Floquet conformal field theories with generally deformed Hamiltonians

Abstract

In this work, we study non-equilibrium dynamics in Floquet conformal field theories (CFTs) in 1+1D, in which the driving Hamiltonian involves the energy-momentum density spatially modulated by an arbitrary smooth function. This generalizes earlier work which was restricted to the sine-square deformed type of Floquet Hamiltonians, operating within a \mathfrak{sl}_2𝔰𝔩2 sub-algebra. Here we show remarkably that the problem remains soluble in this generalized case which involves the full Virasoro algebra, based on a geometrical approach. It is found that the phase diagram is determined by the stroboscopic trajectories of operator evolution. The presence/absence of spatial fixed points in the operator evolution indicates that the driven CFT is in a heating/non-heating phase, in which the entanglement entropy grows/oscillates in time. Additionally, the heating regime is further subdivided into a multitude of phases, with different entanglement patterns and spatial distribution of energy-momentum density, which are characterized by the number of spatial fixed points. Phase transitions between these different heating phases can be achieved simply by changing the duration of application of the driving Hamiltonian. %In general, there are rich internal structures in the heating phase characterized by different numbers of spatial fixed points, which result in different entanglement patterns and distribution of energy-momentum density in space. %Interestingly, after each driving cycle, these spatial fixed points will shuffle to each other in the array, and come back to the original locations after pp (p\ge 1p≥1) driving cycles. We demonstrate the general features with concrete CFT examples and compare the results to lattice calculations and find remarkable agreement.

Conformal Floquet dynamics with a continuous drive protocol

“…In contrast, it shows an oscillatory behavior in the non-heating phase and a logarithmic growth on the phase boundary. Finally, we would like to point out that all of these studies focussed on evolution starting from the CFT vacuum on a strip geometry [32][33][34][35][36][37][38]; the dynamics of the system starting from asymptotic states corresponding to primary operators of the theory, which necessitates computation of four-point correlation functions of the primary fields of the driven CFT, has not been studied so far. This is one of the main goals of this work.…”

“…The exact solution for U (T, 0) for Hamiltonians valued in su (1,1) and driven by discrete protocols were discussed earlier for periodic kicks [46] and square pulse protocols [32][33][34][35][36][37][38]. It was found that such driven system could display two distinct phases depending on the drive parameters; these are termed as the heating (hyperbolic) and the non-heating (elliptic) phase and are found to be separated by a transition line (parabolic phase boundary) [46].…”

“…More recently, several theoretical studies concentrated on the effect of Floquet dynamics on conformal field theories [32][33][34][35][36][37][38][39][40]. Usually driving a CFT is expected to generate an additional scale in the problem which drives the system away from its conformally invariant fixed point.…”

“…The main results obtained in refs. [32][33][34][35][36][37][38][39][40] can be summarized as follows. First, it was noted in ref.…”

Non-Equilibrating a Black Hole with Inhomogeneous Quantum Quench

“…More recently, the Möbius and SSD Hamiltonians have been used to study non-equilibrium processes. In particular, in (1+1)d CFT, solvable models of quantum quench [58] and Floquet dynamics [59,60,61,62,63,64,65] can be constructed using the Möbius and SSD Hamiltonians. They provide rare examples where the dynamics of interacting many-body quantum systems can be solved analytically.…”