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

Hiromichi Ohno

Published Apr 1, 2008• 0 citations
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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 $P( \Phi)=\max\{-w(A_{\Phi})+s(w) : w\in S_{\gamma}(\mathcal{A})\}$ $s(w)=inf\{w(A\otimes)+P(\Phi) : \Phi\}$

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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).

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Zero-Variance of Perturbation Hamiltonian Density in Perturbed Spin Systems

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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
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}
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).