Physica A 184 (1992) 587-597 North-Holland High-temperature expansions for the strongly correlated Falicov-Kimball model Y.S. Yang, C.J. Thompson and A.J. Guttmann Department of Mathematics, The University of Melbourne, Parkville, Victoria 3052, Australia Received 27 November 1991 The possibility of a phase transition in the strongly correlated infinite dimensional spinless Falicov-Kimball model, with hopping interaction scaled by D -l/z, is studied using high- temperature expansions. The Gibbs free energy is expanded to order T -~°. From Pad6 and other methods of analysis, we find no evidence of a non-zero phase transition temperature for any value of the particle density. Our results are consistent with previous analytic predictions. 1. Introduction The nearest neighbour hopping Falicov-Kimball model [1] was proposed as a simple model for a semiconductor-metal transition, based on the existence of both localised (ionic) and band (Bloch) states. The model is described by the following Hamiltonian: N N N Y(= -t Z (d*idj + d~di) + E • nf + U ~ ndinfi -- h ~ ndi , (1.1) (i, j) i=1 i=1 i=1 where d~*, d; and f~, f/are the Fermi creation and annihilation operators for d- and f-electrons at site i respectively and nai ~- dtidi, nh =-f~f are their number operators. The d-electrons are the Bloch electrons with a nearest neighbour hopping term t, and f-electrons represent localised electrons with a binding energy E. d- and f-electrons repel each other with an on-site repulsive energy U. N is the total number of lattice sites. To study the phase transition properties of the model, an "ordering field" h is introduced which resembles the magnetic field. Recently, Brandt and Mielsch suggested [2-5] that, for some parameter values, the model has a non-zero phase transition temperature in high dimen- sions and that in a certain many-body sense the model is exactly soluble in the 0378-4371/92/$05.00 (~ 1992- Elsevier Science Publishers B.V. All rights reserved