Symmetry-Protected Topological States for Interacting Fermions in Alkaline-Earth-Like Atoms

We discuss the quantum simulation of symmetry-protected topological (SPT) states for interacting fermions in quasi-one-dimensional gases of alkaline-earth-like atoms such as ${}^{173}\mathrm{Yb}$. Taking advantage of the separation of orbital and nuclear-spin degrees of freedom in these atoms, we consider Raman-assisted spin-orbit couplings in the clock states, which, together with the spin-exchange interactions in the clock-state manifolds, give rise to SPT states for interacting fermions. We numerically investigate the phase diagram of the system, and study the phase transitions between the SPT phase and the symmetry-breaking phases. The interaction-driven topological phase transition can be probed by measuring local density distribution of the topological edge modes.

Figure 1

Schematics of the system setup. (a) A quasi-1D atomic gas is subject to Raman lasers. (b) Raman level schemes in the clock-states manifold. The green curve indicates the interorbital spin-exchange interaction. The four nuclear spin states from ${}^1{S}_0$ and ${}^3{P}_0$ manifolds are isolated from the other nuclear spins, which can be achieved by imposing spin-dependent laser shifts [38, 47].

Figure 2

(a) The lowest four levels in the entanglement spectrum ${\xi }_i$ ($i=1$, 2, 3, 4), and (b) the second-order Rényi entropy $S_2$ and the von Neumann entropy $S_{\mathrm{vN}}$ as functions of $V_{\text{ex}}/t_s$ for a chain with $N=60$ lattice sites and under open boundary conditions. (c) The bulk energy gap $E_{\text{gap}}$ (the energy difference between the ground state and the first excited state) for a chain with $N=12$ lattice sites under the periodic boundary condition. (Inset) The bulk gap at the critical point as a function of $1/N$. The red solid line is a linear fit, with $E_{\text{gap}}/t_s\sim -0.02\pm 0.05$ in the large-$N$ limit. (d) The von Neumann entropy of a subchain of length $l$ as a function of $\sin (\pi l/N)$ for a chain with $N=120$ sites at the critical point $V_{\text{ex}}/t_s=1.694$. The solid line is the linear fit with $S_{\mathrm{vN}}=(C/6)\ln [\sin (\pi l/N)]+1.87$ with $C=1.018$. The central charge is 6 times the slope of the linear fit. All calculations are performed at half filling, and we fix ${\mathrm{\Gamma }}_z^{g/e}=0$, $U=0$, $t_{\text{so}}/t_s=0.4$.

Figure 3

(a) Phase diagram for a lattice with $N=60$ sites at half filling, with ${\mathrm{\Gamma }}_z^{\alpha }=0$. (b) On-site densities of the states $|\pm ⟩$, with $V_{\text{ex}}/t_s=-11$, $U/t_s=5$ for ORS, and $V_{\text{ex}}/t_s=8$, $U=0$ for SRS, respectively. (c) Local atomic densities in the CDW state, with ${\widehat{n}}_i=\sum _{\alpha \sigma }{\widehat{n}}_{i\alpha \sigma }$, $V_{\text{ex}}=0$, $U/t_s=-4$. (d) Local densities for different orbitals in the ODW state, with ${\widehat{n}}_{i\alpha }=\sum _{\sigma }{\widehat{n}}_{i\alpha \sigma }$, $V_{\text{ex}}=0$, $U/t_s=8$, $t_{\text{so}}/t_s=0.4$.

Figure 4

The edge-mode density distributions $\mathrm{\Delta }{n}_i$ for (a) the $T$ state with $V_{\text{ex}}/t_s=-1$, (b) the $T$ state with $V_{\text{ex}}/t_s=1$, and (c) the SRS state with $V_{\text{ex}}/t_s=2$. Other parameters are the same as those in Fig. 2. The numbers in the figure legends label different degenerate ground states.