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

Abstract

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.

Quantum hypothesis testing in many-body systems

“…We study in detail the case of periodic unitary driving, i.e. Floquet CFTs [62,[89][90][91]. These Type I processes correspond to iterations of Möbius transformations on the BKM geometry and have a solvable albeit rich dynamics.…”

“…We can also consider aperiodic processes [90,91]. In general we a multi-step process is given by a sequence a k ∈ {1, 2} where at the step k we apply the Hamiltonian H a k for a time T a k .…”

“…For unitarily driven CFTs in the Type I setting, the information geometry gives a framework to study complexity and ergodicity. For this purpose, we consider both periodic Floquet driving and aperiodic Fibonacci and random driving, in the context of the models developed in [62,63,[89][90][91]. The prototype examples studied in the literature involve the Möbius subgroup SL(2, R) of Diff + S 1 which already possesses a rather rich dynamics.…”

Thermal and dissipative effects on the heating transition in a driven critical system

“…We will therefore focus our analysis on the heating phase. Note that addition of temporal disorder already erases the non-heating phase in the dissipationless setting [42]. Given the periodicity of the phase diagram along the T 0 L axis, we can also explore a heating phase in the high-frequency limit if T 0 < 0.…”

“…Such hotspots were found to share growing entanglement, and were interpreted as black hole horizons in an effective stroboscopic curved space-time [39]. Different extensions of these drives were then investigated, such as quasi-periodic drives [40,41], random drives [42], non-Hermitian drives [43], continuous drives [44], and general inhomogeneous deformations [45,46]. An important question is whether such phenomenology survives the onset of dissipation and whether dissipation suffices to eliminate this "integrable heating", as opposed to usual ergodic heating.…”