Classification of $SL_2$ deformed Floquet Conformal Field Theories

TL;DR

This work classifies the non-equilibrium dynamics of (1+1)D CFTs under periodic driving by N SL2-deformed Hamiltonians. By mapping operator evolution to SU(1,1) Möbius transformations, the authors distinguish heating and non-heating phases via the Floquet map Π_N and its trace, and derive entanglement and energy growth patterns across elliptic, parabolic, and hyperbolic drives. They provide complete necessary-and-sufficient conditions for non-heating phases at N=2 and sufficient conditions for general N, organized into N layers of constraints built from Casimir data. The findings reveal that heating is generic while non-heating phases can be rare or even absent, depending on the combination and ordering of driving Hamiltonians, with implications for engineered non-equilibrium CFT dynamics and potential extensions to quasi-periodic or random protocols.

Abstract

Classification of the non-equilibrium quantum many-body dynamics is a challenging problem in condensed matter physics and statistical mechanics. In this work, we study the basic question that whether a (1+1) dimensional conformal field theory (CFT) is stable or not under a periodic driving with $N$ non-commuting Hamiltonians. Previous works showed that a Floquet (or periodically driven) CFT driven by certain $SL_2$ deformed Hamiltonians exhibit both non-heating (stable) and heating (unstable) phases. In this work, we show that the phase diagram depends on the types of driving Hamiltonians. In general, the heating phase is generic, but the non-heating phase may be absent in the phase diagram. For the existence of the non-heating phases, we give sufficient and necessary conditions for $N=2$, and sufficient conditions for $N>2$. These conditions are composed of $N$ layers of data, with each layer determined by the types of driving Hamiltonians. Our results also apply to the single quantum quench problem with $N=1$.

Figures (8)

• Figure 1: Different types of manifolds determined by Eq.\ref{['Eq:Casimir']} with different quadratic Casimir $c^{(2)}$. Each single point on the manifold specifies a deformed Hamiltonian through \ref{['Hcfti']} and \ref{['fx_SL2']}. Any point on the manifold is $\operatorname{SL}(2,\mathbb R)$ equivalent to arbitrary points on the same manifold.
• Figure 2: Path integral representation of the correlation function $\langle G|\mathcal{O}(x,\tau)|G\rangle$ in a CFT with periodical boundary conditions. Here $x=0$ and $x=L$ are identified.
• Figure 3: Phase diagrams of a Floquet CFT with the both non-heating (in blue) and heating (in red) phases for the six kinds of pairings with $N=2$ in Table.\ref{['classify']}. The parameters are (from left to right, and then top to bottom): elliptic-elliptic with $\boldsymbol{\mathcal{C}}_1=(1,0,0)$ and $\boldsymbol{\mathcal{C}}_2=(1,0.5,0)$; elliptic-parabolic with $\boldsymbol{\mathcal{C}}_1=(1,0,0)$ and $\boldsymbol{\mathcal{C}}_2=(1,1,0)$; elliptic-hyperbolic with $\boldsymbol{\mathcal{C}}_1=(1,0,0)$ and $\boldsymbol{\mathcal{C}}_2=(0,0.4,0)$; parabolic-parabolic with $\boldsymbol{\mathcal{C}}_1=(1,1,0)$ and $\boldsymbol{\mathcal{C}}_2=(1,0,1)$; parabolic-hyperbolic with $\boldsymbol{\mathcal{C}}_1=(1,1,0)$ and $\boldsymbol{\mathcal{C}}_2=(1,0.6,1)$; hyperbolic-hyperbolic with $\boldsymbol{\mathcal{C}}_1=(1,1.4,0)$ and $\boldsymbol{\mathcal{C}}_2=(1,0,1.4)$.
• Figure 4: $\frac{\boldsymbol{\mathcal{C}}_1}{\mathcal{C}_1}= (\sigma_1^0,\sigma_1^+,\sigma_1^-)=(0,1,0)$ is fixed (the vector in black). The normalized vectors $\frac{\boldsymbol{\mathcal{C}}_2}{\mathcal{C}_2}$ that satisfy the condition in Eq.\ref{['HH_condition']} are in the region in green.
• Figure 5: Phase diagram in a Floquet CFT with $N=2$ driving Hamiltonians, both of which are of hyperbolic types. The corresponding Casimir vectors are $\boldsymbol{\mathcal{C}}_1=(1,\,a,\,0)$ and $\boldsymbol{\mathcal{C}}_2=(1,\,0,\, a)$, where we choose $a=1.4$ (left), $1.41421$ (middle), and $1.41421356$ (right). The location of the non-heating phase in blue will move to infinity as we approach $a=\sqrt{2}$ from $a<\sqrt{2}$. For $a>\sqrt{2}$, the condition in \ref{['HH_condition']} is violated, and the non-heating phase does not exist.