Floquet-Thermalization via Instantons near Dynamical Freezing

TL;DR

The paper develops Floquet flow renormalization (fRG) as a nonperturbative framework to study dynamical freezing in periodically driven quantum many-body systems. It uncovers a universal flow trajectory that passes through an unstable prethermal fixed point with an emergent approximate symmetry, followed by intermediate fixed points and instanton-like transitions leading to late-time thermalization. Using a cosine-drive on a non-integrable spin chain and employing ED and MPO methods, the work shows that freezing delays thermalization with a subleading $1/\Omega^2$ scaling at high frequency, while instanton events govern slow heating beyond the prethermal regime. Analytically, the early-time dynamics reproduce a modified Magnus-like prethermal Hamiltonian, and late-time behavior is captured by nonperturbative instanton solutions that fold the Floquet spectrum into an $\mathcal{O}(\Omega)$ interval. Together, these results provide a gauge-invariant, nonperturbative route to understanding slow heating and memory retention in driven many-body systems, with implications for square-wave drives and long-time dynamics.

Abstract

Periodically driven Floquet quantum many-body systems have revealed new insights into the rich interplay of thermalization, and growth of entanglement. The phenomenology of dynamical freezing, whereby a translationally invariant many-body system exhibits emergent conservation laws and a slow growth of entanglement entropy at certain fixed ratios of a drive amplitude and frequency, presents a novel paradigm for retaining memory of an initial state upto late times. Previous studies of dynamical freezing have largely been restricted to a high-frequency Floquet-Magnus expansion, and numerical exact diagonalization, which are unable to capture the slow approach to thermalization (or lack thereof) in a systematic fashion. By employing Floquet flow-renormalization, where the time-dependent part of the Hamiltonian is gradually decoupled from the effective Hamiltonian using a sequence of unitary transformations, we unveil the universal approach to dynamical freezing and beyond, at asymptotically late times. We analyze the fixed-point behavior associated with the flow-renormalization at and near freezing using both exact-diagonalization and tensor-network based methods, and contrast the results with conventional prethermal phenomenon. For a generic non-integrable spin Hamiltonian with a periodic cosine wave drive, the flow approaches an unstable fixed point with an approximate emergent symmetry. We observe that at freezing the thermalization timescales are delayed compared to away from freezing, and the flow trajectory undergoes a series of instanton events. Our numerical results are supported by analytical solutions to the flow equations.

Figures (10)

• Figure 1: A schematic depiction of the Floquet flow-renormalization trajectory (blue) along the fRG time, $\lambda$. The insets show the evolution of many-body energy-levels between the different fixed points, e.g. from $\lambda=0$ to the first (unstable) prethermal fixed-point ('$P$'), followed by a sequence of instanton events that connect intermediate fixed points ($I_1,~I_2,...$), and final approach to a possibly thermal ('$T$') fixed-point, depending on a non-trivial interplay of the drive frequency and finite system size. The flow starts with the bare Hamiltonian (many-body bandwidth, $W$), and a Floquet drive with frequency, $\Omega$. The flow to $P$ leads to a renormalization of the many-body spectrum. The energy-levels associated with the effective Hamiltonian at the thermal fixed-point $T$ are folded into an interval smaller than $\Omega$. See Table \ref{['tab:timescales']} for a relationship between the fRG time, $\lambda$, and real time, $t$, along with a summary of the different regimes encountered during the flow vis-à-vis these timescales.
• Figure 2: Flow trajectories of a driven harmonic oscillator given by Eq. \ref{['eq:HO_diff']} for $\omega_1=\Omega$, (a) $\omega_0/\Omega=0$, and (b) $\omega_0/\Omega=0.5$ in the $(B_0,B_1)$ plane, depicted for varying $A/\Omega$. The $n-$th freezing point, denoted $\text{FP}_n$, corresponds to distinct values of $A/\Omega$ in (a) and (b), while $\text{NFP}_n$ stands for near-$\text{FP}_n$. ${\mathcal{H}}_\text{pre}$ denotes the subspace of prethermal Hamiltonians (locus $B_1=0$), and ${\mathcal{H}}_\text{sym}$ denotes subspace of $\rm{U}(1)$ symmetric Hamiltonians (locus $B_0=0$). Dynamical freezing corresponds to the flow landing exactly on the intersection of ${\mathcal{H}}_\text{pre}$ and ${\mathcal{H}}_\text{sym}$, as shown by the curves associated with $\text{FP}_1$ (red), $\text{FP}_2$ (orange), and $\text{FP}_3$ (brown).
• Figure 3: (a) $P(\lambda)$ defined in Eq. \ref{['commH0Xnorm']} as a function of $A/\Omega$ for $\lambda\rightarrow\lambda_c$. The sharp drop in $P(\lambda_c)$ marks the freezing points, FP$_1$, FP$_2$, and FP$_3$, in the considered range of $A/\Omega$. The results for $P(\lambda)$ are obtained for system size $L=10$ using exact diagonalization, and for $L=40$ from MPO calculations. (b) $P(\lambda)$ as a function of fRG time, ($\lambda$), for the different freezing points shows a saturation to the plateau at "late" times. Results for all freezing points are shown for $L=40$ from MPO computations, and for non freezing point (NFP) for $L=10$ using ED. Other parameters: $J=1$, $J_{2}=0.2$, $\Omega=10$. We show the results for $P(\lambda)$ for $L=10$ until $\lambda=10$ in the Appendix \ref{['app:Supp']}.
• Figure 4: (a) Half chain entanglement entropy (EE) density, [i.e. $s_{1/2}=S_{1/2}/(L/2))$] plotted stroboscopically as a function of real time starting from a fully polarized state $\ket{\psi}=\ket{x;+}^{\otimes L}$. The results are shown for $L$=10 (using both exact and flow ED), and $L=40$ (using Flow MPO/MPS), at and away from freezing. Inset: Early time evolution of $s_{1/2}$ at FP$_1$. Other parameters are set to $J=1$, $J_{2}=0.2$, and $\Omega=10$. (b) Frobenius norm of $H_{0}(\lambda)-H^{0}_\text{magnus}$ as a function of fRG time for varying $\Omega$. $H^0_\text{magnus}$ is computed analytically in Appendix \ref{['app:Magnus_spinchain']}. Inset: Saturation value at late fRG time falls off as $1/\Omega^2$ at FP$_1$ ($L=10$).
• Figure 5: (a) $P(\lambda \to \lambda_{c})$ obtained from Eq. \ref{['commH0Xnorm']} as a function of $(A/\Omega)$ near FP$_1$ with increasing $B_{x}$ for $L=10$. All other parameters are same as in Fig. \ref{['fig:freezing']}. (b) IPR defined via Eq. \ref{['ipr']} for eigenstates of $H_{0}(\lambda_c)$ in the $\hat{S}_x$ basis plotted in decreasing order for $L=10$, comparing the freezing point (FP1) and the nonfreezing point (NFP), both with and without the magnetic field term in the $x$ direction. Inset: IPR vs $i/D_H$ for increasing system size for FP$_1$ with $B_x\neq0$. Other parameters are: $J=1$, $J_2=0.5$, $\Omega=5$, $\lambda_c=5.0$.