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arXiv:hep-th/9803233

Decay of Spatial Correlations in Thermal States

Christian D. Jäkel

Published Mar 27, 1998• 1 citation
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Abstract

We study the cluster properties of thermal equilibrium states in theories with a maximal propagation velocity (such as relativistic QFT). Our analysis, carried out in the setting of algebraic quantum field theory, shows that there is a tight relation between spectral properties of the generator of time translations and the decay of spatial correlations in thermal equilibrium states, in complete analogy to the well understood case of the vacuum state.

DOI
10.48550/arxiv.hep-th/9803233

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We study the cluster properties of thermal equilibrium states in theories with a maximal propagation velocity (such as relativistic QFT). Our analysis, carried out in the setting of algebraic quantum field theory, shows that there is a tight relation between spectral properties of the generator of time translations and the decay of spatial correlations in thermal equilibrium states, in complete analogy to the well understood case of the vacuum state.
Abstract. We study the cluster properties of thermal equilibrium states in theories with a maximal propagation velocity (such as relativistic QFT). Our analysis, carried out in the setting of algebraic quantum eld theory, shows that there is a tight relation between spectral properties of the generator of time translations and the decay of spatial correlations in thermal equilibrium states, in complete analogy to the well understood case of the vacuum state. Mathematics Subject Classications (1991). 81T05. Decroissance des correlations spatiales pour les etats thermaux. Resume. Nous etudions la propriete de cluster des etats thermaux d'equilibre dans des theories possedant une vitesse maximale de propagation (comme les theories des champs relativistes). Notre analyse, menee dans le cadre de la theorie algebrique des champs, montre qu'il existe une relation etroite entre les proprietes spectrales du generateur des translations temporelles et la decroissance des correlations spatiales pour les etats thermaux dequilibre, en complete analogie avec le cas bien connu de l'etat de vide. 1.
Thermal states are the bedrock of statistical physics. Nevertheless, when and how they actually arise in closed quantum systems is not fully understood. We consider this question for systems with local Hamiltonians on finite quantum lattices. In a first step, we show that states with exponentially decaying correlations equilibrate after a quantum quench. Then, we show that the equilibrium state is locally equivalent to a thermal state, provided that the free energy of the equilibrium state is sufficiently small and the thermal state has exponentially decaying correlations. As an application, we look at a related important question: When are thermal states stable against noise? In other words, if we locally disturb a closed quantum system in a thermal state, will it return to thermal equilibrium? We rigorously show that this occurs when the correlations in the thermal state are exponentially decaying. All our results come with finite-size bounds, which are crucial for the growing field of quantum thermodynamics and other physical applications.
We study the spatial decay of the transversal spin-spin correlations in translation invariant quasifree states on the canonical anticommutation relation algebra on the discrete line. We establish a simple criterion on the spectrum of the density of the quasifree state which ensures the decay to be exponential. Moreover, we illustrate exponential and slower decay with the example of the XY spin chain out of equilibrium, in thermal equilibrium at positive temperature, in the ground state, and in the chaotic state.
In the first part, we analyse the properties of an interacting, massive scalar field in an equilibrium state over Minkowski spacetime. We compare the known real- and imaginary-time formalisms of Thermal Field Theory with the recent construction by Fredenhagen and Lindner of a KMS state for perturbative interacting theories in the context of perturbative Algebraic Quantum Field Theory, in the adiabatic limit. In particular, we show that the construction of Fredenhagen and Lindner reduces to the real-time formalism only if the cocycle which intertwines between the free and interacting dynamics can be neglected. Furthermore, the Fredenhagen and Lindner construction reduces to the ordinary imaginary-time formalism if one considers the expectation value of translation invariant observables. We thus conclude that a complete description of thermal equilibrium for interacting scalar fields is generally obtained only by means of the state constructed by Fredenhagen and Lindner, which combines both formalisms of Thermal Field Theory. We also discuss the properties of the expansion of the Fredenhagen and Lindner construction in terms of Feynman diagrams in the adiabatic limit. We finally provide examples showing that the real- and the imaginary-time formalisms fail to describe thermal equilibrium already at first or second order in perturbation theory. The results presented in this part are summarized in (BDP19). In the second part, we discuss the so-called secular effects, characterized by the appearance of polynomial divergences in the large time limit of truncated perturbative expansions of expectation values in Quantum Field Theory. We show that, although such effect is an artifact of perturbation theory, and thus may not be obtained via exactly solving the dynamical equation if possible, they do not represent the breakdown of perturbation theory itself. Instead, we show that the polynomial divergences follow from a bad choice of state, and we present examples of states which produce expectation values whose perturbative expansion does not present secular effects. In particular, we point that it is possible to obtain non time-divergent perturbative expressions from thermal equilibrium states for the interacting theory. This last part is based on a research project which, by the time this thesis was written, had not been concluded yet.