Exactly Solvable Points and Symmetry Protected Topological Phases of Quantum Spins on a Zig-Zag Lattice

A large number of symmetry-protected topological (SPT) phases have been hypothesized for strongly interacting $\mathrm{spin}\text{−}1/2$ systems in one dimension. Realizing these SPT phases, however, often demands fine-tunings hard to reach experimentally. And the lack of analytical solutions hinders the understanding of their many-body wave functions. Here we show that two kinds of SPT phases naturally arise for ultracold polar molecules confined in a zigzag optical lattice. This system, motivated by recent experiments, is described by a spin model whose exchange couplings can be tuned by an external field to reach parameter regions not studied before for spin chains or ladders. Within the enlarged parameter space, we find the ground state wave function can be obtained exactly along a line and at a special point, for these two phases, respectively. These exact solutions provide a clear physical picture for the SPT phases and their edge excitations. We further obtain the phase diagram by using infinite time-evolving block decimation and discuss the phase transitions between the two SPT phases and their experimental signatures.

Figure 1

(a) Dipolar molecules localized on a zigzag chain. The dipoles point to the $\mathbf{d}$ direction controlled by external electric field, forming an angle $\theta$ with the vertical direction. The exchange couplings $J_1$, $J_1^{\prime }$, and $J_2$ are defined in Eq. (2). (b) The variation of the exchange couplings as functions of $\theta$ ($\gamma =30°$). (c) This highly tunable model contains a few limits, some of which studied before in the literature. (I) a $J_1-J_2$ chain [5]; (II) a coupled antiferromagnetic ladder [6]; (III) a bond alternating chain [7]; (V) a ferromagnetic ladder [9]; (VI) weakly coupled ferromagnetic chains. The solid line and dashed line stand for positive (antiferromagnetic) and negative (ferromagnetic) coupling, respectively. The point (IV), where $J_1^{\prime }=2J_2<0$ and $J_1>0$, is exactly solvable. Here, the thick dashed line indicates $|J_1^{\prime }|>|J_2|$.

Figure 2

The phase diagram of $H$ in the $\theta -\eta$ plane obtained from iTEBD. The insets depict the singlet dimer (SD) and even-parity dimer (ED) phases. The thick solid lines in the SD case indicate a singlet $(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)/\sqrt{2}$, and the thick dashed lines in the ED case stand for an even-parity bond $(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)/\sqrt{2}$. The oval stands for the effective spin-1’s defined in Eq. (4). The dashed line at $\theta \sim 50.9°,\eta \in [0.747,1]$ and a point at $\eta =0,\theta \sim 42.4°$ are exactly solvable, and their singlet or even-parity product state are shown. A chiral phase (Ch) exists in a small $\theta$ regime, and the Tomonaga-Luttinger liquid (TLL) phase occupies the large $\theta$ region. The error bars are due to the finite step size in scanning $\theta$ or $\eta$.

Figure 3

Entanglement entropy $S^{\mathrm{vN}}$, string order parameter $O_1^z$ and $O_2^z$ for (a) the Heisenberg limit $\eta =1$, with $\chi =300$ and (b) along a line at $\theta =40°$, with $\chi =100$. In (b), there is a phase transition at $\eta \sim 0.769$ characterized by the peak of $S^{\mathrm{vN}}$ and jumps of string order parameters.