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.
SciPost Phys. 9, 079 (2020) ·
published 23 November 2020
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We study the quantum dynamics of Bose-Einstein condensates when the
scattering length is modulated periodically or quasi-periodically in time
within the Bogoliubov framework. For the periodically driven case, we consider
two protocols where the modulation is a square-wave or a sine-wave. In both
protocols for each fixed momentum, there are heating and non-heating phases,
and a phase boundary between them. The two phases are distinguished by whether
the number of excited particles grows exponentially or not. For the
quasi-periodically driven case, we again consider two protocols: the
square-wave quasi-periodicity, where the excitations are generated for almost
all parameters as an analog of the Fibonacci-type quasi-crystal; and the
sine-wave quasi-periodicity, where there is a finite measure parameter regime
for the non-heating phase. We also plot the analogs of the Hofstadter butterfly
for both protocols.
SciPost Phys. 9, 071 (2020) ·
published 12 November 2020
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We study quantum many-body systems with a global U(1) conservation law,
focusing on a theory of $N$ interacting fermions with charge conservation, or
$N$ interacting spins with one conserved component of total spin. We define an
effective operator size at finite chemical potential through suitably
regularized out-of-time-ordered correlation functions. The growth rate of this
density-dependent operator size vanishes algebraically with charge density;
hence we obtain new bounds on Lyapunov exponents and butterfly velocities in
charged systems at a given density, which are parametrically stronger than any
Lieb-Robinson bound. We argue that the density dependence of our bound on the
Lyapunov exponent is saturated in the charged Sachdev-Ye-Kitaev model. We also
study random automaton quantum circuits and Brownian Sachdev-Ye-Kitaev models,
each of which exhibit a different density dependence for the Lyapunov exponent,
and explain the discrepancy. We propose that our results are a cartoon for
understanding Planckian-limited energy-conserving dynamics at finite
temperature.
SciPost Phys. 2, 018 (2017) ·
published 28 May 2017
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We compute the energy diffusion constant $D$, Lyapunov time $\tau_{\text{L}}$
and butterfly velocity $v_{\text{B}}$ in an inhomogeneous chain of coupled
Majorana Sachdev-Ye-Kitaev (SYK) models in the large $N$ and strong coupling
limit. We find $D\le v_{\text{B}}^2 \tau_{\text{L}}$ from a combination of
analytical and numerical approaches. Our example necessitates the sharpening of
postulated transport bounds based on quantum chaos.
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