Periodically, Quasi-periodically, and Randomly Driven Conformal Field Theories (II): Furstenberg's Theorem and Exceptions to Heating Phases

TL;DR

This paper analyzes randomly driven (1+1)D conformal field theories using Furstenberg's theorem, classifying phases into heating and exceptional points with distinct energy and entanglement growth behaviors, supported by lattice model comparisons.

Contribution

It applies Furstenberg's theorem to classify dynamical phases in random CFTs and identifies exceptional points with unique entanglement and energy growth patterns.

Findings

Heating phases exhibit exponential energy growth and linear entanglement growth.

Exceptional points can have sublinear entanglement growth while energy still grows exponentially.

Distribution of operator evolution distinguishes between heating and exceptional phases.

Abstract

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 energy-momentum density spatially modulated at a single wavelength and therefore induces a M\"obius transformation on the complex coordinates. The non-equilibrium dynamics is then determined by the corresponding sequence of M\"obius 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 \emph{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 $\sqrt{n}$ 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.