Conformal Floquet dynamics with a continuous drive protocol

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 ($$ {S}_n^m $$ 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 $$ {S}_n^m $$ 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.

Out-of-Time-Order correlators in driven conformal field theories

“…For explicit calculations, we collect some important results of [48]. The evolution operator in (2.6) can be written as U (T, 0) = e −iH F T , where H F = p(T )σ z + iq(T )σ y .…”

“…The evolution operator in (2.6) can be written as U (T, 0) = e −iH F T , where H F = p(T )σ z + iq(T )σ y . Here σ's are the Pauli matrices and p(T ) and q(T ) can either be numerically exactly determined or computed analytically within FPT [48]. The former is obtained by dividing T into N = T /δT steps, within each of which the Hamiltonian in (2.5) remains approximately constant.…”

“…It is usually expected that driving a CFT would lead to generation of a timescale which will in turn spoil its conformal invariance and drive it away from its conformal fixed point [42]. However, recently, it was shown [43][44][45][46][47][48] that this is not necessarily the case; indeed it is possible to subject a CFT to a periodic drive without spoiling the conformal symmetry of the problem. As explained in Sec.…”

“…where one analytically continues to real time at the end of the calculation and (h, h) denotes the conformal dimension of O. Using this prescription, energy density, equal and unequal-time correlation functions, and entanglement entropies of driven CFTs have been computed in both cylindrical and strip geometries [43][44][45][46][47][48]. An interesting property found in all these quantities is the emergence of spatial inhomogeneity due to the drive which is usually not found in typical condensed matter systems.…”

Analytic approaches to periodically driven closed quantum systems: methods and applications

“…Also, it is well-known that quantum systems in the presence of a periodic drive may lead to dynamical localization where the drive leads to suppression of the transport of particles in the system [30,[40][41][42]. Finally, more recently, it was found that there is a class of periodic drives which respects the conformal symmetry of the underlying field theory; such driven conformal field theories lead to drive-induced emergent spatial structures in the energy density and correlation functions that have no analogs in standard non-relativistic systems with external drive [43][44][45].…”