Floquet Symmetry-Protected Topological Phases in Cold-Atom Systems

We propose and analyze two distinct routes toward realizing interacting symmetry-protected topological (SPT) phases via periodic driving. First, we demonstrate that a driven transverse-field Ising model can be used to engineer complex interactions which enable the emulation of an equilibrium SPT phase. This phase remains stable only within a parametric time scale controlled by the driving frequency, beyond which its topological features break down. To overcome this issue, we consider an alternate route based upon realizing an intrinsically Floquet SPT phase that does not have any equilibrium analog. In both cases, we show that disorder, leading to many-body localization, prevents runaway heating and enables the observation of coherent quantum dynamics at high energy densities. Furthermore, we clarify the distinction between the equilibrium and Floquet SPT phases by identifying a unique micromotion-based entanglement spectrum signature of the latter. Finally, we propose a unifying implementation in a one-dimensional chain of Rydberg-dressed atoms and show that protected edge modes are observable on realistic experimental time scales.

Figure 1

A 1D array of atoms is trapped in an optical lattice or tweezer array. Ising interactions for pseudospin states $|\downarrow ⟩$, $|\uparrow ⟩$ are generated by optically coupling $|\uparrow ⟩$ to Rydberg state $|\mathcal{R}⟩$ (solid blue arrows). Random fields $h_i$ are generated by a spatially varying Raman coupling (dotted purple arrows) between $|\downarrow ⟩$ and $|\uparrow ⟩$. While emulating the ESPT phase requires a dimerized chain with Ising couplings $\lambda f(t)$ of dynamically switchable sign, the FSPT phase is simulated simply by alternating between two Hamiltonians consisting of Ising interactions ($H_1$) and a disordered transverse field ($H_2$).

Figure 2

ESPT phase—(a) $F^{\alpha }(t)$ for $N=10$ spins with $\omega =100$, $V_x=0.05$, $V_y=0$, ${\lambda }_1=1.54$ and ${\lambda }_2=0.69$, yielding $b({\lambda }_1,{\lambda }_2)/a({\lambda }_1,{\lambda }_2)\sim 10$. Almost overlapping dotted lines represent the clean undisordered case (black and blue for $F^z$ and $F^x$, respectively). Solid lines correspond to strong on-site disorder, with thick black and blue lines for $F^z$ and $F^x$ in the dimerized case and thin solid yellow and red lines for $F^z$ and $F^x$ in the undimerized case. (inset) Ratio $b(1,{\lambda }_2)/a(1,{\lambda }_2)$ in the dimerized (solid blue) and the undimerized (dotted red) models. The SPT phase corresponds to $b/a>1$ (delimited by the dotted black line). (b) $T_2^{*}$ as a function of frequency and system size [74]. As $\omega$ is increased for $V_x=0.05$ (circles), $T_2^{*}$ saturates consistent with being bounded by $T_2^{*}\sim \min \mathbf{(}\mathcal{O}(\omega ),e^{\mathcal{O}(N)}\mathbf{)}$. Adding generic interactions, $V_y\sum _i{\sigma }_i^y{\sigma }_{i+1}^y$ with $V_y=0.2$ (squares), leads to a breakdown of the edge coherence for all parameters.

Figure 3

FSPT phase—(a) The $⟨r⟩$ ratio as a function of the power law exponent $p$ for a chain with periodic boundary conditions. The $h_i$’s are sampled from the uniform distribution [0.1, 0.9] and $T=\pi$ (in units of $J=1$). There is a MBL-delocalization phase transition around $p_c\approx 3.5$. (inset) $T_2^{*}$ as a function of $N$, where the edge coherence is fit to $\sim N^4$. (b) The entanglement spectrum micromotion for $N=12$. The parameters ($p$, $T$, $J$, $W$) are: (4, $\pi$, 1, 1) for the SPT; (1, $\pi$, 1, 1) for the thermal behavior; (4, $\pi$, 0.05, 0.8) for the paramagnet; $p=4$, $T=\pi$, $J=0.5$, $h\in [0.5,1]$ for the spin glass. (inset) Mutual information $\mathcal{I}(i,j)=S_i+S_j-S_{ij}$ (where $S$ is the von Neumann entropy) within the SPT phase: $\mathcal{I}(1,j)$ (red circles) and $\mathcal{I}(6,j)$ (blue squares) [74]. (c) $F^y(t)$ and $F^z(t)$ for the edge and the bulk in a system of $N=10$ spins for the model in Eq. (3). The bulk curves are almost overlapping. (d) Same as in (c), but with an additional term, $V_x\sum _i{\sigma }_i^x{\sigma }_{i+1}^x$ ($V_x=0.3$) added to $H_1$.