Periodically, quasiperiodically, and randomly driven conformal field theories

In this sequel (to [Phys. Rev. Res. 3, 023044(2021)],

arXiv:2006.10072), we study randomly driven

(1+1)(1+1)

dimensional conformal field theories (CFTs), a family of quantum

many-body systems with soluble non-equilibrium quantum dynamics. The

sequence of driving Hamiltonians is drawn from an independent and

identically distributed random ensemble. At each driving step, the

deformed Hamiltonian only involves the energy-momentum density spatially

modulated at a single wavelength and therefore induces a Möbius

transformation on the complex coordinates. The non-equilibrium dynamics

is then determined by the corresponding sequence of Möbius

transformations, from which the Lyapunov exponent

\lambda_LλL

is defined. We use Furstenberg’s theorem to classify the dynamical

phases and show that except for a few exceptional points that do not satisfy Furstenberg’s

criteria, the random drivings always lead to a heating phase with the

total energy growing exponentially in the number of driving steps

nn

and the subsystem entanglement entropy growing linearly in

nn

with a slope proportional to central charge

cc

and the Lyapunov exponent \lambda_LλL.

On the contrary, the subsystem entanglement entropy at an exceptional

point could grow as \sqrt{n}n

while the total energy remains to grow exponentially. In addition, we

show that the distributions of the operator evolution and the energy

density peaks are also useful characterizations to distinguish the

heating phase from the exceptional points: the heating phase has both

distributions to be continuous, while the exceptional points could

support finite convex combinations of Dirac measures depending on their

specific type. In the end, we compare the field theory results with the

lattice model calculations for both the entanglement and energy

evolution and find remarkably good agreement.