Abstract
We prove a Goldstone-type theorem for a wide class of lattice and continuum quantum systems, both for the ground state and at nonzero temperature. For the ground state (T=0) spontaneous breakdown of a continuous symmetry implies no energy gap. For nonzero temperature, spontaneous symmetry breakdown implies slow clustering (noL 1 clustering). The methods apply also to nonzero-temperature classical systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Goldstone, A. Salam, and S. Weinberg,Phys. Rev. 127:965 (1972).
S. A. Bludman and A. Klein,Phys. Rev. 131:2364 (1964).
H. Araki, K. Hepp, and D. Ruelle,Helv. Phys. Acta 35:164 (1962).
D. Kastler, D. W. Robinson, and J. A. Swieca,Commun. Math. Phys. 2:108 (1966).
H. Ezawa and J. A. Swieca,Commun. Math. Phys. 5:330 (1967).
R. F. Streater,Commun. Math. Phys. 6:233 (1967).
H. Reeh,Nuovo Cimento B17:7 (1973).
W. F. Wreszinski,J. Math. Phys. 17:109 (1976).
W. Driessler, L. Landau, and J. F. Perez,J. Stat. Phys. 20:123 (1979).
M. Hammermesh,Group Theory (Addison-Wesley, New York, 1962).
J. A. Swieca,Commun. Math. Phys. 4:1 (1967).
R. Haag, D. Kastler, and E. B. Trych-Pohlmeyer,Commun. Math. Phys. 38:173 (1974).
R. Haag and E. B. Trych-Pohlmeyer,Commun. Math. Phys. 56:213 (1977).
Ch. Gruber and Ph. A. Martin,Phys. Rev. Lett. 45:853 (1980).
Ch. Gruber and Ph. A. Martin,Ann. Phys. 131:56 (1981).
Author information
Authors and Affiliations
Additional information
Partial financial support by Fundação de Amparo à Pesquisa do Estado de São Paulo.
Partial financial support by CNPq.
Rights and permissions
About this article
Cite this article
Landau, L., Perez, J.F. & Wreszinski, W.F. Energy gap, clustering, and the Goldstone theorem in statistical mechanics. J Stat Phys 26, 755–766 (1981). https://doi.org/10.1007/BF01010937
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01010937