Tunable Floquet selection rules in a driven Ising chain

Abstract

We study a periodically driven spin-$1/2$ Ising chain with a nearest-neighbour coupling and longitudinal field while a weak transverse field induces single-spin flips. Through Floquet perturbation theory (FPT), we obtain signatures of Hilbert space fragmentation (HSF) and an unconventional form of dynamical localisation which we call the Floquet freezing. Our analysis suggests that these observations emerge due to a single Floquet selection rule that dictates the prethermal dynamics. For a special value of the field-to-interaction strength ratio together with commensurate drive periods, this rule permits only a constrained subset of bulk spin flips, leading to prethermal HSF in the full spin-$1/2$ Hilbert space. Under open boundary conditions, the same rule suppresses boundary spin flips up to higher order in perturbation and produces long-lived prethermal edge memory, which is neither topological in origin nor is a strong zero mode. Furthermore, under periodic boundary conditions, the largest surviving fragment is exactly the PXP sector at leading order and therefore exhibits Floquet-inherited scar phenomenology in the prethermal window. At higher commensurate ratios of field strength to interaction strength, all first-order single-spin-flip channels are suppressed and the system enters a regime of Floquet freezing. Hence, our study leverages the selection rules obtained through Floquet perturbation theory to obtain exotic prethermal phenomena at different parameter regimes.

Figures (10)

• Figure 1: Late-time prethermal stroboscopic bulk-site autocorrelation, $C_{\mathrm{bulk}}(200T)$, in the dimensionless parameter plane $(T J_0/\pi,\; h_0/J_0)$ for the periodically driven Ising chain. The neighbourhood of $(T J_0/\pi,h_0/J_0)=(2,2)$ identifies the prethermal HSF regime, where only the $\Delta P=0$ bulk channels survive at first order. Thin dashed lines mark the analytically predicted selection-rule conditions $h_0=2J_0$, $h_0=4J_0$, and $T=2\pi/J_0$, and the arrows indicate the parameter cuts used in the time-trace figures. Red and blue denote positive and negative late-time memory, respectively, while white corresponds to $C_{\mathrm{bulk}}\approx 0$. The data are for a chain of length $L=14$, with $g=1$ and $J_0=10$, averaged over $10$ random computational-basis initial states. The plot is shown on a $69\times 60$ grid without interpolation.
• Figure 2: Bulk single-spin-flip channels for a middle site $i$ with two neighbors, at $h_0=2J_0$. (a) Both neighbors down: $\Delta P_i=0$, so this channel survives the sinc filter for all $T$. (b) Mixed neighbors: $|\Delta P_i|=4J_0$, so this channel is suppressed at $T=\pi/J_0$. (c) Both neighbors up: $|\Delta P_i|=8J_0$, so this channel is suppressed at $T=\pi/(2J_0)$. At the canonical period $T=2\pi/J_0$, channels (b) and (c) lie on sinc zeros, leaving only the $\Delta P=0$ channel in panel (a).
• Figure 3: Edge single spin-flip channels (site $i=1$, OBC) at $h_0=2J_0$. (a) Neighbor down ($s_2=-1$): $|\Delta P_1|=2J_0$. (b) Neighbor up ($s_2=+1$): $|\Delta P_1|=6J_0$. Both channels are absent in PBC where every site has two neighbors. At $T=2\pi/J_0$, the sinc filter suppresses both edge channels ($a_{nm}=\pi$ and $3\pi$, respectively), and they reappear only at $\mathcal{O}(g^3)$ through the third-order FPT term [Eq. \ref{['eq:HF3_edge_proj']}].
• Figure 4: (a) Plot of the stroboscopic bulk-site autocorrelation, $C_{\mathrm{bulk}}(nT)$ with time (in number of Floquet steps, $n$) at the resonant period $T=2\pi/J_0$ for $J_0\in\{1,3,5,10\}$. (b) Plot of $C_{\mathrm{bulk}}(nT)$ with at fixed $J_0=10$ and $h_0=20$, with time (in number of Floquet steps, $n$) driven at driving time periods, $T=\mathrm{factor}\times \pi/J_0$ and $\mathrm{factor}\in\{0.5,1,2,3\}$. Both the plots are for a chain of length, $L=16$ and parameters, $g=1$, and $h_0=2J_0$, averaged over $32$ random initial states. The first $50$ Floquet periods are shown at single-period resolution and later times are block-averaged over 10 periods. The grey dashed line marks the Mazur bound $C_{\mathrm{Mazur}}=0.6$.
• Figure 5: (a) Stroboscopic half-chain entanglement entropy $S_A(nT)$ plotted with time (as number of Floquet steps, $n$) for the fragmented Floquet chain driven with the resonant period $T=2\pi/J_0$ for $J_0\in\{1,3,5,10\}$, plotted as $S_A/S_{\mathrm{Page}}$ using the Page value of the largest fragment. (b) $S_A(nT)$ plotted with time (as number of Floquet steps, $n$) starting from states from three different Krylov fragments of dimensions $\mathcal{D}=987$, $610$, and $377$ at $J_0=10$, $h_0=20$, and $T=2\pi/J_0$. Dashed lines mark the fragment-specific Page values. Both the plots are for chain of length, $L=16$, with parameters $g=1$, and $h_0=2J_0$, with subsystem $A=\{0,1,\ldots,7\}$. The first 50 Floquet periods are shown at single-period resolution and later times are block-averaged over 10 periods.