Abstract
We consider models of independent itinerant fermions interacting with classical continuous or discrete variables (spins), the static Holstein model being a special case. We prove for all values of the fermion-spin coupling and a special value of the fermion chemical potential and classical magnetic field, at which the average fermion density is one-half and the average total spin is zero, that there are two degenerate ground states of period two with antiferromagnetic order for the spins and fermions. The existence of two corresponding low-temperature phases is proven for large coupling and dimension two or more by using a Peierls argument. This generalizes results of Kennedy and Lieb for the Falicov-Kimball model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Aubry, inMicroscopic Aspects of Non Linearity in Condensed Matter Physics, A. R. Bishop, V. L. Pokrovsky, and V. Tognetti, eds. (Plenum Press, New York, 1991), pp. 105–111.
S. Aubry, G. Abramovici, and J. L. Raimbault,J. Stat. Phys. 67:675 (1992).
S. A. Brazovskii, E. Dzyaloshinskii, and I. M. Krichever,Sov. Phys. JETP 56:212 (1982).
H. J. Brascamp and E. H. Lieb, inFunctional Integration and Its Applications, A. M. Arthurs, ed. (Clarendon Press, Oxford, 1975), Chapter 1.
Ph. Choquard,The Anharmonic Crystal (Benjamin, New York, 1967).
J. M. Combes and L. Thomas,Commun. Math. Phys. 34:251 (1973).
J. F. Freericks and L. M. Falicov,Phys. Rev. B 41:2163 (1990).
L. M. Falicov and J. C. Kimball,Phys. Rev. Lett. 22:957 (1967).
J. K. Freericks and E. H. Lieb, The ground state of a general electron-phonon Hamiltonian is a spin singlet, preprint (1994).
G. Gallavotti, private communication.
Ch. Gruber,Helv. Phys. Acta 64:668 (1991).
G. Gallavotti and J. L. Lebowitz,J. Math. Phys. 12:1129 (1971).
Ch. Gruber, J. L. Lebowitz, and N. Macris,Phys. Rev. B 48:4312 (1993).
C. Gruber, D. Ueltschi, and J. Jedrzejewski,J. Stat. Phys. 76:125 (1994).
T. Holstein,Ann. Phys. 8:325 (1959).
T. Kato,Perturbation Theory for Linear Operators (Springer-Verlag, New York, 1966), Chapter 2, p. 282.
T. Kennedy, Some rigorous results on the ground states of the Falicov-Kimball model,Rev. Math. Phys., to appear (1994).
T. Kennedy and E. H. Lieb,Physica 138:320 (1986).
T. Kennedy and E. H. Lieb,Phys. Rev. Lett. 59:1309 (1987).
P. Lemberger,J. Phys. A 25:715 (1992).
E. H. Lieb and M. Loss,Duke Math. J. 71:337–363 (1993).
J. L. Lebowitz and N. Macris, Long range order in the Falicov-Kimball model near the symmetry point: extension of Kennedy-Lieb theorem,Rev. Math. Phys., to appear (1994).
R. Peierls,Quantum Theory of Solids (Oxford University Press, Oxford, 1974).
J. V. Pulé, A. Verbeure, and V. A. Zagrebnov,J. Stat. Phys. 76:159 (1994).
P. G. Van Dongen and D. Vollhardt,Phys. Rev. Lett. 65:1663 (1990).
U. Brandt, R. Schmidt,Z. Phys. B 63:45 (1986).
Author information
Authors and Affiliations
Additional information
Dedicated to Philippe Choquard on the occasion of his 65th birthday.
Rights and permissions
About this article
Cite this article
Lebowitz, J.L., Macris, N. Low-temperature phases of itinerant fermions interacting with classical phonons: The static Holstein model. J Stat Phys 76, 91–123 (1994). https://doi.org/10.1007/BF02188657
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02188657