Abstract
We develop a new method for computing the holographic retarded propagator in generic (non-)equilibrium states using the state/geometry map. We check that our method reproduces the thermal spectral function given by the Son-Starinets prescription. The time-dependence of the spectral function of a relevant scalar operator is studied in a class of non-equilibrium states. The latter are represented by AdS-Vaidya geometries with an arbitrary parameter characterising the timescale for the dual state to transit from an initial thermal equilibrium to another due to a homogeneous quench. For long quench duration, the spectral function indeed follows the thermal form at the instantaneous effective temperature adiabatically, although with a slight initial time delay and a bit premature thermalisation. At shorter quench durations, several new non-adiabatic features appear: (i) time-dependence of the spectral function is seen much before than that in the effective temperature (advanced time-dependence), (ii) a big transfer of spectral weight to frequencies greater than the initial temperature occurs at an intermediate time (kink formation) and (iii) new peaks with decreasing amplitudes but in greater numbers appear even after the effective temperature has stabilised (persistent oscillations). We find four broad routes to thermalisation for lower values of spatial momenta. At higher values of spatial momenta, kink formations and persistent oscillations are suppressed, and thermalisation time decreases. The general thermalisation pattern is globally top-down, but a closer look reveals complexities.
Highlights
The power of holography suggests that we may be able to get some hints for answers to such questions, because both generic initial conditions and couplings to environment can be realised using present numerical gravity techniques
We will find radically different qualitative behaviours of the time dependence of the spectral function which reveal characteristic features of the dynamics of the nonequilibrium state, even when the one-point function of the corresponding operator does not show any distinctive transition in its behaviour as we vary external parameters
This path integral leads to an explicit (d + 1)−Euclidean dual bulk spacetime via the holographic dictionary, which should be glued to a (d + 1)−Lorentzian spacetime that allows us to obtain the dynamics in the real part of the Schwinger-Keldysh contour in the dual field theory
Summary
The retarded propagator is related to linear response even when the system is far away from equilibrium. In order to extract the retarded propagator, we compute (numerically) the new gravitational asymptotically AdS solution with same initial conditions (2.10), and sources (2.12) and (2.13) where the additional pieces hμν and f are adiabatically switched on as mentioned above. We will restrict ourselves to the retarded propagator, it is to be noted that the fully causal multi-point correlation functions involving operators Owhich have vanishing expectation values and those which do not can be extracted from the full non-linear solution of the gravitational equations. In this case we switch on all sources adiabatically and keep all initial conditions unaltered. The retarded correlation function of a probe scalar operator will be found to exhibit diverse qualitative behaviour that will reveal the characteristic features of dynamics of the bulk spacetime
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