Periodically, quasi-periodically, and randomly driven conformal field theories (II): Furstenberg's theorem and exceptions to heating phases

In this sequel (to [Phys. Rev. Res. 3, 023044(2021)], arXiv:2006.10072), we study randomly driven (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 energymomentum density spatially modulated at a single wavelength and therefore induces a Mobius transformation on the complex coordinates. The non-equilibrium dynamics is then determined by the corresponding sequence of Mobius transformations, from which the Lyapunov exponent lambda(L) 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 n and the subsystem entanglement entropy growing linearly in n with a slope proportional to central charge c and the Lyapunov exponent lambda(L). On the contrary, the subsystem entanglement entropy at an exceptional point could grow as root 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.

Automated summary

In this study, we investigate the properties of randomly driven (1 + 1) dimensional conformal field theories (CFTs), revealing the characteristics of non-equilibrium dynamical phases, including the heating phase and exceptional points. In most cases, random drivings lead to exponential growth of total energy and linear growth of subsystem entanglement entropy. However, at exceptional points, the subsystem entanglement entropy may grow as the square root of the number of driving steps, while the total energy still grows exponentially. Furthermore, we distinguish the heating phase from exceptional points by analyzing the distributions of operator evolution and energy density peaks.