Topological phase in a one-dimensional interacting fermion system

We study a one-dimensional (1D) interacting topological model by means of the exact diagonalization method. The topological properties are first examined with the existence of the edge states at half-filling. We find that the topological phases are not only robust to small repulsive interactions, but also are stabilized by small attractive interactions, and also finite repulsive interaction can drive a topological nontrivial phase into a trivial one while the attractive interaction can drive a trivial phase into a nontrivial one. Next we calculate the Berry phase and parity of the bulk system and find that they are equivalent in characterizing the topological phases. With them we obtain the critical interaction strengths and construct part of the phase diagram in the parameters’ space. Finally we discuss the effective Hamiltonian at the large-$U$ limit and provide an additional understanding of the numerical results. These results could be realized experimentally using cold atoms trapped in the 1D optical lattice.

Figure 1

(a) The tight-binding band structure of $\mathcal{H}\left(k\right)$. The low-energy excitation is located at $k_1$ (black line) or $k_2$ (green line) depending on the parameters. (b) Edge modes in the topological phase ($M=-1$) on a chain of length $N=20$ with OBC. (c) Berry phase of the occupied Bloch state of Eq. (1) at different $M$. (d) The plot of the curve $[d_x\left(k\right),d_z\left(k\right)]$ with $k\in [0,2\pi ]$. In all figures $B=1$ is set.

Figure 2

The energies of the quasiparticles added or removed from the half-filling system with OBC and PBC. (a, b): the phase at $U=0$ is topological ($M=-1$); (c) the phase at $U=0$ is trivial ($M=1$). The critical interactions $U_c$ in (a) and (c) can be accurately determined from the parity of the ground-state wave function. Here $B=1$.

Figure 3

The distribution of the quasiparticles added or removed from the half-filling system with OBC. (a, b) $M=-1$; (c) $M=1$ . Here $B=1$ and the system size $N=10$.

Figure 4

The energies and parities of the ground- and the first-excited states. (a) $M=-1$ and (b) $M=1$. The red (black) curve has the parity value $-1$(1). In both figures $B=1$ and $N=8$.

Figure 5

The phase diagram in $(M,U)$ and $(M,B)$ planes. The system size $N=8$.