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

Free energy density for mean field perturbation of states of a one-dimensional spin chain

Fumio Hiai, Milán Mosonyi, Hiromichi Ohno, Dėnes Petz

Published Oct 28, 2009• 15 citations
OpenAlex

Abstract

Motivated by recent developments on large deviations in states of the spin chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the variational expression of free energy density in the presence of a mean field type perturbation. We extend their results from the product state case to the Gibbs state case in the setting of translation-invariant interactions of finite range. In the special case of a locally faithful quantum Markov state, we clarify the relation between two different kinds of free energy densities (or pressure functions).

Related papers

FREE ENERGY DENSITY FOR MEAN FIELD PERTURBATION OF STATES OF A ONE-DIMENSIONAL SPIN CHAIN

FUMIO HIAI, Milán Mosonyi, HIROMICHI OHNO +1 more

Motivated by recent developments on large deviations in states of the spin chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the variational expression of free energy density in the presence of a mean field type perturbation. We extend their results from the product state case to the Gibbs state case in the setting of translation-invariant interactions of finite range. In the special case of a locally faithful quantum Markov state, we clarify the relation between two different kinds of free energy densities (or pressure functions).

arXiv:0706.4148•Reviews in Mathematical Physics•Apr 1, 2008•6 citations

Large deviations and Chernoff bound for certain correlated states on a spin chain

Fumio Hiai, Milán Mosonyi, Tomohiro Ogawa

In this paper we extend the results of Lenci and Rey-Bellet [J. Stat. Phys. 119, 715 (2005)] on the large deviation upper bound of the distribution measures of local Hamiltonians with respect to a Gibbs state in the setting of translation-invariant finite-range interactions. We show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by Gibbs states of the above kind as well as finitely correlated states. As an application of the methods, the Chernoff bound for correlated states with factorization property is studied. In the specific case of the distributions of the ergodic averages of a one-site observable with respect to an ergodic finitely correlated state, the spectral theory of positive maps is applied to prove the full large deviation principle.

arXiv:0706.2141•Journal of Mathematical Physics•Dec 1, 2007•54 citations

Markov states of the quantum electromagnetic field

Luigi Accardi, G. S. Watson

A translation-invariant state (a quantum Markov chain) is associated with a nearest-neighbor interaction on a one-dimensional lattice by a new technique which provides closed forms for all the correlation functions. When applied to an Ising-type perturbation of a chain of harmonic oscillators, the dynamics can be computed explicitly. The resulting translation-invariant distribution is substantially different from the Planck distribution when the temperature and the coupling constant are large. For the evolution of the field operators on a given mode, we obtain a natural nonlinear generalization of the theorem which states that the free evolution of the field operators is obtained by second quantization of the classical free evolution.

Physical Review A•Feb 1, 1987•5 citations

Free energy density for mean field perturbation of states of a one-dimensional spin chain (Inequalities on Linear Operators and its Applications)

博道 大野

Free energy density for mean field perturbation of states of a one-dimensional spin chain九州大学数理学研究院 大野博道 本研究では, スピン系 (UHF 環) 上のシフト不変な状態の, 平均エントロピーや 圧力関数についての考察を行う.まず, 必要な事柄をいくつか定義する.S(w_{n})\underline{1}$ .$\mathfrak{n}arrow\infty n$ ただし, $s(w_{n})=\omega_{n}(-\log_{D}(\omega_{n}))$ である.すると圧力関数と平均エントロピーの間 に変分表現式が成り立つ.すなわち $P( \Phi)=\max\{-w(A_{\Phi})+s(w) : w\in S_{\gamma}(\mathcal{A})\}$ $s(w)=inf\{w(A\otimes)+P(\Phi) : \Phi\}$

数理解析研究所講究録•Apr 1, 2008•0 citations

Free energy density for mean field perturbation of states of a one-dimensional spin chain (Inequalities on Linear Operators and its Applications)

Hiromichi Ohno

•Apr 1, 2008•0 citations

Quantum Large Deviations

T. C. Dorlas

We reconsider the quantum analogue of Varadhans Theorem proved by Petz, Raggio and Verbeure. They proved this theorem using standard techniques in quantum statistical mechanics of lattice systems to arrive at a variational formula over states on an operator algebra, which can subsequently be reduced to a variational formula in terms of a single real variable. In this paper a new proof is given using a quantum version of the large deviation analysis together with the Trotter product formula. The proof is subsequently extended to the general case of q non-commuting variables resulting in a variational formula for general mean-field quantum spin systems as first derived by Raggio and Werner.

arXiv:2511.01686•Nov 3, 2025•0 citations
Motivated by recent developments on large deviations in states of the spin chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the variational expression of free energy density in the presence of a mean field type perturbation. We extend their results from the product state case to the Gibbs state case in the setting of translation-invariant interactions of finite range. In the special case of a locally faithful quantum Markov state, we clarify the relation between two different kinds of free energy densities (or pressure functions).
Motivated by recent developments on large deviations in states of the spin chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the variational expression of free energy density in the presence of a mean field type perturbation. We extend their results from the product state case to the Gibbs state case in the setting of translation-invariant interactions of finite range. In the special case of a locally faithful quantum Markov state, we clarify the relation between two different kinds of free energy densities (or pressure functions).
In this paper we extend the results of Lenci and Rey-Bellet [J. Stat. Phys. 119, 715 (2005)] on the large deviation upper bound of the distribution measures of local Hamiltonians with respect to a Gibbs state in the setting of translation-invariant finite-range interactions. We show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by Gibbs states of the above kind as well as finitely correlated states. As an application of the methods, the Chernoff bound for correlated states with factorization property is studied. In the specific case of the distributions of the ergodic averages of a one-site observable with respect to an ergodic finitely correlated state, the spectral theory of positive maps is applied to prove the full large deviation principle.
A translation-invariant state (a quantum Markov chain) is associated with a nearest-neighbor interaction on a one-dimensional lattice by a new technique which provides closed forms for all the correlation functions. When applied to an Ising-type perturbation of a chain of harmonic oscillators, the dynamics can be computed explicitly. The resulting translation-invariant distribution is substantially different from the Planck distribution when the temperature and the coupling constant are large. For the evolution of the field operators on a given mode, we obtain a natural nonlinear generalization of the theorem which states that the free evolution of the field operators is obtained by second quantization of the classical free evolution.
Free energy density for mean field perturbation of states of a one-dimensional spin chain九州大学数理学研究院 大野博道 本研究では, スピン系 (UHF 環) 上のシフト不変な状態の, 平均エントロピーや 圧力関数についての考察を行う.まず, 必要な事柄をいくつか定義する.S(w_{n})\underline{1}$ .narrow∞n\mathfrak{n}arrow\infty nnarrow∞n ただし, s(wn)=ωn(−log⁡D(ωn))s(w_{n})=\omega_{n}(-\log_{D}(\omega_{n}))s(wn​)=ωn​(−logD​(ωn​)) である.すると圧力関数と平均エントロピーの間 に変分表現式が成り立つ.すなわち P(Φ)=max⁡{−w(AΦ)+s(w):w∈Sγ(A)}P( \Phi)=\max\{-w(A_{\Phi})+s(w) : w\in S_{\gamma}(\mathcal{A})\}P(Φ)=max{−w(AΦ​)+s(w):w∈Sγ​(A)} s(w)=inf{w(A⊗)+P(Φ):Φ}s(w)=inf\{w(A\otimes)+P(\Phi) : \Phi\}s(w)=inf{w(A⊗)+P(Φ):Φ}
We reconsider the quantum analogue of Varadhans Theorem proved by Petz, Raggio and Verbeure. They proved this theorem using standard techniques in quantum statistical mechanics of lattice systems to arrive at a variational formula over states on an operator algebra, which can subsequently be reduced to a variational formula in terms of a single real variable. In this paper a new proof is given using a quantum version of the large deviation analysis together with the Trotter product formula. The proof is subsequently extended to the general case of q non-commuting variables resulting in a variational formula for general mean-field quantum spin systems as first derived by Raggio and Werner.