Abstract
Within a holographic model, we calculate the time evolution of 2-point and 1-point correlation functions (of selected operators) within a charged strongly coupled system of many particles. That system is thermalizing from an anisotropic initial charged state far from equilibrium towards equilibrium while subjected to a constant external magnetic field. One main result is that thermalization times for 2-point functions are significantly (approximately three times) larger than those of 1-point functions. Magnetic field and charge amplify this difference, generally increasing thermalization times. However, there is also a competition of scales between charge density, magnetic field, and initial anisotropy, which leads to an array of qualitative changes on the 2- and 1-point functions. There appears to be a strong effect of the medium on 2-point functions at early times, but approximately none at later times. At strong magnetic fields, an apparently universal thermalization time emerges, at which all 2-point functions appear to thermalize regardless of any other scale in the system. Hence, this time scale is referred to as saturation time scale. As extremality is approached in the purely charged case, 2- and 1-point functions appear to equilibrate at infinitely late time. We also compute 2-point functions of charged operators. Our results can be taken to model thermalization in heavy ion collisions, or thermalization in selected condensed matter systems.
Highlights
The topic of thermalization in the AdS/CFT correspondence is nearly as old as the correspondence itself
This result is seen from the geodesic approximation in Vaiyda setups. 2-point functions of small length separations correspond to geodesics which do not probe deep into the AdS bulk
There is an important distinction to be made here. These equilibrium times that we describe for the background field theory are strictly times at which the 1-point functions of the energy momentum tensor, PT and PL take on the values they would in equilibrium
Summary
We follow the characteristic formulation of general relativity by Bondi and Sachs [82, 83]. In order to solve the Einstein equations we define new variables, corresponding to outgoing and in-falling null hypersurfaces. The radial diffeomorphism symmetry of our metric ansatz allows us to shift the bulk radial coordinate by an arbitrary function of time. This motivates the introduction of a “covariant” derivative under this residual bulk radial shift and defines a derivative in the direction of outgoing bulk radial null geodesics, f. We can see that the eq.’s (2.8a) to (2.8d) can be solved in order for S, S , B , and A given a profile for B on the initial time slice. The residual shift invariance is apparent in an undetermined term when solving the Einstein equations order by order near the boundary. The holographic renormalization procedure yields the dual energy momentum tensor of the field theory [62], the non-zero components are, T00
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