Topological phases with long-range interactions

Topological phases of matter are primarily studied in systems with short-range interactions. In nature, however, nonrelativistic quantum systems often exhibit long-range interactions. Under what conditions topological phases survive such interactions, and how they are modified when they do, is largely unknown. By studying the symmetry-protected topological phase of an antiferromagnetic spin-1 chain with $1/r^{\alpha }$ interactions, we show that two very different outcomes are possible, depending on whether or not the interactions are frustrated. While unfrustrated long-range interactions can destroy the topological phase for $\alpha \lesssim 3$, the topological phase survives frustrated interactions for all $\alpha >0$. Our conclusions are based on strikingly consistent results from large-scale matrix-product-state simulations and effective-field-theory calculations, and we expect them to hold for more general interacting spin systems. The models we study can be naturally realized in trapped-ion quantum simulators, opening the prospect for experimental investigation of the issues confronted here.

Figure 1

(a) Low-lying energy levels of the Haldane chain for even $L$. The entanglement structure of ground states is shown at the bottom. The ground states in the total $S^z=0,1,2$ subspace are named $|0\rangle ,|1\rangle ,|2\rangle$ and have energies $E_0,E_1,E_2$. (b), (c) The $m\text{th}$ largest value ${\lambda }_m$ ($m=1,2,\cdots ,8$) of the ground-state entanglement spectrum for $H_{\alpha }$ (b) and $H_{\alpha }^{\prime }$ (c) using finite-size MPS calculations with $L=200$. We choose the $|1\rangle$ state to avoid extra entanglement between edge spins. For $H_{\alpha }^{\prime }$, the entanglement spectrum for $1.5\le \alpha \le 4$ will exhibit a smooth crossover between the $\alpha =1.5$ and $\alpha =4$ cases due to the finite system size, but we expect a sharp transition at some ${\alpha }_c\lesssim 3$ in the thermodynamic limit. The exact pair degeneracies in $\left\{{\lambda }_m\right\}$ are a result of the spatial-inversion symmetry protecting the topological phase [44, 49].

Figure 2

(a) Bulk gap ${\mathrm{\Delta }}_{\alpha }$ and ground-state correlation length ${\xi }_{\alpha }$ in the $L\to \infty$ limit, obtained by finite-size scaling for $200\le L\le 500$. (b) Bulk gap ${\mathrm{\Delta }}_{\alpha }^{\prime }$ and ${\xi }_{\alpha }^{\prime }$ with $L=100$ and $L=300$. (c) Ground-state string-ordered correlation function ${\mathcal{S}}_{ij}$ for $H_{\alpha }$ and $H_{\alpha }^{\prime }$ with $\alpha =1.5$ and $L=300$. For various $\alpha$ and $200\le L\le 500$, we consistently find that ${\mathcal{S}}_{ij}$ quickly saturates to a finite value for $H_{\alpha }$ at all $\alpha >0$, but vanishes at large distance for $H_{\alpha }^{\prime }$ at $\alpha \lesssim 3$. (d) Ground-state spin-spin correlation ${\mathcal{C}}_{ij}$ for $\alpha =0.5$ and $L=500$. This choice of $\alpha =0.5$ is arbitrary, but assists in a clear presentation of the coexisting exponential and $1/r^{\alpha +4}$ power-law decays.

Figure 3

(a) Distribution of an edge excitation in state $|1\rangle$ for $L=500$ and $\alpha =2$. (b) Edge gap $|E_1-E_0|$ as a function of the chain size $L$ for $\alpha =3$. (c) Lowest-energy magnon probability density distribution for $L=200$ and $\alpha =3.0,0.5$. (d) The finite-size correction to the lowest magnon excitation energy [see Eq. (7)]. For $\alpha =3$, we obtain $v_{\alpha }=2.18$ and $v_{\alpha }/{\mathrm{\Delta }}_{\alpha }\approx 4.51$, in good agreement with the ${\xi }_{\alpha }\approx 4.55$ obtained in Fig. 2.