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arXiv:0803.0922 · Physical Review B

Polychronakos-Frahm spin chain of BCN type and the Berry-Tabor conjecture

J. C. Barba, Federico Finkel, Artemio González-López, Miguel A. Rodrı́guez

Published Jun 17, 2008• 30 citations
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Abstract

We compute the partition function of the $\text{su}(m)$ Polychronakos-Frahm spin chain of $B{C}_{N}$ 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 $B{C}_{N}$-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.

DOI
10.1103/physrevb.77.214422

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We compute the partition function of the su(m)\text{su}(m)su(m) Polychronakos-Frahm spin chain of BCNB{C}_{N}BCN​ 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 BCNB{C}_{N}BCN​-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.
We construct a new exactly solvable supersymmetric spin chain related to the BCN extended root system, which includes as a particular case the BCN version of the Polychronakos–Frahm spin chain. We also introduce a supersymmetric spin dynamical model of Calogero type which yields the new chain in the large coupling limit. This connection is exploited to derive two different closed-form expressions for the chain's partition function by means of Polychronakos's freezing trick. We establish a boson–fermion duality relation for the new chain's spectrum, which is in fact valid for a large class of (not necessarily integrable) spin chains of BCN type. The exact expressions for the partition function are also used to study the chain's spectrum as a whole, showing that the level density is normally distributed even for a moderately large number of particles. We also determine a simple analytic approximation to the distribution of normalized spacings between consecutive levels which fits the numerical data with remarkable accuracy. Our results provide further evidence that spin chains of Haldane–Shastry type are exceptional integrable models, in the sense that their spacings distribution is not Poissonian, as posited by the Berry–Tabor conjecture for “generic” quantum integrable systems.
According to a long-standing conjecture of Berry and Tabor, the distribution of the spacings between consecutive levels of a "generic" integrable model should follow Poisson's law. In contrast, the spacings distribution of chaotic systems typically follows Wigner's law. An important exception to the Berry-Tabor conjecture is the integrable spin chain with long-range interactions introduced by Haldane and Shastry in 1988, whose spacings distribution is neither Poissonian nor of Wigner's type. In this letter we argue that the cumulative spacings distribution of this chain should follow the "square root of a logarithm" law recently proposed by us as a characteristic feature of all spin chains of Haldane-Shastry type. We also show in detail that the latter law is valid for the rational counterpart of the Haldane-Shastry chain introduced by Polychronakos.
According to a long-standing conjecture of Berry and Tabor, the distribution of the spacings between consecutive levels of a “generic” integrable model should follow Poisson's law. In contrast, the spacings distribution of chaotic systems typically follows Wigner's law. An important exception to the Berry-Tabor conjecture is the integrable spin chain with long-range interactions introduced by Haldane and Shastry in 1988, whose spacings distribution is neither Poissonian nor of Wigner's type. In this letter we argue that the cumulative spacings distribution of this chain should follow the “square root of a logarithm” law recently proposed by us as a characteristic feature of all spin chains of Haldane-Shastry type. We also show in detail that the latter law is valid for the rational counterpart of the Haldane-Shastry chain introduced by Polychronakos.
We derive an exact expression for the partition function of the su(m)\mathrm{su}(m)su(m) Haldane-Shastry spin chain, which we use to study the density of levels and the distribution of the spacing between consecutive levels. Our computations show that when the number of sites NNN is large enough, the level density is Gaussian to a very high degree of approximation. More surprisingly, we also find that the nearest-neighbor spacing distribution is not Poissonian, so that this model departs from the typical behavior for an integrable system. We show that the cumulative spacing distribution of the model can be well approximated by a simple functional law involving only three parameters.
We derive the exact spectra as well as partition functions for a class of BC N type of spin Calogero models, whose Hamiltonians are constructed by using supersymmetric analogues of polarized spin reversal operators (SAPSRO). The strong coupling limit of these spin Calogero models yields BC N type of Polychronakos–Frahm (PF) spin chains with SAPSRO. By applying the freezing trick, we obtain an exact expression for the partition functions of such PF spin chains. We also derive a formula which expresses the partition function of any BC N type of PF spin chain with SAPSRO in terms of partition functions of several A K types of supersymmetric PF spin chains, where K ⩽ N − 1 . Subsequently we show that an extended boson–fermion duality relation is obeyed by the partition functions of the BC N type of PF chains with SAPSRO. Some spectral properties of these spin chains, like level density distribution and nearest neighbor spacing distribution, are also studied.