Spectral self-similarity, one-dimensional Ising chains and random matrices

Partition functions of one-dimensional Ising chains with specific long dist ance exchange between N spins are connected to the N-soliton tau-functions of the Korteweg-de Vries (KdV) and B-type Kadomtsev-Petviashvili (BKP) inte grable equations. The condition of translational invariance of the spin lat tice selects infinite-soliton solutions with soliton amplitudes forming a n umber of geometric progressions. The KdV equation generates a spin chain wi th exponentially decaying antiferromagnetic exchange. The BKP case is riche r. It comprises both ferromagnets and antiferromagnets and, as a special ca se, includes an exchange decaying as 1/(i - j)(2) for large /i - j/. The co rresponding partition functions are calculated exactly for a homogeneous ma gnetic field and some fixed values of the temperature. The connection betwe en these Ising chains and random matrix models is considered as well. A sho rt account of the basic ideas underlying; the present work has been publish ed in JETP Lett. 66 (1997) 789. (C) 1999 Elsevier Science B.V.