Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 9

On the equivalence of the KMS condition and the variational principle for quantum lattice systems with mean-field interactions

Pages: 2909 – 2961

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n9.a2

Authors

J.-B. Bru (Department of Mathematics, University of the Basque Country UPV/EHU, Leioa, Spain; and Basque Center for Applied Mathematics, Bilbao, Spain)

W. de Siqueira Pedra (Departamento de Matemática, Instituto de Ciências Matemáticas e da Computação, Universidade de São Paulo, São Carlos, SP, Brazil)

R. S. Yamaguti Miada (Basque Center for Applied Mathematics, Bilbao, Spain; and Instituto de Fíca, Universidade de São Paulo, Brazil)

Abstract

We extend Araki’s well-known results on the equivalence of the KMS condition and the variational principle for equilibrium states of quantum lattice systems with short-range interactions, to a large class of models possibly containing mean-field interactions (representing an extreme form of long-range interactions). This result is reminiscent of van Hemmen’s work on equilibrium states for mean-field models. The extension was made possible by our recent outcomes on states minimizing the free energy density of mean-field models on the lattice, as well as on the infinite volume dynamics for such models.

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W. de Siqueira Pedra has been supported by CNPq (309723/2020-5) and FAPESP (2017/22340-9), and R. S. Yamaguti by CNPq (140782/2020-6). J.-B. Bru is supported by the Basque Government through the grant IT1615-22 and the BERC 2022-2025 program, by the COST Action CA18232 financed by the European Cooperation in Science and Technology (COST), and by the grant PID2020-112948GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”.

Published 30 October 2023