Nonequilibrium Phase Transition in a Periodically Driven $XY$ Spin Chain

We present a general formulation of Floquet states of periodically time-dependent open Markovian quasifree fermionic many-body systems in terms of a discrete Lyapunov equation. Illustrating the technique, we analyze periodically kicked $XY$ spin-$\frac{1}{2}$ chain which is coupled to a pair of Lindblad reservoirs at its ends. A complex phase diagram is reported with reentrant phases of long range and exponentially decaying spin-spin correlations as some of the system’s parameters are varied. The structure of phase diagram is reproduced in terms of counting nontrivial stationary points of Floquet quasiparticle dispersion relation.

Figure 1

Residual correlations $C_{\mathrm{res}}(\tau ,h)$ (plotted in a log-scale, $n=64$, with coupling constants ${\Gamma }_1^L={\Gamma }_1^R=0.5$, ${\Gamma }_2^L=0.3$ and ${\Gamma }_2^R=0.1$) in SFS of kicked Heisenberg $XY$ spin chain for anisotropy $\gamma =0.1$ (top left) and $\gamma =0.9$ (top right). Correlations that fall below ${10}^{-8}$ threshold belong to white colored region and correspond to a phase with exponentially decaying correlations. At higher anisotropy LRMC phase ceases to be a simply connected region. Below we show some typical spin-spin correlation matrices chosen along the red line on $\gamma =0.1$ phase diagram ($n=128$). Leftmost panel clearly shows exponential fall-off from the diagonal, the middle two belong to LRMC phase near the phase boundary, and the rightmost to a kind of antiresonance which occurs for $h=p\pi /(2\tau )$, $p\in \mathbb{Z}$ where the kick has no effect.

Figure 2

Half the total number of nontrivial stationary points (as their number is always even) of quasiparticle dispersion $\theta (\kappa )$ for kicked $XY$ chain (same parameters as in Fig. 1). The structure evidently coincides with the phase diagram of residual correlations $C_{\mathrm{res}}(\tau ,h)$.

Figure 3

Critical behavior of correlations in non-LRMC phase (left) and LRMC phase (right) for kicked $XY$ chain of size $n=480$, and $\tau =0.5$, where $\Delta h=h-h_c$. In non-LRMC phase correlations decay as $C_{\mathrm{res}}(r)\propto \exp (-r/\xi )$. We confirm, for two values of $\gamma =0.1$, 0.5 that correlation length $\xi$ diverges with critical exponent $\nu =1/2$ when approaching the critical point. In the right panel we show an abrupt transition of the order parameter $C_{\mathrm{res}}$ when entering the LRMC phase. In the inset, zooming in the region near the phase transition point, we observe appearance of the correlation resonances (explained in the simplified context in Ref. 20). Couplings ${\Gamma }_{1,2}^{L,R}$ are reduced by a factor of ${10}^{-6}$ with respect to Fig. 1.