The Polychronakos-Frahm spin chain of BC_N type and Berry-Tabor's conjecture

Abstract

We compute the partition function of the su(m) Polychronakos-Frahm spin chain of BC_N type by means of the freezing trick. We use this partition function to study several statistical properties of the spectrum, which turn out to be analogous to those of other spin chains of Haldane-Shastry type. In particular, we find that when the number of particles is sufficiently large the level density follows a Gaussian distribution with great accuracy. We also show that the distribution of (normalized) spacings between consecutive levels is of neither Poisson nor Wigner type, but is qualitatively similar to that of the original Haldane-Shastry spin chain. This suggests that spin chains of Haldane-Shastry type are exceptional integrable models, since they do not satisfy a well-known conjecture of Berry and Tabor according to which the spacings distribution of a generic integrable system should be Poissonian. We derive a simple analytic expression for the cumulative spacings distribution of the BC_N-type Polychronakos-Frahm chain using only a few essential properties of its spectrum, like the Gaussian character of the level density and the fact the energy levels are equally spaced. This expression is in excellent agreement with the numerical data and, moreover, there is strong evidence that it can also be applied to the Haldane-Shastry and the Polychronakos-Frahm spin chains.

arXiv commentrevTeX, 11 pages, 4 figures

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• An exactly solvable supersymmetric spin chain of BC_N type
• Inozemtsev's hyperbolic spin model and its related spin chain
• On R-matrix valued Lax pairs for Calogero-Moser models
• Thermodynamics of spin chains of Haldane-Shastry type and one-dimensional vertex models
• Long-range level correlations in quantum systems with finite Hilbert space dimension
• Spectral properties of supersymmetric Polychronakos spin chain associated with A_{N-1} root system
• The spin Sutherland model of D_N type and its associated spin chain
• The exactly solvable spin Sutherland model of B_N type and its related spin chain
• A new perspective on the integrability of Inozemtsev's elliptic spin chain
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• Superconformal Quantum Mechanics from M2-branes
• Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions
• Level density distribution for one-dimensional vertex models related to Haldane-Shastry like spin chains
• $1/f^{\alpha }$ noise and integrable systems
• Partition functions of Polychronakos like spin chains associated with polarized spin reversal operators
• Exact solution of $D_N$ type quantum Calogero model through a mapping to free harmonic oscillators
• Appearance of branched motifs in the spectra of $B{C}_N$ type Polychronakos spin chains
• Supersymmetric analogue of BC_N type rational integrable models with polarized spin reversal operators
• The open Haldane-Shastry chain: thermodynamics and criticality
• Rational quantum integrable systems of D_N type with polarized spin reversal operators
• Super Rogers-Szegö polynomials associated with $B{C}_N$ type of Polychronakos spin chains
• Generalized Lipkin-Meshkov-Glick models of Haldane-Shastry type