Lyapunov exponent as a signature of dissipative many-body quantum chaos
Antonio M. Garcı́a-Garcı́a, J. J. M. Verbaarschot, Jie Zheng
Abstract
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- <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>q</a:mi> </a:mrow> </a:math> limit of a <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>q</c:mi> </c:math> -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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>q</e:mi> <e:mo>≥</e:mo> <e:mn>4</e:mn> </e:math> 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