LongHin Tang, David M. Long, Anatoli Polkovnikov, Anushya Chandran, Pieter W. Claeys
SciPost Phys. 15, 030 (2023) ·
published 26 July 2023

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Central spin models provide an idealized description of interactions between a central degree of freedom and a mesoscopic environment of surrounding spins. We show that the family of models with a spin1 at the center and XX interactions of arbitrary strength with surrounding spins is integrable. Specifically, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz. As in the homogenous limit, the states divide into two exponentially large classes: bright states, in which the spin1 is entangled with its surroundings, and dark states, in which it is not. On resonance, the bright states further break up into two classes depending on their weight on states with central spin polarization zero. These classes are probed in quench dynamics wherein they prevent the central spin from reaching thermal equilibrium. In the single spinflip sector we explicitly construct the bright states and show that the central spin exhibits oscillatory dynamics as a consequence of the semilocalization of these eigenstates. We relate the integrability to the closely related class of integrable RichardsonGaudin models, and conjecture that the spin$s$ central spin XX model is integrable for any $s$.
SciPost Phys. 10, 014 (2021) ·
published 22 January 2021

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We discuss how the language of wave functions (state vectors) and associated noncommuting Hermitian operators naturally emerges from classical mechanics by applying the inverse WignerWeyl transform to the phase space probability distribution and observables. In this language, the Schr\"odinger equation follows from the Liouville equation, with $\hbar$ now a free parameter. Classical stationary distributions can be represented as sums over stationary states with discrete (quantized) energies, where these states directly correspond to quantum eigenstates. Interestingly, it is now classical mechanics which allows for apparent negative probabilities to occupy eigenstates, dual to the negative probabilities in Wigner's quasiprobability distribution. These negative probabilities are shown to disappear when allowing sufficient uncertainty in the classical distributions.
We show that this correspondence is particularly pronounced for canonical Gibbs ensembles, where classical eigenstates satisfy an integral eigenvalue equation that reduces to the Schr\"odinger equation in a saddlepoint approximation controlled by the inverse temperature. We illustrate this correspondence by showing that some paradigmatic examples such as tunneling, band structures, Berry phases, Landau levels, level statistics and quantum eigenstates in chaotic potentials can be reproduced to a surprising precision from a classical Gibbs ensemble, without any reference to quantum mechanics and with all parameters (including $\hbar$) on the order of unity.
Silvia Pappalardi, Anatoli Polkovnikov, Alessandro Silva
SciPost Phys. 9, 021 (2020) ·
published 19 August 2020

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Understanding the footprints of chaos in quantummanybody systems has been
under debate for a long time. In this work, we study the echo dynamics of the
SherringtonKirkpatrick (SK) model with transverse field under effective time
reversal. We investigate numerically its quantum and semiclassical dynamics. We
explore how chaotic manybody quantum physics can lead to exponential
divergence of the echo of observables and we show that it is a result of three
requirements: i) the collective nature of the observable, ii) a properly chosen
initial state and iii) the existence of a welldefined chaotic semiclassical
(large$N$) limit. Under these conditions, the echo grows exponentially up to
the Ehrenfest time, which scales logarithmically with the number of spins $N$.
In this regime, the echo is well described by the semiclassical (truncated
Wigner) approximation. We also discuss a shortrange version of the SK model,
where the Ehrenfest time does not depend on $N$ and the quantum echo shows only
polynomial growth. Our findings provide new insights on scrambling and echo
dynamics and how to observe it experimentally.
SciPost Phys. 2, 021 (2017) ·
published 16 June 2017

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We discuss several classes of integrable Floquet systems, i.e. systems which
do not exhibit chaotic behavior even under a time dependent perturbation. The
first class is associated with finitedimensional Lie groups and
infinitedimensional generalization thereof. The second class is related to the
row transfer matrices of the 2D statistical mechanics models. The third class
of models, called here "boost models", is constructed as a periodic interchange
of two Hamiltonians  one is the integrable lattice model Hamiltonian, while
the second is the boost operator. The latter for known cases coincides with the
entanglement Hamiltonian and is closely related to the corner transfer matrix
of the corresponding 2D statistical models. We present several explicit
examples. As an interesting application of the boost models we discuss a
possibility of generating periodically oscillating states with the period
different from that of the driving field. In particular, one can realize an
oscillating state by performing a static quench to a boost operator. We term
this state a "Quantum Boost Clock". All analyzed setups can be readily realized
experimentally, for example in cold atoms.