Abstract
We consider quantum unbounded spin systems (lattice boson systems) in ν-dimensional lattice space Zν. Under appropriate conditions on the interactions we prove that in a region of high temperatures the Gibbs state is unique, is translationally invariant, and has clustering properties. The main methods we use are the Wiener integral representation, the cluster expansions for zero boundary conditions and for general Gibbs state, and explicitly β-dependent probability estimates. For one-dimensional systems we show the uniqueness of Gibbs states for any value of temperature by using the method of perturbed states. We also consider classical unbounded spin systems. We derive necessary estimates so that all of the results for the quantum systems hold for the classical systems by straightforward applications of the methods used in the quantum case.
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References
S. Albeverio and R. Høegh-Krohn, Homogeneous random fields and statistical mechanics,J. Funct. Anal. 19:242–272 (1975).
H. Akari, On uniqueness of KMS-states of one-dimensional quantum lattice,Commun. Math. Phys. 44:1–7 (1975).
C. Borgs and R. Waxler, First order phase transition in unbounded spin systems I, II,Commun. Math. Phys. 126:291–324 (1989);126:483–506 (1990).
O. Bratteli and D. W. Robinson,Operator Algebras and Quantum Statistical Mechanics I, II (Springer-Verlag, New York, 1979, 1981).
Ph. Choquard,The Anharmonic Crystals (Benjamin, New York, 1967).
M. Duneau, D. Iagolnitzer, and B. Souillard, Properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems.Commun. Math. Phys. 31:191–208 (1973).
M. Duneau, D. Iagolnitzer, and B. Souillard, Strong cluster properties for classical systems with finite range interaction,Commun. Math. Phys. 35:307–320 (1974).
K. H. Fichtner, Point processes and the position distribution of infinite boson systems,J. Stat. Phys. 47:959–978 (1987).
K. H. Fichtner and W. Freudenberg, Characterization of infinite boson systems, I. On the contruction of states of boson systems,Commun. Math. Phys. 137:315–357 (1991).
W. Freudenberg, Characterization of infinite boson systems, II. On the existence of the conditional reduced density matrix,Commun. Math. Phys. 137:461–472 (1991).
J. Frölich, B. Simon, and T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking,Commun. Math. Phys. 50:79–85 (1976).
H. O. Georgii,Canonical Gibbs Measures (Springer-Verlag, Berlin, 1979).
J. Ginibre, Some applications of functional integration in statistical mechanics, inStatistical Mechanics and Quantum Field Theory, C. Dewitt and R. Stora, eds. (1971), pp. 327–427.
R. B. Israel,Convexity in the Theory of Lattice Gases (Princeton, University Press, Princeton, New Jersey, 1979).
A. Kishimoto, On uniqueness of KMS states of one-dimensional quantum lattice systems,Commun. Math. Phys. 47:167–170 (1976).
Ju. G. Kondratiev, Phase transitions in quantum models of ferroelectrics,BiBos 487 (1991).
H. Kunz, Analyticity and clustering properties of unbounded spin systems,Commun. Math. Phys. 59:53–69 (1976).
O. E. Lanford and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics,Commun. Math. Phys. 13:194–215 (1969).
J. L. Lebowitz and E. Presutti, Statistical mechanics of systems of unbounded spins,Commun. Math. Phys. 50:195–218 (1976).
J. L. Lebowitz and E. Presutti,Commun. Mth. Phys. 78:151 (1980).
E. Nylund, K. Lindenberg, and G. Tsironis, Proton dynamics in hydrogen-bonded systems,J. Stat. Phys. 70:163–181 (1993).
E. Olivieri, P. Picco, and Yu. M. Suhov, On the Gibbs states for one-dimensional lattice boson systems with a long-range interaction,J. Stat. Phys. 70(3/4):985–1028 (1993).
Y. M. Park, The cluster expansion for the classical and quantum lattice systems,J. Stat. Phys. 27(3):553–576 (1982).
Y. M. Park, Quantum statistical mechanics for superstable interactions: Bose-Einstein statistics,J. Stat. Phys. 40:259–302 (1985).
Y. M. Park, Quantum statistical mechanics of unbounded continuous spin systems,J. Korean Math. Soc. 22:43–74 (1985).
Y. M. Park and H. J. Yoo, A characterization of Gibbs states of lattice boson systems,J. Stat. Phys. 75(1/2):215–239 (1994).
S. Pirogov and Ya. G. Sinai, Phase diagram of classical lattice spin systems,Theor. Math. Phys. 25:1185–1192 (1975);26:39–49 (1976).
C. Preston,Random Fields (Springer-Verlag, 1976).
D. Ruelle, Statistical mechanics of a one-dimensional lattice gas,Commun. Math. Phys. 9:267–278 (1968).
D. Ruelle,Statistical Mechanics, Rigorous Results (Benjamin, New York, 1969).
D. Ruelle, Superstable interactions in classical statistical mechanics,Commun. Math. Phys. 18:127–159 (1970).
D. Ruelle, Probability estimates for continuous spin systems,Commun. Math. Phys. 50:189–194 (1976).
B. Simon,Functional Integration and Quantum Physics (Academic Press, New York, 1979).
B. Simon,Trace Ideals and Their Applications (Cambridge University Press, Cambridge, 1979).
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Park, Y.M., Yoo, H.J. Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems. J Stat Phys 80, 223–271 (1995). https://doi.org/10.1007/BF02178359
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DOI: https://doi.org/10.1007/BF02178359