We study the dissipative dynamics of a periodically driven inhomogeneous critical lattice model in one dimension. The closed system dynamics starting from pure initial states is well-described by a driven Conformal Field Theory (CFT), which predicts the existence of both heating and non-heating phases in such systems. Heating is inhomogeneous and is manifested via the emergence of black-hole like horizons in the system. The robustness of this CFT phenomenology when considering thermal initial states and open systems remains elusive. First, we present analytical results for the Floquet CFT time evolution for thermal initial states. Moreover, using exact calculations of the time evolution of the lattice density matrix, we demonstrate that for short and intermediate times, the closed system phase diagram comprising heating and non-heating phases, persists for thermal initial states on the lattice. Secondly, in the fully open system with boundary dissipators, we show that the nontrivial spatial structure of the heating phase survives particle-conserving and non-conserving dissipations through clear signatures in mutual information and energy density evolution.
We study the dissipative dynamics of a periodically driven inhomogeneous critical lattice model in one dimension. The closed system dynamics starting from pure initial states is well-described by a driven Conformal Field Theory (CFT), which predicts the existence of both heating and non-heating phases in such systems. Heating is inhomogeneous and is manifested via the emergence of black-hole like horizons in the system. The robustness of this CFT phenomenology when considering thermal initial states and open systems remains elusive. First, we present analytical results for the Floquet CFT time evolution for thermal initial states. Moreover, using exact calculations of the time evolution of the lattice density matrix, we demonstrate that for short and intermediate times, the closed system phase diagram comprising heating and non-heating phases, persists for thermal initial states on the lattice. Secondly, in the fully open system with boundary dissipators, we show that the nontrivial spatial structure of the heating phase survives particle-conserving and non-conserving dissipations through clear signatures in mutual information and energy density evolution.
While driven interacting quantum matter is generically subject to heating and scrambling, certain classes of systems evade this paradigm. We study such an exceptional class in periodically driven critical (1+1)-dimensional systems with a spatially modulated but disorder-free time evolution operator. Instead of complete scrambling, the excitations of the system remain well defined. Their propagation is analogous to the evolution along light cones in a curved spacetime obtained by two black holes. The Hawking temperature serves as an order parameter which distinguishes between heating and nonheating phases. Beyond a timescale determined by the inverse Hawking temperature, excitations are absorbed by the black holes resulting in a singular concentration of energy at their horizon. We obtain these results analytically within conformal field theory, capitalizing on a mapping to sine-squared deformed field theories. Furthermore, by means of numerical calculations for an interacting XXZ spin-12 chain, we demonstrate that our findings survive lattice regularization.
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 ) 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 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.
We study Krylov construction in periodically driven conformal field theories and their lattice realisations via critical fermions. Two types of driving are considered: a square-wave drive and a continuous sinusoidal drive. Using the Arnoldi construction, we examine Arnoldi coefficients and return amplitudes in periodically driven conformal field theories in the heating and non-heating phases. In the heating phase, the Arnoldi coefficients approach unity exponentially; in contrast, in the non-heating phase, they exhibit oscillatory behaviour. For the lattice realisations, we further analyse the Krylov complexity of the correlation matrix, quasi energy level statistics, and the graph structure induced by the Floquet operator. Although the two drives exhibit similar Krylov growth on the CFT side, their lattice realisations exhibit markedly different spectral and graph signatures, indicating distinct mechanisms governing the transition between the heating and non-heating phases.
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.
In this paper, we analytically study phase transitions in quasiperiodically driven one-dimensional quantum critical systems that are described by conformal field theories (CFTs). The phase diagrams and phase transitions can be analytically obtained by using Avila's global theory in one-frequency quasiperiodic cocycles. Compared to previous works where the quasiperiodicity was introduced in the driving time and no phase transitions were observed [Wen et al., Phys. Rev. Res. 3, 023044 (2021)], here we propose a setup where the quasiperiodicity is introduced in the driving Hamiltonians. In our setup, one can observe the heating phases, nonheating phases, and phase transitions. The phase diagram as well as the Lyapunov exponents that determine the entanglement entropy evolution can be analytically obtained. In addition, based on Avila's theory, we prove there is no phase transition in the previously proposed setup of quasiperiodically driven CFTs [Wen et al., Phys. Rev. Res. 3, 023044 (2021)]. We verify our field theory results by studying the time evolution of entanglement entropy on lattice models.
We investigate the interplay between unitary and non-unitary driven many-body dynamics in (1+1)-dimensional quantum critical systems described by conformal field theory (CFT). By formulating a coherent state approach, we demonstrate that the growth of entanglement entropy and energy can be found analytically for a class of non-unitary driven CFTs, where the evolution alternates between real and imaginary time evolution, the latter corresponding to postselected weak measurements. We find that non-unitary evolution leads to the emergence of steady states at infinite times for the cases of periodic, quasiperiodic, and random drives. In a special class of drives, for mixed initial states, we uncover purification phase transitions that arise as a result of the competition between unitary evolution and weak measurements. We compare the CFT evolution with the corresponding non-unitary dynamics of critical lattice models, finding remarkable agreement.