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arXiv:hep-th/0212308 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Relaxation in conformal field theory, Hawking-Page transition, and quasinormal or normal modes

Danny Birmingham, Ivo Sachs, Sergey N. Solodukhin

Published May 30, 2003• 90 citations
arXivHTMLPDF

Abstract

We study the process of relaxation back to thermal equilibrium in $(1+1)$-dimensional conformal field theory at finite temperature. When the size of the system is much larger than the inverse temperature, perturbations decay exponentially with time. On the other hand, when the inverse temperature is large, the relaxation is oscillatory with the characteristic period set by the size of the system. We then analyze the intermediate regime in two specific models: namely, free fermions, and a strongly coupled large k conformal field theory which is dual to string theory on $(2+1)$-dimensional anti--de Sitter spacetime. In the latter case, there is a sharp transition between the two regimes in the $k=\ensuremath{\infty}$ limit, which is a manifestation of the gravitational Hawking-Page phase transition. In particular, we establish a direct connection between quasinormal and normal modes of the gravity system, and the decaying and oscillating behavior of the conformal field theory.

DOI
10.1103/physrevd.67.104026

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We study the process of relaxation back to thermal equilibrium in (1+1)(1+1)(1+1)-dimensional conformal field theory at finite temperature. When the size of the system is much larger than the inverse temperature, perturbations decay exponentially with time. On the other hand, when the inverse temperature is large, the relaxation is oscillatory with the characteristic period set by the size of the system. We then analyze the intermediate regime in two specific models: namely, free fermions, and a strongly coupled large k conformal field theory which is dual to string theory on (2+1)(2+1)(2+1)-dimensional anti--de Sitter spacetime. In the latter case, there is a sharp transition between the two regimes in the k=\ensuremath∞k=\ensuremath{\infty}k=\ensuremath∞ limit, which is a manifestation of the gravitational Hawking-Page phase transition. In particular, we establish a direct connection between quasinormal and normal modes of the gravity system, and the decaying and oscillating behavior of the conformal field theory.
The Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence that arises in string theory has had implications for the study of phenomena across a range of subfields in physics, from spacetime geometry to the behavior of condensed matter systems. Two major themes that have featured prominently in these investigations have been the behavior of systems out of equilibrium, and the emergence of spacetime. In this thesis, aspects of these themes are considered and analyzed. The question of equilibration and thermalization in 2D conformal field theories is addressed and refined via a number of observations about local versus global thermalization in such systems, the validity of particular diagnostics of thermalization, the dependence of the equilibration behavior of a conformal field theory on its operator spectrum, and the holographic dual of the generalized Gibbs ensemble that is of interest in studies of equilibration in systems with a large number of conserved quantities. A formalism for analyzing the non-equilibrium dynamics of 1+1-dimensional conformal field theories is discussed, and its physical relevance is motivated with an example connecting such a system to an experimental system that exhibited unusual equilibration behavior. Qualitative agreement is demonstrated between the CFT picture and the experimental observations.The emergence of spacetime geometry from quantum entanglement, while largely a byproduct of considerations from holographic dualities, has also been proposed to have a direct, non-holographic manifestation. Here a particular realization of such a direct emergence is presented through a demonstration that, in the presence of quantum entanglement alone, certain observations of electric fields in the entangled system appear qualitatively the same as the corresponding observations in a physically-connected geometric spacetime, so that the entanglement effectively mimics particular features associated with geometric connectivity.
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It was recently found that the quasinormal modes (or Ruelle--Pollicott resonances) of the SYK model at infinite temperature form a Christmas tree shape, reminiscent of AdS black holes. We generalise this computation to finite temperature, allowing us to continuously connect the infinite temperature results to the low temperature regime dual to JT gravity. We contrast the movement of the quasinormal modes with a few examples: various AdS black holes, dynamical phase transitions, and the large ppp SYK chain. We find that the relaxation rate increases monotonically with temperature only at strong coupling, corresponding to the gravitational regime. Byproducts of our investigations are new results on operator growth that may be of independent interest.
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We consider a quantum quench in a finite system of length LLL described by a 1+11+11+1-dimensional conformal field theory (CFT), of central charge ccc, from a state with finite energy density corresponding to an inverse temperature $\ensuremath{\beta}\ensuremath{\ll}L$. For times ttt such that $\ensuremath{\ell}/2<t<(L\ensuremath{-}\ensuremath{\ell})/2$ the reduced density matrix of a subsystem of length $\ensuremath{\ell}$ is exponentially close to a thermal density matrix. We compute exactly the overlap F\mathcal{F}F of the state at time ttt with the initial state and show that in general it is exponentially suppressed at large $L/\ensuremath{\beta}$. However, for minimal models with c<1c<1c<1 (more generally, rational CFTs), at times which are integer multiples of L/2L/2L/2 (for periodic boundary conditions, LLL for open boundary conditions) there are (in general, partial) revivals at which F\mathcal{F}F is O(1)O(1)O(1), leading to an eventual complete revival with F=1\mathcal{F}=1F=1. There is also interesting structure at all rational values of t/Lt/Lt/L, related to properties of the CFT under modular transformations. At early times $t\ensuremath{\ll}(L\ensuremath{\beta}{)}^{1/2}$ there is a universal decay $\mathcal{F}\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}(\ensuremath{\pi}c/3)L{t}^{2}/\ensuremath{\beta}({\ensuremath{\beta}}^{2}+4{t}^{2}))$. The effect of an irrelevant nonintegrable perturbation of the CFT is to progressively broaden each revival at t=nL/2t=nL/2t=nL/2 by an amount O(n1/2)O({n}^{1/2})O(n1/2).