Phase Transitions in Quasi-Periodically Driven Quantum Critical Systems: Analytical Results

TL;DR

The paper investigates non-equilibrium quantum phase transitions in (1+1)D conformal field theories under quasi-periodic driving, introducing a framework based on Avila's global theory to obtain analytic phase diagrams and entanglement dynamics. By modeling driving with $sl(2,\mathbb{R})$ deformations and SU(1,1) Möbius transformations, the authors classify dynamics via the Lyapunov exponent $\lambda_L$ and acceleration $\omega_\lambda$, connecting them to heating and non-heating phases. They distinguish two driving schemes: Type-I, which exhibits only heating, and Type-II, which features heating–non-heating transitions with a finite phase boundary and logarithmic entanglement growth at criticality; both analytical predictions and lattice verifications are presented. The results establish a tractable, analytic route to understand phase behavior in driven CFTs and point to future extensions to Virasoro deformations and non-unitary dynamics for deeper physical interpretation of $\lambda_L(\epsilon)$ and $\omega_\lambda$.

Abstract

In this work, we study analytically the phase transitions in quasi-periodically driven one dimensional quantum critical systems that are described by conformal field theories (CFTs). The phase diagrams and phase transitions can be analytically obtained by using Avila's global theory in one-frequency quasiperiodic cocycles. Compared to the previous works where the quasiperiodicity was introduced in the driving time and no phase transitions were observed [1], here we propose a setup where the quasiperiodicity is introduced in the driving Hamiltonians. In our setup, one can observe the heating phases, non-heating phases, and the phase transitions. The phase diagram as well as the Lyapunov exponents that determine the entanglement entropy evolution can be analytically obtained. In addition, based on Avila's theory, we prove there is no phase transition in the previously proposed setup of quasi-periodically driven CFTs [1]. We verify our field theory results by studying the time evolution of entanglement entropy on lattice models.

Figures (15)

• Figure 1: Two different ways of doing quasiperiodic drivings as considered in this work. In the type-I driving (top), we fix the two driving Hamiltonians and introduce the quasi-periodicity in the time interval for $H_0$. The time intervals for $H_0$ evolution in the $n$-th step is $n\omega l$, where $\omega$ is an irrational number, while the time intervals for $H_1$ evolution are constant. This protocol was numerically studied in Ref.QuasiPeriodic. In the type-II driving (bottom), we fix the time intervals of driving in every other step. The driving Hamiltonian $H_1$ in every other step is fixed, while the driving Hamiltonians $H_{0,n}$ depend on $n$ in a quasiperiodic way. See the main text for more details.
• Figure 2: Complexified Lyapunov exponents $\lambda_L(\epsilon)$ in the subcritical, critical, and supercritical regimes. Near $\epsilon=0$, one has $\lambda_L=0$ and $\omega_\lambda=0$ in the subcritical case, $\lambda_L=0$ and $\omega_\lambda\in \mathbb Z^+$ in the critical case, and $\lambda_L>0$ and $\omega_\lambda\in \mathbb Z^+$ in the supercritical case. In driven CFTs, these three cases correspond to the non-heating phase, phase transition, and the heating phase, respectively. Note the function $\epsilon\mapsto \lambda_L(\epsilon)$ is an even function when the matrices are $\operatorname{SL}(2,\mathbb R)$ or $\operatorname{SU}(1,1)$.
• Figure 3: Complexified Lyapunov exponents $\lambda_L(\epsilon)$ in the uniformly hyperbolic case, where one has $\lambda_L>0$ and $\omega_\lambda=0$ near $\epsilon=0$. This case corresponds to the heating phase in our driven CFT.
• Figure 4: Lyapunov exponents $\lambda_L$ in the type-I driving in Fig.\ref{['Fig:Driving_Protocol']} (see also Sec.\ref{['subsec:quasiSetup1']}) as a function of $\theta$ that characterizes the deformation of $H_1$ in \ref{['tanh_deform']}, with $q=L/l=2$. From bottom to top, we fix $T_1/L_{\text{eff},0}=0.2$, $0.3$, $0.4$, and $0.5$. Here $L_{\text{eff},0}=L\cosh(2\theta_0)$ with $\theta_0=0.1$. The numerical results are obtained from \ref{['Def:Lyapunov']}, \ref{['Theta_0']}, and \ref{['M1_theta_maintext']}, and we choose $\omega=(\sqrt 5-1)/2$ in \ref{['Theta_0']}. The analytical results are obtained from \ref{['lambda_epsilon_example']} by setting $\epsilon=0$.
• Figure 5: Complexified Lyapunov exponents $\lambda_L(\epsilon)$ in \ref{['Complex_Lyapunov_0']} as a function of $\epsilon$ in the type-I quasiperiodic driving in Fig.\ref{['Fig:Driving_Protocol']}. We fix $T_1/L_{\text{eff},0}=0.5$, and choose $\theta=0.2$, $0.3$, and $0.4$ in \ref{['tanh_deform']} from bottom to top. Here $L_{\text{eff},0}=L\cosh(2\theta_0)$ with $\theta_0=0.1$. The dashed red lines correspond to $\lambda_L(\epsilon)=\lambda_L(\epsilon=0)+|\epsilon|$. The features here correspond to the supercritical case in Fig.\ref{['Lyapunov_Phase']}.