Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Korean Mathematical Society

VARIATIONAL PRINCIPLE FOR QUANTUM UNBOUNDED SPIN SYSTEMS

Soonwon Choi, Sung-Ho Jo, H.I. Kim, H.H. Lee, Hyunsik Yoo

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

We study the variational principle for quantum unbounded spin systems interacting via superstable and regular interactions. We show that the (weak) KMS state constructed via the thermodynamic limit of finite volume Green's functions satisfies the Gibbs variational equality.

Related papers

ENTROPY DENSITIES FOR GIBBS STATES OF QUANTUM SPIN SYSTEMS

Fumio Hiai, Dėnes Petz

This paper is a contribution to the general theory of quantum spin systems. We deal with a general theory in the sense that no concrete interaction shows up but an arbitrary relatively short-range interaction is chosen. It is well known that the mean entropy plays an important role in the thermodynamics of quantum spin systems: it is one of the ingredients of the Lanford–Ruelle–Robinson variational principle. We show that in the background of the existence of the mean entropy, there is an operator convergence which resembles the McMillan theorem from information theory. Asymptotic equipartition property and several entropy densities are investigated, in particular, the relation of the Gibbs condition to relative entropies. Although this paper is not intended to be a review, we try to give an overview of the theory of quantum Gibbs states in the mathematical sense. The paper is organized as follows. Section 1 provides an introduction and further sections contain our new results. Each section is closed with a short discussion on sources and references.

Reviews in Mathematical Physics•Dec 1, 1993•14 citations

VARIATIONAL PRINCIPLE FOR NON-EQUILIBRIUM STEADY STATES OF THE XX MODEL

Taku Matsui, Yoshiko Ogata

We show that non-equilibrium steady states of the one-dimensional exactly-solved XX model can be characterized by the variational principle of free energy of a long range interaction and that they cannot be a KMS state for any C*-dynamical system of the UHF algebra.

Reviews in Mathematical Physics•Oct 1, 2003•19 citations

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

J.-B. Bru, W. de Siqueira Pedra, Rafael Sussumu Yamaguti Miada

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.

arXiv:2103.00901•Advances in Theoretical and Mathematical Physics•Jan 1, 2022•1 citation

Entropy Density of One-Dimensional Quantum Lattice Systems

Hajime Moriya

We prove that the entropy density of a KMS state of one-dimensional quantum lattice systems is equal to the thermodynamical limit of the entropy of local Gibbs states.

Reviews in Mathematical Physics•Apr 1, 1997•2 citations

GENERALIZED VARIATIONAL PRINCIPLE IN QUANTUM MECHANICS

Andrey Vladimirovich Soldatov

An algorithm is proposed that allows us to derive the convergent sequence of upper bounds for the ground state energy of a quantum system. The algorithm generalizes the well-known variational principle of quantum mechanics and moreover provides qualitative, and under some additional conditions even quantitative, characteristics of the spectrum of a quantum system as a whole.

International Journal of Modern Physics B•Oct 10, 1995•13 citations

VARIATIONAL PRINCIPLE AND THE DYNAMICAL ENTROPY OF SPACE TRANSLATION

Hajime Moriya

The dynamical entropy of space translations is used in the variational calculus of the pressure. It is shown that for quantum spin systems on a lattice of an arbitrary dimension, the pressure obtained in this way coincides with the usual one.

Reviews in Mathematical Physics•Nov 1, 1999•3 citations
This paper is a contribution to the general theory of quantum spin systems. We deal with a general theory in the sense that no concrete interaction shows up but an arbitrary relatively short-range interaction is chosen. It is well known that the mean entropy plays an important role in the thermodynamics of quantum spin systems: it is one of the ingredients of the Lanford–Ruelle–Robinson variational principle. We show that in the background of the existence of the mean entropy, there is an operator convergence which resembles the McMillan theorem from information theory. Asymptotic equipartition property and several entropy densities are investigated, in particular, the relation of the Gibbs condition to relative entropies. Although this paper is not intended to be a review, we try to give an overview of the theory of quantum Gibbs states in the mathematical sense. The paper is organized as follows. Section 1 provides an introduction and further sections contain our new results. Each section is closed with a short discussion on sources and references.
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.