Polychronakos-Frahm spin chain ofBCNtype and the Berry-Tabor conjecture

Abstract

We compute the partition function of the su͑m͒ 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 ͑normalized͒ 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 such as the Gaussian character of the level density and the fact that the energy levels are equally spaced. This expression is shown to be in excellent agreement with the numerical data.

The spin Sutherland model of type and its associated spin chain

“…In fact, this simplified formula turns out to be quite efficient for the numerical computation of the chain's spectrum, making it possible to perform a statistical analysis of the spectrum when the number of particles becomes very large. It would be worthwhile to carry out such a study, and compare its results with the corresponding ones for other spin chains of HS type [37,43,44,[56][57][58][59].…”

“…However, for completeness' sake we shall next provide an elementary proof of this fact. We first note that, since the set P (C) of complex-valued continuous functions periodic in C contains the dense set C 0 (C), to prove that (42) is a basis of L 2 (C) we need only show that every f ∈ P (C) can be uniquely represented by a Fourier series of the form (43). Let, then, f : C → C be a continuous function periodic in C, and denote bȳ f : R N → C its T-periodic extension.…”

“…Indeed, Olshanetsky and Perelomov established the existence of an underlying A N root system structure for both the spinless Calogero and Sutherland models, and constructed generalizations thereof associated with any (extended) root system [4]. Spin generalizations of the BC N Calogero and Sutherland models have also been proposed, and various properties of the related lattice models of HS type have been studied with the help of the freezing trick [38][39][40][41][42][43]. Among the other classical root systems, the exceptional ones are comparatively less interesting, since their associated models consist of at most 8 particles.…”

The exactly solvable spin Sutherland model of type and its related spin chain

“…In the following discussion, we shall label the basis functions ϕ (δ) n simply by ϕ ν , with ν defined by (43). As in Ref.…”

“…Spin generalizations of the BC N Calogero-Sutherland model have been extensively studied in the last few years, and various properties of their related spin chains of HS type have been analyzed with the help of the freezing trick [37][38][39][40][41][42][43][44]. Among the other classical root systems, the exceptional ones are comparatively less interesting in this context, since their associated models consist of at most 8 particles.…”