# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Ellis, R.S.: Entropy, large deviations, and statistical mechanics. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0566.60097 [2] Lewis, J.T.: The large deviation principle in statistical mechanics: an expository account. In: Stochastic mechanics and stochastic processes, Swansea Conference Proceedings 1986. Davies, I., Truman, A. (eds.). Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer (in press) [3] van den Berg, M., Lewis, J.T., Pulè, J.V.: The large deviation principle and some models of an interacting boson gas. Commun. Math. Phys.118, 61–85 (1988) · Zbl 0679.76124 · doi:10.1007/BF01218477 [4] van den Berg, M., Lewis, J.T., Pulè, J.V.: Large deviations and the boson gas. In: Stochastic mechanics and stochastic processes, Swansea Conference Proceedings 1986. Davies, I., Truman, A. (eds.). Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer (in press) · Zbl 0648.60107 [5] Duffield, N.G., Pulè, J.V.: Thermodynamics of the B.C.S. model through large deviations. Lett. Math. Phys.14, 329–331 (1987) · Zbl 0637.58040 · doi:10.1007/BF00402142 [6] Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math.19, 261 (1966) · Zbl 0147.15503 · doi:10.1002/cpa.3160190303 [7] Dixmier, J.:C. Amsterdam, New York, Oxford: North-Holland 1977 [8] Lewis, J.T., Pulè, J.V.: The equivalence of ensembles in statistical mechanics. In: Stochastic analysis and its applications, Swansea Conference Proceedings 1983. Truman, A., Williams, D. (eds.). Lecture Notes in Mathematics, Vol. 1095. Berlin, Heidelberg, New York: Springer 1984 [9] Wada, Y., Takano, F., Fukuda, N.: Exact treatment of Bardeen’s theory of superconductivity in the strong coupling limit. Progr. Theoret. Phys. (Kyoto)19, 597 (1958) · Zbl 0083.45503 · doi:10.1143/PTP.19.597 [10] Thouless, D.J.: Strong-coupling limit in the theory of superconductivity. Phys. Rev.117, 1256 (1960) · Zbl 0091.23204 · doi:10.1103/PhysRev.117.1256 [11] Thirring, W., Wehrl, A.: On the mathematical structure of the B.C.S. model. Commun. Math. Phys.4, 303 (1967) · Zbl 0163.23302 · doi:10.1007/BF01653644 [12] Thouless, D.J.: The quantum mechanics of many-body systems. New York: Academic Press 1961 · Zbl 0103.23502 [13] Lieb, E.H.: The classical limit of quantum spin systems. Commun. Math. Phys.31, 327 (1973) · Zbl 1125.82305 · doi:10.1007/BF01646493 [14] Simon, B.: The classical limit of quantum partition functions. Commun. Math. Phys.71, 247 (1980) · Zbl 0436.22012 · doi:10.1007/BF01197294 [15] Berezin, F.A.: Covariant and contravariant symbols of operators. (Russian) Izv. Akad. Nauk SSSR Ser. Mat.36, 1134 (1972) [English transl.: Math. USSR Izv.6, 117 (1972)] [16] Hepp, K., Lieb, E.H.: Equilibrium statistical mechanics of matter interacting with the quantized radiation field. Phys. Rev. A8, 2517 (1973) [17] Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. II. Berlin, Heidelberg, New York: Springer 1981 · Zbl 0463.46052 [18] Cook, J.M.: Asymptotic properties of a boson field with given source. J. Math. Phys.2, 33 (1961) · doi:10.1063/1.1724210 [19] Lewis, J.T., Raggio, G.A.: Equilibrium thermodynamics of a spin-boson model. J. Stat. Phys.50, 1201–1220 (1988) · Zbl 1084.82507 · doi:10.1007/BF01019161 [20] Fannes, M., Sisson, P.N.M., Verbeure, A., Wolfe, J.C.: Equilibrium state and free energy of an infinite mode Dicke maser model. Ann. Phys.98, 38 (1976) · doi:10.1016/0003-4916(76)90236-0 [21] Zagrebnov, V.A.: The approximating Hamiltonian method for an infinite-mode Dicke maser model withA 2-term. Z. Phys. B55, 75 (1984) · doi:10.1007/BF01307504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.