Detecting edge degeneracy in interacting topological insulators through entanglement entropy

The existence of degenerate or gapless edge states is a characteristic feature of topological insulators, but is difficult to detect in the presence of interactions. We propose a new method to obtain the degeneracy of the edge states from the perspective of entanglement entropy, which is very useful to identify interacting topological states. Employing the determinant quantum Monte Carlo technique, we investigate the interaction effect on two representative models of fermionic topological insulators in one and two dimensions, respectively. In the two topologically nontrivial phases, the edge degeneracies are reduced by interactions but remain to be nontrivial.

Figure 1

The second-order Renyi entanglement entropy of the 1D SSHH model. $S_{2,\mathrm{pbc}}$, $S_{2,\mathrm{obc}}$, and $S_{2,\mathrm{edge}}$ are plotted in (a) ($U=0$) and (b) ($U=1$) as functions of $\delta t$ and in (c) ($\delta t=-0.2$) as functions of $U$. Parameter values for the QMC simulations are the projection time $\beta =120$ (200) for $U\ge 1$ ($U=0.5$) and the discrete time step $\Delta \tau =0.1$. The chain length is $L=100$ in (a), and $L=40$ in (b),(c).

Figure 2

The phase diagram of the SSHH model determined from the edge entanglement entropy $S_{n,\mathrm{edge}}$ and edge degeneracy $\mathcal{D}$.

Figure 3

The second-order Renyi entanglement entropy of the KMH model vs $L_y$. $S_{2,\mathrm{edge}}$ is plotted in the inset. Parameter values are $U=1$, $\lambda =0.2$, $\beta =10L_y$, $\Delta \tau =0.1$, and $L_x=6$.

Figure 4

The helical edge behavior of the KMH model. (a) The single-particle energy spectra at $U=0$. Red and blue symbols mark the edge spectra with even ($L_y=10$) and odd ($L_y=9$) values of $L_y$, respectively. (b) The many-body energy gap $\Delta$ vs $L_y$ for the effective model Eq. (11), with $U=1$ and $\Lambda =\pi v_F/2=1$. The inset plots $\Delta$ vs $1/L_y$ for only odd $L_y$.