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arXiv:1805.00031

Floquet conformal field theory

Xueda Wen, Jie-qiang Wu

Published Apr 30, 2018• 43 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.1805.00031

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Given a genericgenericgeneric 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 Mo¨\ddot{\text{o}}o¨bius transformations.
A bstract We find classes of driven conformal field theories (CFT) in d + 1 dimensions with d > 1, whose quench and floquet dynamics can be computed exactly. The setup is suitable for studying periodic drives, consisting of square pulse protocols for which Hamiltonian evolution takes place with different deformations of the original CFT Hamiltonian in successive time intervals. These deformations are realized by specific combinations of conformal generators with a deformation parameter β ; the β < 1 ( β > 1) Hamiltonians can be unitarily related to the standard (Lüscher-Mack) CFT Hamiltonians. The resulting time evolution can be then calculated by performing appropriate conformal transformations. For d ≤ 3 we show that the transformations can be easily obtained in a quaternion formalism. Evolution with such a single Hamiltonian yields qualitatively different time dependences of observables depending on the value of β , with exponential decays characteristic of heating for β > 1, oscillations for β < 1 and power law decays for β = 1. This manifests itself in the behavior of the fidelity, unequal-time correlator, and the energy density at the end of a single cycle of a square pulse protocol with different hamiltonians in successive time intervals. When the Hamiltonians in a cycle involve generators of a single SU(1, 1) subalgebra we calculate the Floquet Hamiltonian. We show that one can get dynamical phase transitions for any β by varying the time period of a cycle, where the system can go from a non-heating phase which is oscillatory as a function of the time period to a heating phase with an exponentially damped behavior. Our methods can be generalized to other discrete and continuous protocols. We also point out that our results are expected to hold for a broader class of QFTs that possesses an SL(2, C ) symmetry with fields that transform as quasi-primaries under this. As an example, we briefly comment on celestial CFTs in this context.
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}_2 𝔰 𝔩 2 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 p p ( p\ge 1 p ≥ 1 ) driving cycles. We demonstrate the general features with concrete CFT examples and compare the results to lattice calculations and find remarkable agreement.
This paper investigates the dynamical phases of Floquet Conformal Field Theories (CFTs) in space-time dimensions greater than two. Building upon our previous work 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) S U ( 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.
A quantum critical system described at low energy by a conformal field theory (CFT) and subjected to a time-periodic boundary drive displays multiple dynamical regimes, depending on the drive frequency. We compute the behavior of quantities including the entanglement entropy and Loschmidt echo, confirming analytic predictions from field theory by exact numerics on the transverse field Ising model and demonstrate universality by adding nonintegrable perturbations. The dynamics naturally separate into three regimes: a slow-driving limit, which has an interpretation as multiple quantum quenches with amplitude corrections from CFT; a fast-driving limit, in which the system behaves as though subject to a single quantum quench; and a crossover regime displaying heating. The universal Floquet dynamics in all regimes can be understood using a combination of boundary CFT and Kibble-Zurek scaling arguments.
Abstract We study the properties of a conformal field theory (CFT) driven periodically with a continuous protocol characterized by a frequency ωD. Such a drive, in contrast to its discrete counterparts (such as square pulses or periodic kicks), does not admit exact analytical solution for the evolution operator U. In this work, we develop a Floquet perturbation theory which provides an analytic, albeit perturbative, result for U that matches exact numerics in the large drive amplitude limit. We find that the drive yields the well-known heating (hyperbolic) and non-heating (elliptic) phases separated by transition lines (parabolic phase boundary). Using this and starting from a primary state of the CFT, we compute the return probability (Pn), equal (Cn) and unequal (Gn) time two-point primary correlators, energy density(En), and the mth Renyi entropy ( Snm {S}_n^m Snm​ S n m ) after n drive cycles. Our results show that below a crossover stroboscopic time scale nc, Pn, En and Gn exhibits universal power law behavior as the transition is approached either from the heating or the non-heating phase; this crossover scale diverges at the transition. We also study the emergent spatial structure of Cn, Gn and En for the continuous protocol and find emergence of spatial divergences of Cn and Gn in both the heating and non-heating phases. We express our results for Snm {S}_n^m Snm​ S n m and Cn in terms of conformal blocks and provide analytic expressions for these quantities in several limiting cases. Finally we relate our results to those obtained from exact numerics of a driven lattice model.
In this paper and its upcoming sequel, we study nonequilibrium dynamics in driven (1 + 1)-dimensional conformal field theories (CFTs) with periodic, quasiperiodic, and random driving.We study a soluble family of drives in which the Hamiltonian only involves the energy-momentum density spatially modulated at a single wavelength.The resulting time evolution is then captured by a Möbius coordinate transformation.In this paper, we establish the general framework and focus on the first two classes.In periodically driven CFTs, we generalize earlier work and study the generic features of entanglement and energy evolution in different phases, i.e., the heating and nonheating phases and the phase transition between them.In quasiperiodically driven CFTs, we mainly focus on the case of driving with a Fibonacci sequence.We find that (i) the nonheating phases form a Cantor set of measure zero; (ii) in the heating phase, the Lyapunov exponents (which characterize the growth rate of the entanglement entropy and energy) exhibit self-similarity, and can be arbitrarily small; (iii) the heating phase exhibits periodicity in the location of spatial structures at the Fibonacci times; (iv) one can find exactly the nonheating fixed point, where the entanglement entropy and energy oscillate at the Fibonacci numbers, but grow logarithmically and polynomially at the non-Fibonacci numbers; (v) for certain choices of driving Hamiltonians, the nonheating phases of the Fibonacci driving CFT can be mapped to the energy spectrum of electrons propagating in a Fibonacci quasicrystal.In addition, another quasiperiodically driven CFT with an Aubry-André-type sequence is also studied.We compare the CFT results to lattice calculations and find remarkable agreement.