ANNALS OF PHYSICS 108, 448-453 (1977) High-Temperature Expansion for the Coulomb Lattice* JOSEPH E. AVRON+.* Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540 Received December 3, 1976 It is shown that the high-temperature expansion for the Coulomb lattice is Tr exp{ -/3H) = d-31*[1 - C&I + C# - C$+ + O(jla log fl)]. C, and C, are the usual Korteweg-de Vries constants of motion. CII reflects the local singularity of the Coulomb potential and is absent for smooth periodic fields. C, turns out to be independent of the details of the potential. In this work, a derivation of the high-temperature expansion for the partition function, Z(p) = Tr exp{ -/IH}, to the third order in /3 is presented. His a one-particle (Bloch) Hamiltonian describing the motion in a periodic medium with local I/] x 1 singularities. This espansion is analologous to the semiclassical Wigner-Kirkwood expansions [l], and its theoretical as well as practical importance is obvious. (Since p = l,.,. - 27.2 eV, the high-order terms in the expansion are negligible only at extremely high temperatures.) This problem has recently attracted considerable attention among mathematicians [2, 31, because the expansion coefficients turn out to be the constants of motion for Korteweg-de Vries flows (and is therefore intimately connected with the inverse scattering problem). However, since applications to solid- state physics were not considered, all existing treatments have assumed smooth potentials. The local l/] x 1 singularity of the Couolomb or Yukawa crystal leads to formal divergencies for all terms higher than p2. As is shown, this is due to the fact that the asymptotic expansion picks nonanalytic terms in higher orders. There is also a perturbation theoretic interest in this problem. (The asymptotic expansion of the partition function is essentially the perturbation series for semi- groups.) In particular, the failure of the usual expansion and the emergence of non- analytic terms strongly resemble the Klauder phenomenon in the perturbation theory of eigenvalues with very strong local singularities [4, 51. (The weak-coupling limit for eignevalues about to be absorbed in the continuum displays similar features [6].) Let H(k) = -A/h + v(x) in L2(Q) with k quasiperiodic boundary conditions (i.e., Y(x + a) = eipaY(x), a E L, x E a@. JJ is the unit cell of the crystal, D = R3/L, and L is the lattice. k is the usual quasimomentum and v is assumed real, v E L2(Q). The spectrum of H(k) is discrete (bands), bounded below, and & / En(k)l-2 < co. * Supported in part by NSF Grant MPS 74-22844. t Wigner fund fellow of the Technische Universitlt, Berlin, West Germany. r On leave of absence from NRCN, I&r-Sheva, Israel. 448 Copyright 0 1977 by Academic Press, Inc. All rights of reproduction in any form reserved. ISSN 000349 16