Inozemtsevs hyperbolic spin model and its related spin chain

Over the last few years a significant amount of effort has been devoted to the study of spin chains of Haldane-Shastry (HS) type, due to their remarkable integrability properties and their interest in connection with several important conjectures in quantum chaos. This class of chains, intimately related to integrable dynamical models of Calogero-Sutherland type [1][2][3], are characterized by the fact that the interactions between the spins are both long-ranged and position-dependent. For instance, in the original Haldane-Shastry chain [4,5] the spins occupy equidistant positions on a circle, the strength of the interactions being inversely proportional to the square of the distance between the spins measured along the chord. Historically, this model was introduced while searching for a spin chain whose exact ground state coincided with Gutzwiller's variational wavefunction for the one-dimensional Hubbard model in the limit of large on-site interaction [6][7][8]. In fact, the HS chain can be obtained in this limit from the Hubbard model with longrange hopping in the half-filling regime [9]. In Haldane's original paper [4], the spectrum of the HS chain with spin 1/2 was inferred on the basis of numerical calculations. In particular, it was observed that the levels are highly degenerate, which suggests the presence of a large underlying symmetry group and the possible integrability of the model. This symmetry group was subsequently identified [10] as the Yangian Y(sl m ), where m is the number of internal degrees of freedom. As to the model's integrability, it was established around the same time by Fowler and Minahan [11] using Polychronakos's exchange operator formalism [12].

The rigorous derivation of the spectrum of the HS chain with arbitrary su(m) spin was carried out by Bernard et al. [13] by taking advantage of its connection with the generalization of Sutherland’s model to particles with spin [14]. At the heart of this connection is the mechanism known as Polychronakos’s “freezing trick” [15]. The physical idea behind this mechanism is that when the coupling constant of the spin Sutherland (trigonometric) model tends to infinity, the particles concentrate around the equilibrium of the scalar part of the potential, so that the dynamical and internal degrees of freedom decouple. It can be shown that the coordinates of this equilibrium are essentially the HS chain sites, and that in this limit the internal degrees of freedom are governed by the chain’s Hamiltonian. The freezing trick can also be applied to the spin Calogero (rational) model [16], obtaining in this way a spin chain -the so-called Polychronakos-Frahm (PF) chain [15,17]-with non-equidistant sites given by the zeros of the N -th degree Hermite polynomial, N being the number of spins.

The original Calogero and Sutherland models mentioned above are both based on the root system A N -1 , in the sense that the interaction between the particles depends only on their relative distance. As shown by Olshanetsky and Perelomov [18], there are integrable generalizations of these models associated with any (extended) root system, the rank of the root system basically coinciding with the number of particles. For this reason, the most studied models of Calogero-Sutherland (CS) type are by far those associated with the BC N root system (including the hyperbolic Sutherland model) [18][19][20][21][22], and to a lesser extent with the D N system [23,24]. A hyperbolic variant of the Sutherland model of A N -1 type with an external confining potential of Morse type has also been considered in the literature, in both the scalar [25] and the spin [26] cases.

For all the spin CS models mentioned in the previous paragraphs, a corresponding spin chain of Haldane-Shastry type has been constructed by means of the freezing trick [19,21,23,24,27,28]. In the case of the rational and trigonometric chains, this mechanism has been applied to derive a closed-form expression for the partition function in terms of the quotient of the partition functions of the corresponding spin and scalar dynamical models [22][23][24][29][30][31], whose spectrum can be easily computed. Expanding the partition function in powers of q ≡ e -1/(kBT ) , one can compute the chain’s spectrum for relatively large values of N and determine some of its statistical properties. A common feature of all of these chains is the fact that when the number of sites is sufficiently large the level density is approximately Gaussian. This result, for which there is ample numerical evidence, has also been rigorously established in some cases [32]. The knowledge of a continuous approximation to the (cumulative) level density is of great importance in the context of quantum chaos, as it is used to transform the raw energies so that the resulting “unfolded” spectrum has an approximately uniform level density [33]. The distribution of spacings between consecutive unfolded levels is widely used for testing the inte

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