Periodically, quasiperiodically, and randomly driven conformal field theories

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.