Abstract
Recently, a direct signature of chaos in many body system has been realized from the energy density retarded Green’s function using the phenomenon of ‘pole skipping’. Moreover, special locations in the complex frequency and momentum plane are found, known as the pole skipping points such that the retarded Green’s function can not be defined uniquely there. In this paper, we compute the correction/shift to the pole skipping points due to a spatial anisotropy in a holographic system by performing near horizon analysis of EOMs involving different bulk field perturbations, namely the scalar, the axion and the metric field. For vector and scalar modes of metric perturbations we construct the gauge invariant variable in order to obtain the master equation. Two separate cases for every bulk field EOMs is considered with the fluctuation propagating parallel and perpendicular to the direction of anisotropy. We compute the dispersion relation for momentum diffusion along the transverse direction in the shear channel and show that it passes through the first three successive pole skipping points. The pole skipping phenomenon in the sound channel is found to occur in the upper half plane such that the parameters Lyapunov exponent λL and the butterfly velocity vB are explicitly obtained thus establishing the connection with many body chaos.
Highlights
Time t) of separation between two phase space trajectories which were infinitesimally close at some initial time t0
Using the tools of gauge/gravity duality one can perform the gravitational shock wave analysis [6,7,8,9,10] to calculate the out-of-time ordered correlation function (OTOC) which is regarded as the measure of chaos in quantum systems
The OTOC characterises the chaotic behavior in many body quantum systems in terms of two parameters, namely the lyapunov exponent λL and the butterfly velocity vB
Summary
We will briefly describe the supergravity solution as obtained by the authors in [40, 41] which is dual to a spatially anisotropic strongly coupled SYM theory at finite temperature. In relativistic heavy ion collision the plasma that is created has been found to be locally anisotropic for a very short time period due to the pressure difference along the longitudinal and transverse direction. This motivates the authors towards a dual gravitational background with anisotropy along a spatial direction. The explicit form of the different metric components in (2.2) is given in [40, 41], where the authors have introduced an anisotropy parameter a. We will do the near horizon analysis of the EOM for different field perturbation to obtain the special points in the complex (w − q) plane where the pole skipping phenomenon will be explicit
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