Xueda Wen, Yingfei Gu, Ashvin Vishwanath, Ruihua Fan
SciPost Phys. 13, 082 (2022) ·
published 5 October 2022
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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.
SciPost Phys. 10, 049 (2021) ·
published 25 February 2021
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In this work, we study non-equilibrium dynamics in Floquet conformal field
theories (CFTs) in 1+1D, in which the driving Hamiltonian involves the
energy-momentum density spatially modulated by an arbitrary smooth function.
This generalizes earlier work which was restricted to the sine-square deformed
type of Floquet Hamiltonians, operating within a $\mathfrak{sl}_2$ sub-algebra.
Here we show remarkably that the problem remains soluble in this generalized
case which involves the full Virasoro algebra, based on a geometrical approach.
It is found that the phase diagram is determined by the stroboscopic
trajectories of operator evolution. The presence/absence of spatial fixed
points in the operator evolution indicates that the driven CFT is in a
heating/non-heating phase, in which the entanglement entropy grows/oscillates
in time. Additionally, the heating regime is further subdivided into a
multitude of phases, with different entanglement patterns and spatial
distribution of energy-momentum density, which are characterized by the number
of spatial fixed points. Phase transitions between these different heating
phases can be achieved simply by changing the duration of application of the
driving Hamiltonian. We demonstrate the general features with concrete CFT
examples and compare the results to lattice calculations and find remarkable
agreement.
