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 Ramanujan Mathematical Society

Uniqueness of Equilibrium State for A Quantum Spin System On a Graph

Stephen Dias Barreto

Published Jun 1, 2012• 0 citations
OpenAlex

Abstract

In this paper we study a quantum spin glass [3,17] as a quantum spin system on an infinite graph. We construct the global dynamics for the quantum spin system on an infinite graph with deterministic nearest neighbour type of interactions. As expected, at a fixed temperature T , the equilibrium state [6,7,12] satisfies the kms [14,18,12] condition and is invariant under the action of the time evolution group. Furthermore, it is shown that the equilibrium state is unique if additional assumptions are made regarding the structure of the graph. Such results have been proved in the past mainly for translation invariant interactions on lattices, see [1,2,8,13]. More recently Matsui has established the uniqueness of the the Gibbs States for quantum spin models on lattices based on the ergodicity of Markov semigroups through the purification of quantum Gibbs states [10]. In our case the nearest neighbour type of interaction we have considered is more general than similar type of interactions considered in some earlier works referred to above.

Related papers

A note on adiabatic time evolution and quasi-static processes in translation-invariant quantum systems

Jak\v{s}i\'c, Vojkan, Claude‐Alain Pillet, Clément Tauber

We study the slowly varying, non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice $\mathbb Z^d$. This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in [Jak\v{s}i\'c, Pillet, Tauber, arXiv:2204.00440], we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work.

arXiv:2204.02177•Apr 5, 2022•0 citations

Approach to Equilibrium in Translation-Invariant Quantum Systems: Structural Results II

Vojkan Jakšić, Claude‐Alain Pillet, Anna Szczepanek

We continue the development, started in [AHP 25 (2023), 715-749], of a structural theory of approach to equilibrium in translation-invariant quantum spin systems. Dynamical Steady States (DSS) are defined as w*-limit points of the time-averaged evolution of an initial state. We show that if a DSS is an equilibrium state of a sufficiently regular interaction, then the specific energy of this interaction is necessarily a constant of the motion. Two distinct mechanisms lead to this conclusion, one based on dynamical surface energies and the other on conservation of specific relative entropy. We conjecture that these properties hold for a natural class of interactions. For a DSS admitting a physical equilibrium description, the associated equilibrium states are characterized by a dynamical Gibbs variational principle: they are precisely the minimizers of the specific relative entropy from the initial state over the physical equilibrium states. The achieved minimum equals the increase of specific entropy. The structural theory also yields characterizations of approach to thermal equilibrium, based on two model-dependent questions: restoration of a sufficiently local thermodynamic description at large times and classification of the corresponding local constants of motion. Once locality is restored, the constants of motion determine the possible asymptotic equilibrium states. In particular, if finite-range locality is restored and the energy is the only finite-range constant of the motion, then every DSS is a thermal equilibrium state at the temperature fixed by energy conservation. The finite-time dynamical input underlying this theory is established in a companion paper.

arXiv:2610.21475•Oct 15, 2026•0 citations

Extremal States of Multipartite Quantum Spin Systems

Oskar Olander

We consider two quantum spin models on $k$-partite graphs. The system is assumed to be invariant under all permutations within any of the $k$ sets. In the limit of infinitely many particles, the Gibbs state is described as a mixture of independent spins due to a version of the Quantum de Finetti theorem. First, we study a spin-1/2 antiferromagnetic Heisenberg model and identify its three phases. At low temperature all $k$ sets are magnetised but cancel each other, at medium temperature there is a net magnetic field and at high temperature there is no magnetisation. The magnetisation temperature is a solution to a $k$-degree polynomial. Second, we allow a general spin $s$ and identify all orthogonally invariant models which have frustration. The criterium for a frustration-free model is that the $k$-sets can be partitioned into two parts, where the only antiferromagnetic interaction is between the two parts.

arXiv:2609.12873•Sep 11, 2026•0 citations

Dynamics and phase transitions for a continuous system of quantum particles in a box

Guy Battle

A particular type of continuous quantum system with infinitely many particles is analyzed, and the existence of dynamics is proven in the GNS representations of certain states. The dynamics is not a group of automorphisms on the original algebra, so equilibrium states are defined in terms of the KMS condition in the representations of the states. The basic theorems about KMS states do not apply here. Nevertheless, for a special class of interactions it is proven that the central decomposition of an equilibrium state is concentrated on a Borel set of equilibrium factor states and that such factor states are precisely the extremal equilibrium states. Furthermore, the equilibrium factor states are in one-to-one correspondence with sets of functions satisfying a certain system of trace equations. This explicit correspondence is then used to show that there are no phase transitions for high temperature, and an example of a phase transition is constructed for low temperature. The phase transition also provides an example of continuous symmetry breaking.

Journal of Mathematical Physics•Jan 1, 1978•3 citations

Phase diagrams of quantum spin glass models

Ferenc Pázmándi, Zbigniew Domański

Infinite-range quantum spin glass models with arbitrary spin are studied within the Matsubara time formalism. As an example the dynamic spin self-interaction R(t, t') and the resulting phase diagrams are presented for the quantum Ising model in a transverse field. Some results, such as the phase transition temperatures, are also reported for the XY and the Heisenberg models.

Journal of Physics Condensed Matter•Mar 1, 1993•6 citations

EQUILIBRIUM STATES AND THEIR ENTROPY DENSITIES IN GAUGE-INVARIANT C*-SYSTEMS

Nobuyuki AKIHO, Fumio Hiai, Dėnes Petz

A gauge-invariant C*-system is obtained as the fixed point subalgebra of the infinite tensor product of full matrix algebras under the tensor product unitary action of a compact group. In this paper, thermodynamics is studied in such systems and the chemical potential theory developed by Araki, Haag, Kastler and Takesaki is used. As a generalization of quantum spin system, the equivalence of the KMS condition, the Gibbs condition and the variational principle is shown for translation-invariant states. The entropy density of extremal equilibrium states is also investigated in relation to macroscopic uniformity.

arXiv:math/0409601•Reviews in Mathematical Physics•May 1, 2005•3 citations
In this paper we study a quantum spin glass [3,17] as a quantum spin system on an infinite graph. We construct the global dynamics for the quantum spin system on an infinite graph with deterministic nearest neighbour type of interactions. As expected, at a fixed temperature T , the equilibrium state [6,7,12] satisfies the kms [14,18,12] condition and is invariant under the action of the time evolution group. Furthermore, it is shown that the equilibrium state is unique if additional assumptions are made regarding the structure of the graph. Such results have been proved in the past mainly for translation invariant interactions on lattices, see [1,2,8,13]. More recently Matsui has established the uniqueness of the the Gibbs States for quantum spin models on lattices based on the ergodicity of Markov semigroups through the purification of quantum Gibbs states [10]. In our case the nearest neighbour type of interaction we have considered is more general than similar type of interactions considered in some earlier works referred to above.
We study the slowly varying, non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice Zd\mathbb Z^dZd. This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in [Jak\v{s}i\'c, Pillet, Tauber, arXiv:2204.00440], we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work.
We continue the development, started in [AHP 25 (2023), 715-749], of a structural theory of approach to equilibrium in translation-invariant quantum spin systems. Dynamical Steady States (DSS) are defined as w*-limit points of the time-averaged evolution of an initial state. We show that if a DSS is an equilibrium state of a sufficiently regular interaction, then the specific energy of this interaction is necessarily a constant of the motion. Two distinct mechanisms lead to this conclusion, one based on dynamical surface energies and the other on conservation of specific relative entropy. We conjecture that these properties hold for a natural class of interactions. For a DSS admitting a physical equilibrium description, the associated equilibrium states are characterized by a dynamical Gibbs variational principle: they are precisely the minimizers of the specific relative entropy from the initial state over the physical equilibrium states. The achieved minimum equals the increase of specific entropy. The structural theory also yields characterizations of approach to thermal equilibrium, based on two model-dependent questions: restoration of a sufficiently local thermodynamic description at large times and classification of the corresponding local constants of motion. Once locality is restored, the constants of motion determine the possible asymptotic equilibrium states. In particular, if finite-range locality is restored and the energy is the only finite-range constant of the motion, then every DSS is a thermal equilibrium state at the temperature fixed by energy conservation. The finite-time dynamical input underlying this theory is established in a companion paper.
We consider two quantum spin models on kkk-partite graphs. The system is assumed to be invariant under all permutations within any of the kkk sets. In the limit of infinitely many particles, the Gibbs state is described as a mixture of independent spins due to a version of the Quantum de Finetti theorem. First, we study a spin-1/2 antiferromagnetic Heisenberg model and identify its three phases. At low temperature all kkk sets are magnetised but cancel each other, at medium temperature there is a net magnetic field and at high temperature there is no magnetisation. The magnetisation temperature is a solution to a kkk-degree polynomial. Second, we allow a general spin sss and identify all orthogonally invariant models which have frustration. The criterium for a frustration-free model is that the kkk-sets can be partitioned into two parts, where the only antiferromagnetic interaction is between the two parts.
A particular type of continuous quantum system with infinitely many particles is analyzed, and the existence of dynamics is proven in the GNS representations of certain states. The dynamics is not a group of automorphisms on the original algebra, so equilibrium states are defined in terms of the KMS condition in the representations of the states. The basic theorems about KMS states do not apply here. Nevertheless, for a special class of interactions it is proven that the central decomposition of an equilibrium state is concentrated on a Borel set of equilibrium factor states and that such factor states are precisely the extremal equilibrium states. Furthermore, the equilibrium factor states are in one-to-one correspondence with sets of functions satisfying a certain system of trace equations. This explicit correspondence is then used to show that there are no phase transitions for high temperature, and an example of a phase transition is constructed for low temperature. The phase transition also provides an example of continuous symmetry breaking.
A gauge-invariant C*-system is obtained as the fixed point subalgebra of the infinite tensor product of full matrix algebras under the tensor product unitary action of a compact group. In this paper, thermodynamics is studied in such systems and the chemical potential theory developed by Araki, Haag, Kastler and Takesaki is used. As a generalization of quantum spin system, the equivalence of the KMS condition, the Gibbs condition and the variational principle is shown for translation-invariant states. The entropy density of extremal equilibrium states is also investigated in relation to macroscopic uniformity.