The free energy of quantum spin systems and large deviations
type: Item
instance of: scholarly article
title: The free energy of quantum spin systems and large deviations
zbMATH Open document ID: 0657.60041
DOI: 10.1007/BF01218583
author: S. H. Smith
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: The paper contains the application of Varadhan's large deviation principle [\textit{S. R. S. Varadhan}, Commun. Pure Appl. Math. 19, 261-286 (1966; Zbl 0147.155)] to the derivation of the expression for the specific free energy of the spins interacting with a second quantum system. (For an expository account see \textit{J. T. Lewis} [Stochastic mechanics and stochastic processes, Proc. Conf., Swansea/UK 1986, Lect. Notes Math. 1325, 141-155 (1988)]. This principle is a rigorous version of the Laplace (largest term) method. The important constructive ingredient of such an approach is the calculation of the so called rate function I(\(\circ)\) (specific ``entropy''). In this paper it is done for the system of spins. Basing themselves on this result the authors rederive the expression for the free energy of the BCS model in the strong coupling limit as a first application. The second application is given to the Dicke model of \(n(Z_ j+1)\) level atoms (spins) interacting with a quantal radiation field in a region \({\mathcal A}_ n\in {\mathbb{R}}^ d\) of finite volume \(V_ n\). The Hamiltonian of this model is \[ H_ n=H_ n^{(0)}+\epsilon \sum^{n}_{k=1}S^ z_ k+V_ n^{- 1/2}\sum^{n}_{k=1}(a^+(\lambda_ n)+a(\lambda_ n))S^ x_ k, \] where \(H_ n^{(0)}\) is the one-particle (free) boson Hamiltonian which is the second quantized form of an operator \(h_ n\) acting in \(L^ 2({\mathcal A}_ n)\), \(S_ k^{x,z}\) are the x or z components of spin operators, \(\lambda_ n\in L^ 2({\mathcal A}_ n)\), \(a^+(\cdot)\) and a(\(\cdot)\) are boson creation and annihilation operators. If \(\lim_{n\to \infty}\| h_ n^{-1/2}\lambda_ n\| =\Lambda <\infty\), \(\rho =\lim_{n\to \infty}n/V_ n\) and \(f_ 0\) is the free energy determined by the \(H_ n^{(0)}\), then the free energy of the Dicke model is \[ f=f_ 0+\rho \inf_{u\in [0,1]}\{\epsilon jux-\rho \Lambda (ju)^ 2\otimes (1-x^ 2)-\beta^{-1}I^ j_ 0(u)\} \] where \(\beta\) is the inverse temperature and \(I^ j_ 0(u)=I^ j(1)-I^ j(u)\), \[ I^ j(u)=…
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 82C70
zbMATH DE Number: 4074080
zbMATH Keywords: exactly solvable models
zbMATH Keywords: mean-field type models
zbMATH Keywords: large deviation principle
zbMATH Keywords: spins interacting with a second quantum system
zbMATH Keywords: creation and annihilation operators
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4074080