Emergent Spatial Structure and Entanglement Localization in Floquet Conformal Field Theory

Abstract

We study the energy and entanglement dynamics of ð1 þ 1ÞD conformal field theories (CFTs) under a Floquet drive with the sine-square deformed (SSD) Hamiltonian. Previous work has shown that this model supports both a nonheating and a heating phase. Here, we analytically establish several robust and "superuniversal" features of the heating phase which rely on conformal invariance but not on the details of the CFT involved. First, we show the energy density is concentrated in two peaks in real space, a chiral and an antichiral peak, which leads to an exponential growth in the total energy. The peak locations are set by fixed points of the Möbius transformation. Second, all of the quantum entanglement is shared between these two peaks. In each driving period, a number of Bell pairs are generated, with one member pumped to the chiral peak and the other member pumped to the antichiral peak. These Bell pairs are localized, accumulate at these two peaks, and can serve as a source of quantum entanglement. Third, in both the heating and nonheating phases, we find that the total energy is related to the half system entanglement entropy by a simple relation EðtÞ ∝ c exp ½ð6=cÞSðtÞ with c being the central charge. In addition, we show that the nonheating phase, in which the energy and entanglement oscillate in time, is unstable to small fluctuations of the driving frequency in contrast to the heating phase. Finally, we point out an analogy to the periodically driven harmonic oscillator which allows us to understand global features of the phases and introduce a quasiparticle picture to explain the spatial structure, which can be generalized to setups beyond the SSD construction.

Fine structure of heating in a quasiperiodically driven critical quantum system

“…Therefore the asymptotic formula (14) is not valid on the transition lines T = kL. We note that this quadratic growth of the total energy was already observed in periodic drive at T = kL [30,31], together with a logarithmic growth of entanglement entropy. This dynamics effectively corresponds to a single quantum quench with H. This can be understood from the quasiparticle picture: if T = kL, after the time evolution with H, the quasiparticles will go back to their initial positions.…”

“…The full time evolution under the quasiperiodic drive is obtained in a similar way as for the periodic case [29][30][31]: we first note that in the Heisenberg picture the time evolution of any primary field φ(x, T ) = e i H T φ(x, 0)e −i H T amounts to a simple conformal mapping. This can be seen by (i) rotating to imaginary time t → τ and (ii) mapping the space-time manifold to the complex plane with the exponential mapping z = e 2π (τ +ix) L…”

“…Recently, in Refs. [29][30][31], the Floquet dynamics of an interacting critical field theory based on a step drive alternating periodically between the undeformed H and the deformed Hamiltonian H was studied. Based on a classification of Möbius transformations which encode the time evolution of the system over one period via a conformal mapping [29], a phase diagram comprising both heating and nonheating phases was obtained.…”

“…The heating was shown to be related to the emergence of black hole horizons in space-time and inherently nonergodic [30]. In the extreme limit of a purely random drive, the system was shown to lead to always heat up [31]. A natural question is what happens in the intermediate case, where the driving protocol is neither periodic nor completely random; i.e., the case of quasiperiodic driving where the drive is determined by a recursion relation, but for which one cannot extract any periodic pattern.…”

“…Concretely, we study a driven conformal field theory (CFT), where the time-evolution operator alternates between a uniform (1 + 1)-dimensional CFT and one of its nonhomogeneous versions known as sine-square deformation (SSD) [17][18][19][20][21][22], first introduced in the context of lattice systems as an efficient way to suppress boundary effects [23][24][25][26][27][28]. This setup has been previously studied with a periodic Floquet drive, where it displays a rich phase diagram with both heating and nonheating phases [29][30][31]. Our quasiperiodic drive sequence is generated by a deterministic recursion relation that does not contain any periodic pattern.…”