Floquet conformal field theory

TL;DR

This work presents an analytically solvable setup for the Floquet dynamics of a generic (1+1)D CFT driven by a sine-square deformed Hamiltonian, allowing exact results for correlation functions and entanglement entropy across the full phase diagram. The dynamics split into heating and non-heating phases, mapped to hyperbolic and elliptic Möbius transformations, with a universal logarithmic growth of entanglement at the phase transition and a critical exponent ζ = 1/2. High-frequency driving yields an effective single-quench description that reproduces known results, while lattice simulations with a critical free-fermion chain validate the CFT predictions in the non-heating regime. The study connects Floquet CFT behavior to SL(2,R) structure and Möbius transformations, offering a solid analytic framework for bulk-driven quantum critical dynamics and potential holographic interpretations.

Abstract

Given a $generic$ two-dimensional conformal field theory (CFT), we propose an analytically solvable setup to study the Floquet dynamics of the CFT, i.e., the dynamics of a CFT subject to a periodic driving. A complete phase diagram in the parameter space can be analytically obtained within our setup. We find two phases: the heating phase and the non-heating phase. In the heating phase, the entanglement entropy keeps growing linearly in time, indicating that the system keeps absorbing energy; in the non-heating phase, the entanglement entropy oscillates periodically in time, i.e., the system is not heated. At the phase transition, the entanglement entropy grows logarithmically in time in a universal way. Furthermore, we can obtain the critical exponent by studying the entanglement evolution near the phase transition. Mathematically, different phases (and phase transition) in a Floquet CFT correspond to different types of M$\ddot{\text{o}}$bius transformations.

Figures (12)

• Figure 1: Path integral representation of the single-point correlation function $\langle \psi(t)|\mathcal{O}(x)|\psi(t)\rangle$ in $w$-plane, with $w=\tau+ix$.
• Figure 2: (Part of) Phase diagram for a Floquet CFT, plotted according to Eq.\ref{['ThreeMobiusTransformation']}. The solid dots are obtained from a numerical simulation based on a free fermion chain with $L=500$ and $T_0/L<2$.
• Figure 3: Trajectory of $z_n$ in $z$-plane as a function of $n$ in the non-heating phase (top) and the heating phase (bottom). We choose $T_0=T_1=L/10$ in the non-heating phase and $T_0=T_1=L/2$ in the heating phase. The subsystem length is chosen as $l=L/2$.
• Figure 4: Comparison of entanglement entropy evolution between CFT calculations and numerical simulations in heating phase, non-heating phase, and at the phase transition (inset). We choose $T_0=T_1=T$, $L=500$ and $l=L/2$. The phase transition happens at $T^{\ast}/L\simeq 0.416$ in CFT prediction and at $T^{\ast}/L\simeq 0.418$ in the numerical simulation for $T<L$.
• Figure 5: (Top) Oscillation period $T_E$ of the entanglement entropy for $A=[0,L/2]$ as a function of the driving period $T$. (We choose $T_0=T_1=T$.) (Bottom) Scaling behavior of $T_E$ near the phase transition.