Dynamical Phases of Higher Dimensional Floquet CFTs

TL;DR

This paper extends Floquet conformal dynamics to higher dimensions (d=2,3) by employing a quaternionic representation of conformal transformations to analyze multi-step square-pulse drives beyond SU(1,1). It identifies dynamical phases through the eigenstructure of the one-cycle evolution operator and provides a geometric interpretation in terms of Killing horizons on the CFT base space and its AdS bulk, with heating corresponding to non-extremal horizons and critical driving to extremal horizons. Perturbative methods (Magnus expansion and Floquet perturbation theory) are developed to compute Floquet Hamiltonians in high-frequency and high-amplitude regimes, illustrating how drive parameters tune horizon formation. The work also analyzes bulk AdS horizons for holographic CFTs and discusses hybrid dynamical phases that arise in higher-dimensional drives, highlighting a rich interplay between symmetry, geometry, and driven critical dynamics with potential implications for prethermal behavior and holographic duals.

Abstract

This paper investigates the dynamical phases of Floquet Conformal Field Theories (CFTs) in space-time dimensions greater than two. Building upon our previous work [1] which introduced quaternionic representations for studying Floquet dynamics in higher dimensional CFTs, we now explore more general square pulse drive protocols that go beyond a single SU(1,1) subgroup. We demonstrate that, for multi-step drive protocols, the system exhibits distinct dynamical phases characterized by the nature of the eigenvalues of the quaternionic matrix representing time evolution in a single cycle, leading to different stroboscopic responses. Our analysis establishes a fundamental geometric interpretation where these dynamical phases directly correspond to the presence or absence of Killing horizons in the base space of the CFT and in a higher dimensional AdS space on which a putative dual lives. The heating phase is associated with a non-extremal horizon, the critical phase with an extremal horizon which disappears in the non-heating phase. We develop perturbative approaches to compute the Floquet Hamiltonians in different regimes and show, how tuning drive parameters can lead to horizons, providing a geometric framework for understanding heating phenomena in driven conformal systems.

Figures (7)

• Figure 1: A contour plot showing the explicit coordinate dependence of the extremal horizon obtained in equation (\ref{['exthorizon']}). We have chosen here $s=0$, which corresponds to a simple shift in the $t$ coordinate. The unshaded (white) region corresponds to regions where the horizon does not exist. Evidently, these regions are bounded by null rays which yields causal diamonds in this figure. Along the $\theta$ direction, the causal diamonds are identified at $\theta=0$ and $\theta= 2\pi$. There is a periodic pattern of the causal diamonds along the Lorentzian time direction.
• Figure 2: Left : Pure phase eigenvalue plots for parameters $\beta_1 = 0.8, \beta_0 = 0.6$ are shown in the blue region. Right : Blue region indicates pure phase eigenvalues for $\beta_0 = \beta_1 = \sqrt{2} - 10^{-6}$, orange for $\beta_0 = \beta_1 = \sqrt{2} - 10^{-5}$ and green when $\beta_0 = \beta_1 = \sqrt{2} - 10^{-4}$.
• Figure 3: Eigenvalue plots for parameters $\beta_0 = 0.9, \beta_1 = 0.8, T_0 = 11$. Left : Blue regions indicate $\lambda_\pm$ are pure phase, whereas Orange indicates $\mu_\pm$ are pure phase. Right : The intersection region where all 4 eigenvalues are pure phase. These are the regions in the parameter space which will result in elliptic steady state.
• Figure 4: Correlation function in the hybrid phase. The parameters take values, $p_1 = \tfrac{1}{2} + 2\,i , \, q_1 = -i, \, p_2 = 1 + \sqrt{3}\, i, \, q_2 = \tfrac{i}{2}$.
• Figure 5: We have chosen $i = x, k = y, p = z$ directions for the tridirectional Floquet drive. The parameters are chosen to be $\beta_1 = 0.001, \beta_2 = 0.8, \beta_3 = 0.9, T_0 = 7$. The contours are for the quantity $\sum\limits_{i=1}^4 | 1 - |\lambda_i ||$, which should vanish when all the eigenvalues are pure phases. Left : A larger area of the $T_1-T_2$ plane with varying values of the eigenvalues. We zoom into the yellow rectangular region on the right plot. Right : Zoomed in region in the $T_1-T_2$ plane with higher resolution of $\sum\limits_{i=1}^4 | 1 - |\lambda_i ||$.