Spectral self-similarity,one-dimensional Ising chains and random matrices |
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| Institution: | 1. Princeton University, Department of Physics, Joseph Henry Laboratories, Jadwin Hall, P.O. Box 708, Princeton, New Jersey 08544-0708, USA;2. Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, Russia;1. Plant Sciences Department, University of Saskatchewan, 51 Campus Drive, Saskatoon, SK, Canada S7N 5A8;2. National Research Council of Canada, 110 Gymnasium Place, Saskatoon, SK, Canada S7N 0W9;3. Department of Biology, University of Saskatchewan, 112 Science Place, Saskatoon, SK, Canada S7N 5E2;4. Department of Biological Sciences, University of Manitoba, 50 Sifton Road, Winnipeg, MB, Canada R3T 2N2;5. Department of Botany, The University of British Columbia, #3529-6270 University Boulevard, Vancouver, BC, Canada V6T 1Z4;1. Department of Applied Mathematics, School of Mathematical Sciences, Tel Aviv University, P.O. Box 39040, Tel Aviv 6997801, Israel;2. Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel |
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| Abstract: | Partition functions of one-dimensional Ising chains with specific long distance exchange between N spins are connected to the N-soliton τ-functions of the Korteweg-de Vries (KdV) and B-type Kadomtsev-Petviashvili (BKP) integrable equations. The condition of translational invariance of the spin lattice selects infinite-soliton solutions with soliton amplitudes forming a number of geometric progressions. The KdV equation generates a spin chain with exponentially decaying antiferromagnetic exchange. The BKP case is richer. It comprises both ferromagnets and anti ferromagnets and, as a special case, includes an exchange decaying as 1/(i ? j)2 for large |i ? j|. The corresponding partition functions are calculated exactly for a homogeneous magnetic field and some fixed values of the temperature. The connection between these Ising chains and random matrix models is considered as well. A short account of the basic ideas underlying the present work has been published in JETP Lett. 66 (1997) 789. |
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