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The free energy of quantum spin systems and large deviations. (English) Zbl 0657.60041
The paper contains the application of Varadhan’s large deviation principle [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 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 \(I^ j_ 0(u)=I^ j(1)-I^ j(u)\), \[ I^ j(u)=\ln (2j+1)+\sup_{\alpha \geq 0}\{\alpha u-\log \{\sinh (\alpha (2j+1)/2j)/\sinh \alpha /2j\}\} \] is the rate function, calculated by the authors. For the best previous result see V. A. Zagrebnov, Z. Phys. B55, 75-85 (1984).
Reviewer: L.Pastur

MSC:
60F10 Large deviations
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B10 Quantum equilibrium statistical mechanics (general)
82C70 Transport processes in time-dependent statistical mechanics
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