Complex and tunable heating in conformal field theories with structured drives via classical ergodicity breaking

TL;DR

This work analyzes heating in gapless conformal field theories under structured, non-periodic drives, revealing that operator dynamics reduce to products of Möbius transformations governed by a classical map $\mathcal{K}$ with invariant regions. By studying the Thue-Morse (TM) sequence and $\eta$-random multipolar driving, the authors classify heating, non-heating, and prethermal phases, with a triply tunable prethermal lifetime $t^*\sim K^{-2(\eta-\xi)}$ (for $\eta>\xi$) and $t^*\sim K^{-2\eta-2}$ near fixed points; a long-lived prethermal regime emerges due to proximity to preimages of a fixed point. Extending to non-Hermitian regimes via $\mathrm{SL}(2,\mathbb{C})$ and SU(2) deformations, they uncover a non-heating region of nonzero measure bounded by $\mathrm{tr}(\mathcal{M}_0^2\mathcal{N}_0^2)=2$, underpinned by an emergent compact SU(2) subspace that explains robustness under randomness. Free-fermion numerics corroborate the CFT predictions, and the work discusses experimental prospects in platforms like trapped ions and Rydberg atoms, with broader implications for stabilizing gapless quantum systems under complex drives.

Abstract

Emission and absorption of energy are fundamental aspects of non-equilibrium dynamics. The heating induced by driving a many-body system is perhaps the most straightforward diagnostic of the process of equilibration, or the lack thereof. Gapless systems are particularly susceptible to drive-induced heating, and the capacity to control such heating is of experimental importance. Our study addresses this challenge in the framework of conformal field theory (CFT), for which we study families of structured drives up to the aperiodic Thue-Morse sequence. Concretely, we consider a class of spatially inhomogeneous Hamiltonians, where the operator evolution is governed by a non-linear classical dynamical system $\mathcal{K}$. The existence of invariant regions and fixed points of $\mathcal{K}$ leads to different levels of ergodicity breaking. Upon bridging the gap between this dynamical system and the driven CFT, we classify various dynamical phases of matter, including the heating and non-heating phases, as well as a prethermal phase with a controllably slow heating rate. We further generalize the discussion to $η-$random multipolar driving, characterized by $η-$th order multipolar correlation in time. A ``triply tunable'' parametric dependence of the prethermal lifetime arises as $K^{-2(η-ξ)}$, where $K$ quantifies the deviation from the preimages of the fixed points of $\mathcal{K}$, the multipolar order $η$, and the order of the preimages $ξ$. Upon sacrificing Hermiticity by considering SU(2) deformed CFTs, we find another non-heating phase with a non-zero measure, inaccessible via purely unitary CFTs. This is underpinned by an emergent compact subspace in the generic $\mathrm{SL}(2,\mathbb{C})$ group structure, which we also identify in the transfer matrix in non-Hermitian systems with binary disorder.

Figures (9)

• Figure 1: Trace space for TM trace map showing three invariant regions, I-III. Region I is bounded, while Regions II and III are not. The gray line $p=q-2,p\in[0,4]$, as an intersection between Region I and Region III, is naturally invariant. Point $(4,2)$ labeled by a black star is a fixed point, which crucially possesses infinite preimages. For example, the three orange lines are the first three preimages of the fixed point. Higher order preimages lead to fractal fine structures, as shown in Fig. \ref{['Fig:preimageandentropy']}.
• Figure 2: Schematic of the Thue-Morse driving protocol, where the temporal modulation follows the binary sequence shown in the top row. "0" corresponds to applying the Hamiltonian $H_a$ for a duration of $T_a$, while "1" corresponds to applying $H_b$ for a duration of $T_b$.
• Figure 3: Heating dynamics in $\mathrm{SL}(2,\mathbb{R})$ CFTs driven by a Thue-Morse sequence. (a) Heating time in the 2D parameter space. Black lines correspond to the preimages of a fixed point, obtained by solving the inverse map $\mathcal{K}^{-1}$ 8 times. By construction, preimages have a zero Lyapunov exponent. Bright areas correspond to slow heating, concentrated around the preimages. (b) Heating time in a smaller parameter space with clearer fine structures. (c) Entanglement growth versus stroboscopic time $n$ for different driving parameters. The heating rate is significantly suppressed for parameters close to the preimages. (d) Free fermion numerical verification(dots) of the CFT prediction(solid line). We use $L=3000$, $r=1$ and parameters $T_0,T_1$ are determined by the position of initial points. Coordinates of sampled points: A $(4,1.5)$; B $(4,1.45)$; C $(4,1.4)$; D $(5.9,-1.8)$; F $(4.8,1.2)$.
• Figure 4: (a) Entanglement entropy growth for $\eta$-RMD and TM drive with $K=0.05,\ell_1=0$ for driving parameters close to the fixed point. The prethermal lifetime increases for larger $\eta$, highlighting the importance of temporal correlation in stabilizing the system. (b) Dependence of the prethermal lifetime $t^*$ on $K^{-1}$ in a log-log scale. The slope of the power law dependence strongly depends on the multipolar order as $2\eta+2$. (c) Free fermion numerical comparison of the RMD driving (circles) and quench dynamics (triangles) generated by the effective Hamiltonian $H_{\text{eff}}$. The effective Hamiltonian accurately captures the early time evolution of the RMD system. We use $\sigma_0 = 1$, $\sigma^+ = 1/2$ and $\sigma_0 = 2$, $\sigma^+ = 1$ for $\eta=0$ and $\eta=1$, respectively. We also use ${T}{L^{-1}} = K$, $K=0.02$, $\ell_1=0$ and $L=1000$ for the simulation.
• Figure 5: Trace trajectory of $\eta$-RMD for driving parameters around the fixed point. The red curves denote the analytical trajectories derived from the quenched dynamics generated by $\bar{M}_{\eta}'$, while the blue curves denote the RMD trajectories. As $\eta$ increases, the agreement between the two improves. Here we set $\ell_1=0,K=0.04$.

Theorems & Definitions (4)

• proof
• proof
• proof
• proof