Advances in Theoretical and Mathematical Physics

Volume 28 (2024)

Number 1

From short-range to mean-field models in quantum lattices

Pages: 69 – 159

DOI: https://dx.doi.org/10.4310/ATMP.2024.v28.n1.a3

Authors

J.-B. Bru (Departamento de Matemáticas and EHU Quantum Center, Universidad del País Vasco, Bilbao, Spain; Basque Center for Applied Mathematics, Bilbao, Spain; and IKERBASQUE, Basque Foundation for Science, Bilbao, Spain)

Pedra W. de Siqueira (Departamento de Matemática, Instituto de Ciências Matemáticas e da Computação, Universidade de São Paulo, São Carlos, SP, Brazil; and Basque Center for Applied Mathematics, Bilbao, Spain)

K. Rodrigues Alves (Basque Center for Applied Mathematics, Bilbao, Spain; Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Bilbao, Spain)

Abstract

Realistic effective interparticle interactions of quantum many-body systems are widely seen as being short-range. However, the rigorous mathematical analysis of this type of model turns out to be extremely difficult, in general, with many important fundamental questions remaining open still nowadays. By contrast, mean-field models come from different approximations or Ansätze, and are thus less realistic, in a sense, but are technically advantageous, by allowing explicit computations while capturing surprisingly well many real physical phenomena. Here, we establish a precise mathematical relation between mean-field and short-range models, by using the long-range limit that is known in the literature as the Kac, or van der Waals, limit. If both attractive and repulsive longrange forces are present then it turns out that the limit mean-field model is not necessarily what one traditionally guesses. One important innovation of our study, in contrast with previous works on the subject, is the fact that we are able to show the convergence of equilibrium states, i.e., of all correlation functions. This paves the way for studying phase transitions, or at least important fingerprints of them like strong correlations at long distances, for models having interactions whose ranges are finite, but very large. It also sheds a new light on mean-field models. Even on the level of pressures, our results go considerably further than previous ones, by allowing, for instance, a continuum of long-range interaction components, as well as very general short-range Hamiltonians for the “free” part of the model. The present results were made possible by the variational approach of $\href{https://doi.org/10.1090/S0065-9266-2012-00666-6}{[1]}$ for equilibrium states of mean-field models, as well as the game theoretical characterization of these states. Our results are obtained in an abstract, model-independent, way.

Published 14 August 2024