Inozemtsev's hyperbolic spin model and its related spin chain

Physical Review B, 2005

We derive an exact expression for the partition function of the 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 N is large enough the level density is Gaussian to a very high degree of approximation. More surprisingly, we also find that the nearestneighbor 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.

Nuclear Physics B, 2005

We introduce four types of SU(2M + 1) spin chains which can be regarded as the BC N versions of the celebrated Haldane-Shastry chain. These chains depend on two free parameters and, unlike the original Haldane-Shastry chain, their sites need not be equally spaced. We prove that all four chains are solvable by deriving an exact expression for their partition function using Polychronakos's "freezing trick". From this expression we deduce several properties of the spectrum, and advance a number of conjectures that hold for a wide range of values of the spin M and the number of particles. In particular, we conjecture that the level density is Gaussian, and provide a heuristic derivation of general formulas for the mean and the standard deviation of the energy.

EPL (Europhysics Letters), 2008

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.

Annals of Physics, 2014

The aim of this paper is studying from an alternative point of view the integrability of the spin chain with long-range elliptic interactions introduced by Inozemtsev. Our analysis relies on some well-established conjectures characterizing the chaotic vs. integrable behavior of a quantum system, formulated in terms of statistical properties of its spectrum. More precisely, we study the distribution of consecutive levels of the (unfolded) spectrum, the power spectrum of the spectral fluctuations, the average degeneracy, and the equivalence to a classical vertex model. Our results are consistent with the general consensus that this model is integrable, and that it is closer in this respect to the Heisenberg chain than to its trigonometric limit (the Haldane-Shastry chain). On the other hand, we present some numerical and analytical evidence showing that the level density of Inozemtsev's chain is asymptotically Gaussian as the number of spins tends to infinity, as is the case with the Haldane-Shastry chain. We are also able to compute analytically the mean and the standard deviation of the spectrum, showing that their asymptotic behavior coincides with that of the Haldane-Shastry chain.

Physical Review E, 2010

We provide a rigorous proof of the fact that the level density of all su(m) spin chains of Haldane-Shastry type associated with the AN−1 root system approaches a Gaussian distribution as the number of spins N tends to infinity. Our approach is based on the study of the large N limit of the characteristic function of the level density, using the description of the spectrum in terms of motifs and the asymptotic behavior of the dispersion relation.

Physical Review E, 2009

We show that the density of energy levels of a wide class of finite-dimensional quantum systems tends to a Gaussian distribution as the number of degrees of freedom increases. Our result is based on a nontrivial modification of the classical central limit theorem, and is especially suited to models whose partition function is explicitly known. In particular, we provide the first theoretical explanation of the fact that the level density of several spin chains of Haldane-Shastry type is asymptotically Gaussian when the number of sites tends to infinity. PACS numbers: 75.10.Pq, 02.50.Cw Spin chains of Haldane-Shastry (HS) type [1, 2] are the prime example of integrable spin chains with long-range interactions, having close connections with several topics of current interest such as strongly correlated systems [3], generalized exclusion statistics , and the AdS-CFT correspondence . Recent numerical computations have shown that a common feature of these chains and their supersymmetric extensions ] is the fact that the level density (normalized to 1) becomes Gaussian when the number of sites N tends to infinity. The knowledge of the continuous part of the level density is a key ingredient in the theory of quantum chaos, since it is used to rescale ("unfold") the spectrum as a preliminary step in the study of important statistical properties such as the distribution of spacings between consecutive levels . In fact, a long-standing conjecture of Berry and Tabor posits that the spacings distribution of a "generic" integrable system should be Poissonian, while in quantum chaotic systems like polygonal billiards this distribution is given by Wigner's surmise , characteristic of the Gaussian ensembles in random matrix theory. For many spin chains of HS type, it can be shown that the Gaussian character of the level density implies that the spacings distribution obeys neither Poisson's nor Wigner's law, but is rather given by a simple "squareroot of a logarithm" formula . In this letter we develop a generalization of the standard central limit theorem to show that the level density of a wide class of finite-dimensional quantum systems must be asymptotically Gaussian. This class includes in particular a supersymmetric version of the original (trigonometric) Haldane-Shastry spin chain, as well the BC N version [9] of the spin 1/2 Polychronakos-Frahm (rational) chain .

Physical Review B, 2008

We compute the partition function of the su(m) Polychronakos-Frahm spin chain of 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 BCN-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.

Journal of Nonlinear Mathematical Physics, 2005

We compute the spectrum of the trigonometric Sutherland spin model of BC N type in the presence of a constant magnetic field. Using Polychronakos's freezing trick, we derive an exact formula for the partition function of its associated Haldane-Shastry spin chain.

Nuclear Physics B, 2011

In this paper we study the su(m) spin Sutherland (trigonometric) model of D N type and its related spin chain of Haldane-Shastry type obtained by means of Polychronakos's freezing trick. As in the rational case recently studied by the authors, we show that these are new models, whose properties cannot be simply deduced from those of their well-known BC N counterparts by taking a suitable limit. We identify the Weyl-invariant extended configuration space of the spin dynamical model, which turns out to be the Ndimensional generalization of a rhombic dodecahedron. This is in fact one of the reasons underlying the greater complexity of the models studied in this paper in comparison with both their rational and BC N counterparts. By constructing a non-orthogonal basis of the Hilbert space of the spin dynamical model on which its Hamiltonian acts triangularly, we compute its spectrum in closed form. Using this result and applying the freezing trick, we derive an exact expression for the partition function of the associated Haldane-Shastry spin chain of D N type.

1984

We study the ground state of one-dimensional classical spin chains with exchange and dipole interactions between nearest neighbors and with an anisotropy field at an arbitrary angle to the axis of the chain. We construct a mapping which connects the spin states at neighboring chain sites and we consider its properties in various limiting cases. We show that for a sufficiently high anisotropy field chaotic structures may appear. We give the spectral characteristics of these structures for the ferro-and antiferromagnetic cases.