Holographic Equilibration of Nonrelativistic Plasmas

Abstract

We study far-from-equilibrium physics of strongly interacting plasmas at criticality and zero charge density for a wide range of dynamical scaling exponents z in d dimensions using holographic methods. In particular, we consider homogeneous isotropization of asymptotically Lifshitz black branes with full backreaction. We find stable evolution and equilibration times that exhibit small dependence of z and are of the order of the inverse temperature. Performing a quasinormal mode analysis, we find a corresponding narrow range of relaxation times, fully characterized by the fraction z=ðd − 1Þ. For z ≥ d − 1, equilibration is overdamped, whereas for z < d − 1, we find oscillatory behavior. Finally, and most interestingly, we observe that also the nonlinear evolution, although differing significantly from a quasinormal mode fit, is to a high degree of precision characterized by the fraction z=ðd − 1Þ. Introduction.-Quantum criticality has been a focus of interest both in theoretical and experimental physics over the past few decades. It is believed to be a key ingredient in the solution to the various yet unsolved problems such as the high T c superconductivity [1]. In particular, the dynamics of a system near a continuous quantum phase transition is governed by a universal, scale invariant theory characterized by the dimensionality d, the dynamical scaling exponent z, and the various other critical exponents that are independent of the microscopic Hamiltonian of the system. In these systems, the characteristic energy scale Δ, such as the gap separating the first excited state from the ground state, vanishes as the correlation length ξ diverges as Δ ∼ ξ −z . For instance, z ¼ 1 occurs at the touching points of the band structure of monolayer graphene, z ¼ 2 can describe the case of bilayer graphene, and z ¼ 2, 3 occur in heavy fermion systems; see, e.g., Refs. [2][3][4][5][6].The existence of a quantum critical point at vanishing temperature determines the behavior of observables also at finite temperature, and even beyond the thermal equilibrium, in the so-called quantum critical region of the parameter space. In fact, a basic way to characterize this quantum critical region is to consider the response of the system to a small disturbance, determined by the equilibration time τ eq [7]. The quantum critical region corresponds to short relaxation times τ eq ∼ 1=T [8], whereas local equilibrium is reached much more slowly as τ eq ≫ 1=T outside the quantum critical region [1].In this Letter, we want to go one step further and ask the question, what happens when such a quantum critical system is taken completely out of equilibrium, when the perturbation is not small, but of the same order as the Hamiltonian? We answer this question partially in the particular situation when the collective excitations of the system are characterized by global, hydrodynamic

Overdamped modes in Schwarzschild-de Sitter and a Mathematica package for the numerical computation of quasinormal modes

“…Two additional examples, rederiving the QNM's of [9] and [35], are included in [1]. Although we did not discuss a case where the background is known only numerically, it should be clear that as long as the numerical background is known to a high enough precision this will not give any problems, as an analytical background has to be converted to a numerical one in any case.…”

Quasi-normal modes of holographic system with Weyl correction and momentum dissipation

“…in complex frequency plane. We shall particularly focus on the properties of the quasinormal modes (QNMs), which corresponds to the poles of the retarded Green's function for the dual boundary CFT (see [21][22][23][24][25][26][27][28][29][30][31][32][33] and references therein). Our paper is organized as follows.…”

Correlation functions in theories with Lifshitz scaling

“…The ultraviolet divergences are then efficiently handled by introducing a regularised two-point function with the equal time coincident point two-point function subtracted off. Starting from (2.6) we obtain 19) where in the last step we have introduced a scaling variable, 20) which is invariant under the Lifshitz scaling transformation (1.1) with z = d.…”

“…A fast varying source induces a perturbation in some (combination of) field(s) close to the boundary of the space, which subsequently falls into the bulk and forms a black hole. Generically the time evolution has to be followed numerically by solving the dynamical Einstein's equations coupled to whatever matter is present [19]. A simple metric that can be used to model the collapsing configuration is the Vaidya spacetime, which corresponds to null and pressureless matter sourcing Einstein's equations.…”

“…Thermal correlators obtained from geodesics in Lifshitz black hole backgrounds will be considered in section 4. We orient the spatial coordinates on the boundary so that the geodesic endpoints to lie on the x-axis, in which case the geodesic equations becomė 19) whereḟ ≡ df /dλ and E, p are constants. Using (3.19) the affine parameter can be expressed as an integral over the radial coordinate,…”