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Periodically, Quasi-periodically, and Randomly Driven Conformal Field Theories (II): Furstenberg's Theorem and Exceptions to Heating Phases
by Xueda Wen, Yingfei Gu, Ashvin Vishwanath, Ruihua Fan
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Submission summary
| Authors (as registered SciPost users): | Xueda Wen |
| Submission information | |
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| Preprint Link: | scipost_202110_00020v2 (pdf) |
| Date accepted: | 2022-08-23 |
| Date submitted: | 2022-08-02 15:47 |
| Submitted by: | Wen, Xueda |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
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.
Author comments upon resubmission
List of changes
-- We added appendix B.1.2 discussing the early time evolution of the entanglement entropy in the heating phase in the updated draft.
-- We update Fig.23 and the corresponding descriptions in appendix A.5 by including a sample plot of scaling behavior of Lyapunov exponents near the accidental exceptional points.
-- We add a footnote to discuss the possibility of the existence of exceptional points in randomly driven CFTs with general deformations. It is added in footnote 22 in Sec.4 in the part ``Random drivings beyond SL$_2$ deformations" (Page 45 and 46).
Published as SciPost Phys. 13, 082 (2022)
Reports on this Submission
Report #1 by Bastien Lapierre (Referee 2) on 2022-8-2 (Invited Report)
Report
The authors have implemented the requested changes and discussed in great details the different questions, therefore I think the paper is ready for publication in Scipost Physics.