periodic boundary condition is enough to obtain the phase diagram of the model, without the consideration of the open boundary condition case. By analyzing the states corresponding to the degenerate entanglement spectrum and showing the related edge states in real space, physical interpretation about the relation between bulk-edge correspondence and the entanglement spectrum is presented. The method of the entanglement spectrum can be applicable in studying other topological phases of matter.

Figure 1

The solid and dashed lines represent two different bonds. One site in black and its next site in white are treated as a unit cell. The original system is in PBC, and the subsystem we cut off should be a chain of unit cells in OBC, so it consists of an even number of sites.

Figure 2

The chain length $L$ is 200, and the subsystem is cut off with the length of 100. The spectrum are plotted with different $\delta t$. The number above each lowest spectral line is the degree of degeneracy, so the total degree of degeneracy in the ground spectrum is the sum of these numbers.

Figure 3

The chain length $L$ is 16. The spectrum are plotted with different combinations of $\delta t$ and $U$. The number above each ground spectral line is also the degree of degeneracy.

Figure 4

Different colors (gray levels), also labeled as I, II, III, and IV, represent different phases according to the degeneracy of the ground entanglement spectrum.

Figure 5

(a) The chain length $L$ is 12. We compare it with Fig. 3. It shows that the degeneracy is robust when the size of the system changes. (b) The length $L$ of the original chain is 200, and the subsystem is cut off with the length of 50 in both figures. We compare it with Fig. 2. It shows that the degeneracy is robust when the ratio of the subsystem in the whole system changes.

Figure 6

The numbers in the box show the degree of degeneracy corresponding to different numbers of remaining particles with spin-up and spin-down in subsystem, and $n_0$ represents the number of spin-up or spin-down particles in the ground bulk state of the subsystem. The distributions are in (a) free fermion case: $U=0$, and (b) interacting fermion case: $U<0$.

Figure 7

A single-particle state that differs the states corresponding to two of the degenerate ground spectral lines. The original system has 200 sites, and the subsystem is cut off with 100 sites. The figure is plotted with $\delta t=-0.2$ in a single-particle case.