Prethermalization in periodically driven nonreciprocal many-body spin systems

We analyze a new class of time-periodic nonreciprocal dynamics in interacting chaotic classical spin systems, whose equations of motion are conservative (phase-space-volume-preserving) yet possess no symplectic structure. As a result, the dynamics of the system cannot be derived from any time-dependent Hamiltonian. In the high-frequency limit, we find that the magnetization dynamics features a long-lived metastable plateau, whose duration is controlled by the fourth power of the drive frequency. However, due to the lack of an effective Hamiltonian, the prethermal state the system evolves into cannot be understood within the framework of the canonical ensemble. We propose a Hamiltonian extension of the system using auxiliary degrees of freedom, in which the original spins constitute an open yet nondissipative subsystem. This allows us to perturbatively derive effective equations of motion that manifestly display symplecticity breaking at leading order in the inverse frequency. We thus extend the notion of prethermal dynamics, observed in the high-frequency limit of periodically driven systems, to nonreciprocal systems.

Figure 1

(a) An interacting nonintegrable spin chain is subject to time-periodic two-step dynamics that breaks the reciprocity of the interactions and the symplectic structure of phase space. In the first half-cycle spins on one sublattice are held fixed, while the other spins precess in the exchange field of their neighbors, and vice versa during the second half-cycle. (b) The spin chain can be embedded in a larger Hamiltonian system comprised of two interacting dynamical spin degrees of freedom ${\mathit{S}}_j$ and ${\mathit{a}}_j$, which restores symplecticity (see text). (c) The initial condition ${\mathit{a}}_j=-{\mathit{S}}_j$ confines the phase space dynamics of the $(\mathit{S},\mathit{a})$ system to a subspace for all time; the $\mathit{S}$ subsystem is exactly described by the nonsymplectic drive from (a). (d) The nonreciprocal drive breaks magnetization conservation: at large drive frequencies $\omega$, the magnetization of an initial ensemble (shown by collection of points) exhibits a prethermal plateau whose lifetime scales as $t_M\sim {\omega }^4$, before it relaxes in a diffusionlike process (colorcode shows arrow of time from purple to yellow).

Figure 2

Magnetization relaxation in the nonreciprocal drive, Eq. (1), occurs via a prethermal plateau whose lifetime is controlled by the drive frequency $\omega$. The models shown are the XXZ spin chain ($J^x=J^y=1, J^z=\mathrm{\Delta }$) and the Heisenberg model ($J^x=J^y=J^z=1$) on the chain, square lattice, and triangular lattice. The inset shows the relaxation curves for the Heisenberg chain for drive periods $\tau =0.1$ (blue, rightmost) to 1.0 (cyan, leftmost), in steps of 0.1—their intersection with the horizontal line defines $t_M$. The main plot shows that the relaxation timescale $t_M\sim {\omega }^4$ scales with the fourth power of the drive frequency $\omega$ (dotted lines), regardless of the particular spin model. The data have been shifted vertically for clarity. All ensemble averages are taken over 2000 initial states, at temperature $\beta =1$ and initial applied field $h_z=0.7$. The system size for the chains is $L=128$, and the linear sizes for the 2D lattices are $L=16$ (square) and 15 (triangular). The nonreciprocal drive for the triangular lattice does not follow Eq. (1) (it is not bipartite), but follows an analogous three-step protocol (Appendix pp3).

Figure 3

Comparison between the effective stroboscopic dynamics and the exact dynamics in the isotropic Heisenberg chain. Main figure shows the magnetization dynamics within the first three orders of the inverse-frequency expansion (IFE): $M^z$ is conserved at zeroth-order; the first-order curve relaxes faster than the exact curve; whilst the second-order curve takes longer, hinting at a possible oscillatory convergence. Even at first order, the IFE captures the short-time quenched dynamics that drives the state into the “prethermal” plateau. The inset shows that the power-law scaling of the relaxation time, $t_M\sim {\omega }^4$ (dotted lines). is also captured at first order in the IFE. Simulation parameters are the same as in Fig. 2, with $\beta =1$. The curves in the main figure correspond to $\tau =0.8$.

Figure 4

Time dependence of the periodic driving function $g\left(t\right)$ (period $\tau$) and the antiderivative $G\left(t\right)$, as well as the product $g\left(t\right)G\left(t\right)$.

Figure 5

Comparison between the magnetization loss in the nonreciprocal drive (left) and Floquet heating in the reciprocal (Hamiltonian) drive (right), for the same initial ensemble (2000 states, $\beta =1, L=128$). (Top) Both drives exhibit a prethermal plateau (see insets), but heating in the reciprocal drive is suppressed exponentially, compared to the power-law suppression in the nonreciprocal drive. Inset for the nonreciprocal drive shows the curves from $\tau =0.1$ (rightmost) to $\tau =1.0$ (leftmost); inset for the reciprocal drive shows the curves from $\tau =0.8$ (rightmost) to $\tau =1.8$ (leftmost). The functional form of the suppression is independent of the threshold value (controlled by $\kappa$) used to define $t_M$ and $t_E$—the dotted lines in the insets correspond to $\kappa =0.5$ (top) and 0.9 (bottom). (Bottom) show convincingly that the nonreciprocal drive is not described by an exponential suppression, and, conversely, that the reciprocal drive is not described by a power-law suppression.

Figure 6

The tripartite drive for the triangular lattice. (Top left) An example tripartite decomposition of the triangular lattice. (Top right) Configuration of couplings $f$ and $g$ (blue and orange lines, respectively), for each step of the drive ($\mathcal{A}\mathcal{B}\mathcal{C}$, from the top down); the black lines show the $\mathit{S}-\mathit{S}$ couplings, which are always on. $\mathit{S}$ degrees of freedom are represented by filled circles; the auxiliaries by empty circles. (Bottom) The values of the couplings $f$ and $g$ across one drive period for the tripartite drive.