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arXiv:2002.05759 · Physical review. B./Physical review. B

Quantum chaos of unitary Fermi gases in the strong pairing fluctuation region

Xinloong Han, Boyang Liu

Published Jul 17, 2020• 1 citation
arXivHTMLPDF

Abstract

The growth rate of the out-of-time-ordered correlator in a $N$-flavor Fermi gas is investigated and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ is calculated to the order of $1/N$. We find that the Lyapunov exponent monotonically increases as the interaction strength increases from the BCS limit to the unitary region. At the unitarity, the Lyapunov exponent increases while the temperature drops and it can reach to the order of ${\ensuremath{\lambda}}_{L}\ensuremath{\sim}T$ around the critical temperature for the $N=1$ case. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the Lyapunov exponent reaches its maximum ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}4.35T/N$ at the critical temperature. The system scrambles faster for stronger pairing fluctuations. At the BCS limit, the Lyapunov exponent behaviors as ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}{e}^{\ensuremath{\mu}/T}{a}_{s}^{2}{T}^{2}/N$.

DOI
10.1103/physrevb.102.045123

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We study the quantum chaos in the Bose-Fermi Kondo model in which the impurity spin interacts with conduction electrons and a bosonic bath at the intermediate temperature in the large-NNN limit. The out-of-time-ordered correlator is calculated based on the Bethe-Salpeter equation and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ is extracted. Our calculation shows that the Lyapunov exponent monotonically increases as the Kondo coupling JK{J}_{K}JK​ increases, and it can reach an order of ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}T$ as JK{J}_{K}JK​ approaches the multichannel Kondo fixed point (MCKMCKMCK). Furthermore, we also demonstrate that ${\ensuremath{\lambda}}_{L}$ decreases monotonically as the impurity and bosonic bath coupling ggg increases, which is contrary to the general expectation that the most chaotic property occurs at the quantum critical point with the non-Fermi-liquid nature.
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We investigate the many-body quantum chaos of non-Fermi liquid states with Fermi surfaces in two spatial dimensions by computing their out-of-time-order correlation functions. Using a recently proposed large N theory for the critical Fermi surface, and the ladder identity of Gu and Kitaev, we show that the chaos Lyapunov exponent takes the maximal value of 2πk_{B}T/ℏ, where T is the absolute temperature. We also examine a phenomenological model that can be continuously tuned between a non-Fermi liquid without quasiparticles and a Fermi liquid with quasiparticles. We find that the Lyapunov exponent becomes smaller than the maximal value precisely when quasiparticles are restored.
A distinct feature of Hermitian quantum chaotic dynamics is the exponential increase of certain out-of-time-order correlation (OTOC) functions around the Ehrenfest time with a rate given by a Lyapunov exponent. Physically, the OTOCs describe the growth of quantum uncertainty that crucially depends on the nature of the quantum motion. Here, we employ the OTOC in order to provide a precise definition of dissipative quantum chaos. For this purpose, we compute analytically the Lyapunov exponent for the vectorized formulation of the large- q limit of a q -body Sachdev-Ye-Kitaev model coupled to a Markovian bath. These analytic results are confirmed by an explicit numerical calculation of the Lyapunov exponent for several values of q ≥ 4 based on the solutions of the Schwinger-Dyson and Bethe-Salpeter equations. We show that the Lyapunov exponent decreases monotonically as the coupling to the bath increases and eventually becomes negative at a critical value of the coupling signaling a transition to a dynamics which is no longer quantum chaotic. Therefore, a positive Lyapunov exponent is a defining feature of dissipative many-body quantum chaos. The observation of the breaking of the exponential growth for sufficiently strong coupling suggests that dissipative quantum chaos may require in certain cases a sufficiently weak coupling to the environment. Published by the American Physical Society 2024