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数理解析研究所講究録

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

Published Apr 1, 2008• 0 citations
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Abstract

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,ただし,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}$
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).
We present analytical results for the strongly anisotropic random-field Ising model, consisting of weakly interacting spin chains. We combine the mean-field treatment of interchain interactions with an analytical calculation of the average chain free energy (``chain mean-field'' approach). The free energy is found using a mapping on a Brownian motion model. We calculate the order parameter and give expressions for the critical random magnetic-field strength below which the ground state exhibits long-range order and for the critical temperature as a function of the random magnetic-field strength. In the limit of vanishing interchain interactions, we obtain corrections to the zero-temperature estimate by Imry and Ma [Phys. Rev. Lett. 35, 1399 (1975)] of the ground-state density of domain walls (spin flips) in the one-dimensional random-field Ising model. One of the problems to which our model has direct relevance is the lattice dimerization in disordered quasi-one-dimensional Peierls materials, such as the conjugated polymer trans-polyacetylene.
The authors present an exact solution for the one-dimensional Ising model in a random field. The distribution of field strengths is site independent, symmetric and has a three-peak structure. The free energy is obtained for all finite temperatures. The low-temperature behaviour is studied in detail. They find an expansion for the free energy in integer powers of temperature. The ground-state energy and the zero-point entropy are calculated explicitly. The specific heat is linear for low temperatures, in contradiction with mean-field theory. The origin of an additional exponential parameter is discussed.
We consider a one-dimensional Ising model (chain) with the the nearest-neighbor interaction and with a random nonmagnetic dilution. We find the exact free energy of such a chain as a function of the impurity concentration, temperature, and the external magnetic field. In the case of antiferromagnetic interaction in the chain, we find the specific magnetization, the mean value of the product of neighboring spins, and the entropy as functions of these parameters. We study the residual system entropy.