Thermalization of holographic excited states

Published for SISSA by Springer Received: January 1, 2022 Accepted: February 14, 2022 Published: March 1, 2022 Pedro Jorge Martíneza and Guillermo A Silvab a Instituto Balseiro, Centro Atómico Bariloche, 8400-S.C. de Bariloche, Río Negro, Argentina b Instituto de Física La Plata — CONICET and Departamento de Física, Universidad Nacional de La Plata, C.C. 67, 1900, La Plata, Argentina E-mail: , Abstract: We propose a real time holographic framework to study thermalization processes of a family of QFT excited states. The construction builds on Skenderis-van Rees’s holographic duals to QFT Schwinger-Keldysh complex-time ordered paths. Thermalization is explored choosing a set of observables Fn which essentially isolate the excited state contribution. Focusing on theories defined on compact manifolds and with excited states defined in terms of Euclidean path integrals, we identify boundary conditions that allow to avoid any number of modes in the initial field state. In the large conformal dimensions regime, we give precise prescriptions on how to compute the observables in terms of bulk geodesics. Keywords: AdS-CFT Correspondence, Black Holes ArXiv ePrint: 2110.07555 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP03(2022)003 JHEP03(2022)003 Thermalization of holographic excited states Contents 1 2 Framework 2.1 From SK paths to geometries: SvR prescription 2.2 Holographic excited states 2.3 Mode-skipping sources 2.4 The Fn family of observables 4 4 5 7 9 3 Case study I: pure AdS 3.1 Study of F1 3.1.1 Skipping N-modes 3.1.2 Paradigmatic simple sources 3.2 Study of F2 3.2.1 HHL regime: half-bred geodesics 3.2.2 HHH: a geodesic warm up 11 15 15 17 18 19 20 4 Case study II: BTZ 4.1 SK path and geometry 4.2 Study of F1 4.2.1 Relevant sources 4.2.2 Geodesic approximation 4.3 Study of F2 23 24 26 26 27 29 5 Case study III: AdS4+1 BH 5.1 Study of F1 5.2 Study of F2 32 33 35 6 Discussion and conclusions 37 A Finite energy of Holographic excited states 39 B On complexified geodesics 40 1 Introduction The first concrete formulation of the holographic correspondence was made in Euclidean signature. This realization proposed the identification of the AdS gravitational partition function with the QFT generating functional. External CFT sources were equated with the asymptotic boundary conditions for bulk fields [1, 2]. By using the asymptotic sources as –1– JHEP03(2022)003 1 Introduction –2– JHEP03(2022)003 auxiliary tools and considering different topologies, the framework allowed the computation of vacuum and thermal n-point correlators [1–3]. It soon became clear that keeping nonzero sources at the AdS boundary corresponded to CFT deformations, generically triggering RG flows [4]. However, intrinsic real-time phenomenology was out of reach. In particular, a strong interest in the physics of strongly coupled quark-gluon plasma [5], alongside the general quest for a holographic description of QFT hydrodynamics [6, 7] as well as to finding a QFT perspective of black hole interior physics [8, 9], revealed the necessity of a real-time formulation of the holographic dictionary. From the outset, a Lorentzian formulation of AdS/CFT requires to deal with: (i) the correct prescription for determining time-ordering in the correlators (Feynmann, Causal, etc.) as well as, (ii) imposing initial/final conditions in time. In Euclidean signature, these issues were absent since only the asymptotic AdS boundary shows up, a manifestation of the uniqueness of the Euclidean correlator. Important efforts in formalizing the real time scenario [10–12] identified timelike (asymptotic) and spacelike (initial/final times) boundaries. The latter of these are tricky to interpret in the holographic setup if we adopt the philosophy of describing everything from the asymptotic boundary. Further generalizations allowed to compute retarded Green functions for thermal systems [13, 14]. Here, non-trivial chemical potentials on the CFT translated into non-zero bulk gauge fields profiles at the AdS boundary [15]. Again, asymptotic boundary conditions on the timelike boundary were used either as auxiliary tools to compute correlation functions [15] or as deformations of the CFT [4]. Despite some Euclidean computations being successfully carried over to real-time via analytic continuation, applications were fairly restrictive and usually required physical input to obtain the correct result [16]. Moreover, this method did not conceptually explained how initial/final conditions and causality issues were encoded in the holographic map. A full systematic approach to attack real-time problems addressing the above issues was developed by Skenderis and van Rees (SvR) in a series of works [17, 18]. Following original ideas of Schwinger, Keldysh, Hartle and Hawking, SvR proposed to describe SchwingerKeldysh complex t-contours in QFT in terms of glued AdS geometries of mixed signature (see [19] for a recent review). For example, the Euclidean AdS-prescription for the standard solid ball was viewed as dual to an ordered straight vertical path (pure imaginary) in the QFT complex t-plane. Real-time physics was then obtained by deforming the initial vertical contour to the real axis. General curves in the complex t-plane becomes dual to several AdS geometries glued together. In what follows we will collectively refer to the complex tcontours as Schwinger-Keldysh (SK) paths, we will denote them by C. The main advantage of the framework is that the ordering along the SK-contour fixes automatically the correct analytic extension of all real-time correlators requiring no further input. Although some years have passed since its formulation, the potential of the SvR viewpoint has not been fully explored yet. In this work, we aim to make a step forward in this direction by studying thermalization processes. In SK formalism, initial/final QFT wavefunctions are described in terms of Euclidean evolution (pure imaginary time segments) with appropriate operator insertions along it. A specific wavefunction arises in standard fashion as a cut open Euclidean path integral. SvR Initial studies of these states [27, 29] revealed that, via a limiting process, one could seemingly create an initial state as localized in the bulk as desired. However, it was noted recently [30] that not all sets of initial data (in particular non-analytic profiles) can be reached via asymptotic sources using Euclidean path integrals. More precisely, it was shown that the problem of finding asymptotic sources for a given general set of initial data is itself ill-posed. A second goal of this work is to explore a related question: is there a precise formula for asymptotic boundary conditions to the bulk path integral such that a single normal (or quasi-normal) state is given as an initial condition? In this context we find an interesting and reassuring answer: by a limiting process, we will build asymptot (...truncated)