Dynamical Freezing and Scar Points in Strongly Driven Floquet Matter: Resonance vs Emergent Conservation Laws

We consider a clean quantum system subject to strong periodic driving. The existence of a dominant energy scale, $h_D^x$, can generate considerable structure in an effective description of a system that, in the absence of the drive, is nonintegrable and interacting, and does not host localization. In particular, we uncover points of freezing in the space of drive parameters (frequency and amplitude). At those points, the dynamics is severely constrained due to the emergence of an almost exact, local conserved quantity, which scars the entire Floquet spectrum by preventing the system from heating up ergodically, starting from any generic state, even though it delocalizes over an appropriate subspace. At large drive frequencies, where a naïve Magnus expansion would predict a vanishing effective (average) drive, we devise instead a strong-drive Magnus expansion in a moving frame. There, the emergent conservation law is reflected in the appearance of the “integrability” of an effective Hamiltonian. These results hold for a wide variety of Hamiltonians, including the Ising model in a transverse field in any dimension and for any form of Ising interaction. The phenomenon is also shown to be robust in the presence of two-body Heisenberg interactions with any arbitrary choice of couplings. Furthermore, we construct a real-time perturbation theory that captures resonance phenomena where the conservation breaks down, giving way to unbounded heating. This approach opens a window on the low-frequency regime where the Magnus expansion fails.

Popular Summary

Periodic shaking of a bunch of interacting particles triggers chaos and disorder: The system absorbs energy without bound and reaches a state that locally resembles one with infinite temperature and maximum entropy. This age-old folklore has long been a hindrance to realizing new and exotic nonequilibrium phases of matter, as well as controlling chaos in interacting quantum systems. Here, we show that this featureless, infinite-temperature state is not inevitable if the drive strength is beyond a certain threshold.

We show that under such a drive, the memory of an arbitrary initial state can instead remain “dynamically frozen” for all time because of quantum interference, allowing for the stabilization of interesting dynamical phases of quantum matter. This behavior, which we show to hold in a very large class of interacting quantum spin systems, is explained by the emergence of new drive-induced conservation laws, not present in the undriven system.

Our result opens up an avenue for realizing new nonequilibrium phases of interacting quantum matter and controlling quantum chaos in complex many-body systems.

Figure 1

Scars, resonances, and emergent conservation law. (a) $m_{\mathrm{DE}}^x/m_0^x$, the ratio of magnetizations after infinite (diagonal ensemble average) and 0 (initial state) cycles versus drive frequency $\omega$. Freezing, reflected in a large value of this ratio, occurs over a broad range of $\omega$ and is strongest at particular “scar” points (marked with arrows) $h_D^x=k\omega$, where $k$ is an integer (for $h_D^x=-40$ here, the ten arrows mark $\omega =40/k;k=1,2,\dots ,10$). Results are shown for zero and high-temperature initial states: the former being the ground state of $H(0)$ [which gives an initial magnetization $m^x(0)\lesssim 1$], and the latter being the Gibbs state with $\beta ={10}^{-2}$ [$m^x(0)\approx 0.05$] for $H_I$ of the form $H(0)$ but with $h_D^x=5$; all other parameters are the same as the driven Hamiltonian, namely, $J=1,\kappa =0.7\pi /3,h_0^x=e/10,h_D^x=40,h^z=1.2,L=14$. The sharp dips in the green lines represent resonances, discussed in detail in the main text on Floquet-Dyson perturbation theory. Parameters are chosen to avoid these resonances. (b) $m^x$ as a function of $h_D^x$ for a fixed ratio $|h_D^x/\omega |=4$, showing the freezing cutoff $(h_D^x\approx 18)$ above which a stable regime of freezing sets in. Inset: finite-size behavior of the freezing. For intermediate strengths of the driving field, higher-order resonances lead to nonmonotonic behavior of $m_{\mathrm{DE}}^x$ on $L$, which is then sensitively dependent on small variations of $h_D^x$. (c) $⟨m^x⟩$ of the Floquet states plotted against the serial number (normalized by the Hilbert space dimension $D_H$) of the Floquet states, arranged in decreasing order of $⟨m^x⟩$. At the scar points ($\omega =10$, 20, and 40), the $⟨m^x⟩$ values form steps coinciding with the eigenvalues of $m^x$ arranged and plotted in the same order: $m^x$ emerges as a emergent conserved quantity, hence the freezing of $m^x$ for any generic initial state.

Figure 2

(De)localization of the wave function over the $x$ basis (simultaneous eigenstates of all of the ${\sigma }_i^x$’s) as evidenced by the half-chain entanglement entropy ($E_{\frac{1}{2}}$) versus system size $L$, for different driving strengths $h_D^x$ (rows) and initial states (left column: maximally $m^x$ polarized; middle: $L/2$-domain-pair state with vanishing total $m^x$; right: Néel state). Top row (small $h_D^x=5$): $E_{\frac{1}{2}}$ entropy grows linearly with system size for all initial states, signaling ergodicity. For stronger drives ($h_D^x=20$ and 40 in the middle and bottom rows, respectively), scars appear, and $E_{\frac{1}{2}}$ depends strongly on the initial states, reflecting the size of the emergent magnetization sectors: For the fully polarized initial states (left column), $E_{\frac{1}{2}}$ does not grow at all for the freezing or scar points [marked as (F) in the figure legends and represented by almost indistinguishably coincidental black and violet triangles], while for the Néel and the $L/2$-domain-pair initial states, there is considerable growth in $E_{\frac{1}{2}}$ even at the scar points, reflecting (at least partial) delocalization over the large concomitant magnetization sectors. The results are for $J=1,\kappa =0.7,h_0^x=e/10,h^z=1.2,L=14$, averaged over ${10}^4$ cycles after driving for ${10}^{10}$ cycles.

Figure 3

Plots of the maximum expectation value of $S^x$ versus $h^x$, for $S=20,T=10,h^z=1$, and (a) $h_D^x=40$ and (b) $h_D^x=12.8\pi \simeq 40.212$. In panel (a), we see pronounced dips for $h^x$ equal to all integer multiples of $2\pi /T$, while in panel (b) we see pronounced dips only when $h^x$ is equal to odd integer multiples of $2\pi /T$, as predicted by the FDPT result, Eq. (33).

Figure 4

Interacting case: Freezing and resonances in the magnetization ratio $m_{\mathrm{DE}}^x/m_0^x$ versus $h_0^x$. The observable, initial states at zero [panels (a,b)] and high temperature (inverse temperature $\beta ={10}^{-2}$) [panel (c)], and other parameters are as described in Fig. 1. Results shown for slow [(a) $\omega =0.4$] and very slow [(b,c) $\omega =0.04$] drives (points on the green lines). The resonances obtained from first-order FDPT, Eq. (42) (purple vertical lines), show a remarkable match with the numerical values of dips in $m^x$. [Some higher-order resonances are also visible at $\omega =0.4$ in panel (a)]. The other parameters are the same as in Fig. 1.

Figure 5

Periodicity in drive strength, $h_D^x$, of the magnetization response [diagonal ensemble average $m_{\mathrm{DE}}^x$, Eq. (3)]. The top row shows periodicity for both off-resonance (left, $h_0^x=-0.2$) and on-resonance (right, $h_0^x=-0.21$) drives. Other parameters and the initial low-temperature state, as in Fig. 2. In both cases, the leading frequency of oscillations is $\mathrm{\Omega }\approx 157.08\approx 2\pi /\omega$, visible in the bottom panel, as predicted by Eq. (43). The other parameters are the same as in Fig. 1.

Figure 6

Infinite-time limit of the magnetization, $m_{\mathrm{DE}}^x$, versus driving frequency $\omega$ for Hamiltonians of the form in Eq. (6) but with different types of $H_0^x$, namely, three-spin interactions [(a) $H_0^x=H_0^{x(3\text{Spin})}$, Eq. (45)] and long-range interactions [(b) $H_0^x=H_0^{x(\mathrm{L}\mathrm{R})}$, Eq. (46)]. The strong freezing of $m_{\mathrm{DE}}^x$ near the freezing points ($\omega =h_D^x/k$) is observed in agreement with the prediction of the Magnus expansion up to the two leading orders in both cases. Interestingly, small corrections due to the higher-order terms are also observed, namely, small deviations of the peak heights from unity for the three-spin case, and tiny shifts of the peak from the freezing condition for the long-range case. Apart from these corrections, the higher-order terms do not appear to change any of the key aspects of the phenomenon (strong emergent conservation of $m^x$ at all times and consequent lack of unbounded heating). Here, the parameters are $J_{xxx}=0.5,J=1,\kappa =0.7\pi /3,h_0^x=e/10,h^z=1.2,h_D^x=40,L=20$.

Figure 7

Infinite-time limit of the magnetization, $m_{\mathrm{DE}}^x$, versus driving frequency $\omega$ for Heisenberg interactions. (a) For the Hamiltonian in [Eq. (6)], with $V$ replaced by $V^{\mathrm{HB}}$ [Eq. (47)] and $H_0^x$ by $H_0^{x(\mathrm{HB})}$ [Eq. (48)], with translationally invariant isotropic interaction strengths ($J_i^x=J_i^y=J_i^z=1.0$). (b) Same model as in the left panel, but with anisotropic interaction strengths $J_i^x=1.0,J_i^y=0.5,J_i^z=0.6$. In both cases, the emergence conservation of $m^x$ and the concomitant absence of thermalization are clearly visible. Slight shifts of the peaks from the predicted values are observed as in Fig. 6. Data are shown for $J=1,\kappa =0.7\pi /3,h_0^x=e/10,h^z=1.2,h_D^x=40,L=20$.

Figure 8

$L$ dependence of $m_{\mathrm{DE}}^x$ at the freezing peaks corresponding to $h_D^x=40$ and for $\omega =1.0$ and 10.0, plotted against the system size $L$ for different variants of the drive Hamiltonian $H(t)$ [Eq. (6)]. Panels (a)–(c) show different variants of Ising interactions: (a) $H_0^x$ contains the nearest- and next-nearest-neighbor interactions and on-site fields [Figs. 1]. (b) $H_0^x=H_0^{x(3\text{Spin})}$ [three-spin interactions; Fig. 6]. (c) $H_0^x=H_0^{x(\mathrm{L}\mathrm{R})}$ [additional long-range interactions; Fig. 6]. Panels (d) and (e) show Heisenberg interactions: $H(t)=H^{\mathrm{HB}}(t)$. Results are shown for (d) translationally invariant clean chains and for the isotropic case [Fig. 7] and (e) for the anisotropic case [Fig. 7].

Figure 9

Mismatches between the numerical dips in $m_{\mathrm{DE}}^x$ and the first-order FDPT prediction [vertical lines obtained from Eq. (42)]. The parameters are chosen such that the condition for a scar is not satisfied, i.e., $h_D^x\ne n\omega$.

Figure 10

Freezing of the entanglement growth of the $L/2$-domain-pair $x$-basis state under a half-up half-down field drive [Eq. (c1)]. The fate of a fully polarized state under the same drive is also shown for comparison. The main frame shows $E_{\frac{1}{2}}$, while the inset shows $m^x$, averaged over ${10}^4$ cycles, after driving for ${10}^{10}$ cycles. The results are for $J=1,\kappa =0.7,h_0^x=e/10,h_D^x=40,L=14$.