Aperiodically Driven Integrable Systems and Their Emergent Steady States

Does a closed quantum many-body system that is continually driven with a time-dependent Hamiltonian finally reach a steady state? This question has only recently been answered for driving protocols that are periodic in time, where the long-time behavior of the local properties synchronizes with the drive and can be described by an appropriate periodic ensemble. Here, we explore the consequences of breaking the time-periodic structure of the drive with additional aperiodic noise in a class of integrable systems. We show that the resulting unitary dynamics leads to new emergent steady states in at least two cases. While any typical realization of random noise causes eventual heating to an infinite-temperature ensemble for all local properties in spite of the system being integrable, noise that is self-similar in time leads to an entirely different steady state (which we dub the “geometric generalized Gibbs ensemble”) that emerges only after an astronomically large time scale. To understand the approach to the steady state, we study the temporal behavior of certain coarse-grained quantities in momentum space that fully determine the reduced density matrix for a subsystem with size much smaller than the total system. Such quantities provide a concise description for any drive protocol in integrable systems that are reducible to a free-fermion representation.

Popular Summary

Exposing an ensemble of atoms or molecules to a changing field—such as pulses from a laser or a varying magnetic field—can lead to some intriguing behavior that is not seen when the system is left alone. In particular, if the changes are periodic in time, that can be a useful experimental tool for generating novel nonequilibrium phases. Much work has been devoted to understanding the steady-state behavior of periodically driven systems, but the consequences of breaking that periodicity are largely unexplored. Aperiodic perturbations are also inevitable in experimental setups because of imperfect control over the time variation of the driving field. Our mathematical analysis reveals the emergence of steady states that arise only in the presence of aperiodic driving.

We consider a class of minimal models of interacting quantum spins that allow for a tractable analysis when driven aperiodically in time. We find that new steady states occur in at least two cases. Along the way, we also introduce new coarse-grained quantities in momentum space that completely determine the local properties of the system and are useful for studying driven systems in general.

Our work shows that aperiodically driven systems can have much richer behaviors over long time scales than their periodically driven counterparts and that classifying the full range of possible steady states is a worthwhile endeavor. One interesting research direction would be to study if quasiperiodic (i.e., neither fully random nor periodic) driving can avoid heating that is seen in generic driven systems.

Figure 1

The separation of a closed system into a subsystem with $l$ sites (indicated in black) and the environment with $L-l$ sites (indicated in red). Local properties only involve degrees of freedom on the sites of a subsystem that satisfies $l\ll L$ with $L\gg 1$.

Figure 2

Behavior of the coarse-grained quantities $(|u_k(n){|}^2{)}_c$ as a function of $n$ for a perfectly periodic drive ($dT=0$). The parameters used are $L=131072$, $g_i=4$, $g_f=2$, and $T=0.2$. Each coarse-grained cell contains $N_c=2048$ consecutive momenta, and $k_c$ denotes the average momentum of such a cell. Results for $k_c=31\pi /64$ (lower line in red) and $k_c=15\pi /64$ (upper line in blue) are shown.

Figure 3

Plot of $P_n(\cos {\theta }_k)$ calculated at $n=5\times 10^5$ for a system size of $L=131072$ spins, where the number of coarse-grained cells equals ${\mathcal{N}}_{\mathrm{cell}}=32$ and each cell contains $N_c=2048$ consecutive momenta. Results for $k_c=31\pi /64$ (red solid line) and $k_c=15\pi /64$ (blue solid line) are shown, where $k_c$ denotes the average momentum of the corresponding cell. Here, $\overline{P}(\cos {\theta }_k)$ at $k=31\pi /64$ (red dashed line) and $k=15\pi /64$ (blue dashed line) are also shown using the analytical result in Eq. (33). The drive parameters are $g_i=4$, $g_f=2$, $T=0.2$, and $dT=0$.

Figure 4

The overlap $|⟨{\psi }_k(n)|{\psi }_{k+\delta k}(n)⟩{|}^2$ for two different kinds of unitary processes as a function of $n$. Here, $dT=0$ (black lines) denotes a perfectly periodic drive protocol, while $dT=0.05$ (red lines) is a typical realization of a random process where ${\tau }_i$ are chosen to be $\pm 1$ with equal probability [Eq. (15)]. Note that $T=0.2$ in both cases, and the starting wave function equals $\mathbf{(}{u}_k(0),v_k(0){\mathbf{)}}^T=(0,1{)}^T$ for all momenta. Here, $k=\pi /2$ is taken as the reference mode, and its overlap calculated with the next ten consecutive momenta is shown, where the momentum spacing equals $\pi /65536$.

Figure 5

Plots of $P_n(\cos {\theta }_k)$ for various values of $n$. The distributions are constructed using $N_c=8192$ points in a coarse-grained momentum cell with average momentum $k_c=31\pi /64$; the system size equals $L=524288$, and the number of coarse-grained momentum cells equals ${\mathcal{N}}_{\mathrm{cell}}=32$. The drive parameters used are $g_i=4$, $g_f=2$, $T=0.2$, and $dT=0.05$. For small $n(=100)$ (red solid line), $P_n(\cos {\theta }_k)$ approaches the $dT=0$ analytical result in Eq. (33) (red dashed line). At larger values of $n(=10000)$ (green solid line), the distribution strongly deviates from the $dT=0$ form, and it approaches a constant function equal to $1/2$ at still larger values of $n(=200000)$ (blue solid line). The lower panel shows the positions of the $N_c=8192$ points (in blue) on the unit sphere for different values of $n$.

Figure 6

(a) Behavior of the coarse-grained quantities $1/2-(|u_k(n){|}^2{)}_c$ as a function of $n$ for different values of $dT$ and $k_c$ (average momentum of the coarse-grained cell). The same random sequence ${\tau }_i$ has been used in all the cases. The other parameters are $L=524288$, ${\mathcal{N}}_{\mathrm{cell}}=32$, and $N_c=32$, with $g_i=4$, $g_f=2$, and $T=0.2$. (b) Plot of $1/2-(|u_k(n){|}^2{)}_c$ as a function of $n(dT{)}^2(\sin k_c{)}^2$, which shows that the decay of $(|u_k(n){|}^2{)}_c$ to $1/2$ is consistent with an exponential decay with a time scale ${\tau }_{k_c,dT}=1/[(dT{)}^2(\sin k_c{)}^2]$. The fit to an exponential function is displayed in panel (b) using open circles. The color scheme used in both panels is as follows: $dT=0.05$, $k_c=15\pi /64$ (in black), $dT=0.05$, $k_c=31\pi /64$ (in red), $dT=0.04$, $k_c=5\pi /64$ (in green), $dT=0.04$, $k_c=15\pi /64$ (in blue), $dT=0.02$, $k_c=15\pi /64$ (in magenta), and $dT=0.02$, $k_c=31\pi /64$ (in orange).

Figure 7

Plot of ${\mathcal{D}}_{n,\mathrm{I}\mathrm{T}\mathrm{E}}$ as a function of $n(dT{)}^2$ for different values of $dT$ ($dT=0.05$ in black, $dT=0.04$ in red, and $dT=0.02$ in green), where the other drive parameters are $g_i=4$, $g_f=2$, and $T=0.2$. The system size equals $L=524288$ spins, and the subsystem consists of $l=16$ consecutive spins. The same random sequence ${\tau }_i$ has been used in all the cases. (Inset) Plot of ${\mathcal{D}}_{n,\mathrm{G}\mathrm{G}\mathrm{E}}$ as a function of $n$.

Figure 8

Behavior of the coarse-grained quantities $(|u_k(n){|}^2{)}_c$ as a function of $n$ when the ${\tau }_i$’s are chosen according to the Thue-Morse sequence. The parameters used are $L=131072$, $g_i=4$, $g_f=2$, $T=0.2$, and $dT=0.05$. Each coarse-grained cell contains $N_c=2048$ consecutive momenta, and $k_c$ denotes the average momentum of such a cell. Results for $k_c=31\pi /64$ (lower red curve) and $k_c=15\pi /64$ (upper blue curve) are shown.

Figure 9

Behavior of ${\widehat{a}}_m·{\widehat{b}}_m$ as a function of $m$ for a unitary process with $k=\pi /2$, $T=0.2$, and $dT=0.05$, where the ${\tau }_i$’s are chosen according to the Thue-Morse sequence. (Inset) Behaviors of the three components of ${\widehat{a}}_m$ [$a_{1m}$ (red), $a_{2m}$ (green), $a_{3m}$ (blue)] as functions of $m$ for the same unitary process. The unit vector ${\widehat{a}}_m$ points in the direction of ${\widehat{a}}_1+{\widehat{b}}_1$ for large $m$ [from Eq. (52)] since ${\widehat{a}}_m={\widehat{b}}_m$ for $m\gg 1$.

Figure 10

The positions of the $N_c$ points (in blue) contained in a coarse-grained momentum cell on the unit sphere for various values of $n=2^m$, where the system is viewed geometrically, and the ${\tau }_i$’s are chosen according to the Thue-Morse sequence. The parameters used are $L=131072$, $g_i=4$, $g_f=2$, $T=0.2$, and $dT=0.05$. Each coarse-grained momentum cell contains $N_c=2048$ consecutive momenta, and the results displayed are for a cell with average momentum of $k_c=31\pi /64$.

Figure 11

We show $\delta n_k={\widehat{a}}_k(T,dT)-{\widehat{a}}_k(T,0)$ as a function of $k$. Here, ${\widehat{a}}_k(T,dT)$ refers to the final value ${\widehat{a}}_{\infty }={\widehat{b}}_{\infty }$ to which ${\widehat{a}}_m$ and ${\widehat{b}}_m$ settle down at large $m$, where the ${\tau }_i$’s in the unitary process are chosen according to the Thue-Morse sequence with given values of $T$ and $dT$. The drive parameters are (a) $g_i=4$, $g_f=2$, $T=0.2$, and $dT=0.05$, and (b) $g_i=4$, $g_f=2$, $T=0.2$, and $dT=0.8$. The components of $\delta n_k$ are indicated as follows: $\delta n_{1k}$ (red line), $\delta n_{2k}$ (green line), and $\delta n_{3k}$ (blue line).

Figure 12

(a) Convergence of $⟨\psi (n)|s^x|\psi (n)⟩$ to the final steady-state value calculated using the g-GGE. The steady-state values are shown as dashed lines and are for drive parameters $g_i=4$, $g_f=2$, $T=0.2$, and $dT=0.05$ (in black) and $dT=0.8$ (in red), with a system size equal to $L=131072$. Note that $m$ denotes exponentially growing time steps since $n=2^m$. The inset of (a) shows the rapid convergence of the same local quantity as a function of $n$ for a perfectly periodic protocol with $T=0.2$. (b) The data for $dT=0.01$ (in green) and $dT=0.05$ (in black) now shown as a function of $n$. The dashed line represents the value obtained from the g-GGE (the difference between the steady-state values at $dT=0.01$ and $dT=0.05$ is small and not visible on this scale). The open circles represent the calculation at $n=2^m$ using the unitary matrix $A_m$ at each $k$ for $dT=0.05$. The direct calculation using Eq. (15) at each $n$ (black line) agrees perfectly with the results at $n=2^m$ (open circles) using $A_m$.