Interaction-induced topological charge pump

Based on a topological transition of the symmetry-protected topological phase (SPT), an interaction-induced topological charge pump (iTCP) is proposed with the symmetry-breaking parameter as a synthetic dimension. It implies that the phase boundary of the SPT is a topological obstruction, although iTCP and the gap closing singularity are stable for symmetry-breaking perturbations. As for the iTCP, an interaction is essential since the pumped charge is trivial for a noninteracting system. We have confirmed the bulk-edge correspondence for this iTCP using density matrix renormalization group for the Rice-Mele model with nearest-neighbor interactions. As for a realization in optical lattices, an interaction sweeping pump protocol is proposed as well.

Figure 1

A schematic of a topological pump protocol (red line) protected by a gapless phase boundary passing through the protocol loop $\ell$. $\left|g\right(t,\theta )\rangle$ is a ground state of the snapshot Hamiltonian $H(t,\theta )$.

Figure 2

(a) MF phase diagram for the interacting SSH model. ${\gamma }_2$ is the CDW order parameter (see Supplemental Material for details [42]). We set $J_2=1$. (b) The phase diagram for the interacting SSH model is given by the entanglement spectrum of the open system and the Berry phase associated with the twisted boundary condition. The phase boundary of the open boundary condition is fixed by the entanglement spectrum by the DMRG with $L=64$ and the Berry phase using the ED. (c) The Berry phase $\gamma$ in an $L/2$-particle system. The sharp transition is due to the gap closing by the twist boundary condition [29].

Figure 3

Excitation spectrum $\mathrm{\Delta }E$: (a) (D1)-pump protocol. (b) (D2)-pump protocol. The behavior of the c.m. with fixed $\mu$: (c) (D1)-pump protocol with $\mu =-0.15$. (d) (D2)-pump protocol with $\mu =-0.26$. In the red shaded area, the system includes $L/2-1$ particles, and in other regions, $L/2$ particles. For all data, $L=64$.

Figure 4

The density distribution including the left and right edge state at $t/T=0.49865$ (a) and 0.501 83 (b) in (D1)-pump protocol. The density distribution is defined by $\langle \mathrm{\Psi }(L/2)\left|n_j\right|\mathrm{\Psi }(L/2)\rangle$, where $\left|\mathrm{\Psi }\right(L/2)\rangle$ is a ground state with $L/2$ particles. The value of c.m. is $\pm 0.42287$. The system size is $L=64$. (c) System size dependence of the total jump of the c.m. with $\mu =-0.15$. Each of the jumps of the c.m. are induced by changing the particle number and occur at $t/T=0.46272$ and 0.537 76.

Figure 5

(a) A schematic figure of the move of the pump loop as varying $J_1$ and the topological obstruction for a finite $\delta V$. Here, ${\mathrm{\Delta }}_d\equiv {\mathrm{\Delta }}_{j=\text{odd}}-{\mathrm{\Delta }}_{j=\text{even}}$. ${\mathrm{\Delta }}_d$ is a symmetry-breaking parameter (${\lambda }_3$). The blue (dotted) line is a topological obstruction. (b) The behavior of $C_N$ of the (D1)-pump protocol as varying $J_1$. (c) The distribution of $C_N$ in the (D1)-pump protocol under the perturbation $\delta V$. For all results, there is almost no system size dependence.