F. Finkel

AN-Type Dunkl Operators and New Spin Calogero–Sutherland Models

Abstract: A new family of AN-type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero–Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians.

New spin Calogero–Sutherland models related to BN-type Dunkl operators

Abstract We construct several new families of exactly and quasi-exactly solvable BCN-type Calogero–Sutherland models with internal degrees of freedom. Our approach is based on the introduction of a new family of Dunkl operators of BN type which, together with the original BN-type Dunkl operators, are shown to preserve certain polynomial subspaces of finite dimension. We prove that a wide class of quadratic combinations involving these three sets of Dunkl operators always yields a spin Calogero–Sutherland model, which is (quasi-)exactly solvable by construction. We show that all the spin Calogero–Sutherland models obtainable within this framework can be expressed in a unified way in terms of a Weierstrass ℘ function with suitable half-periods. This provides a natural spin counterpart of the well-known general formula for a scalar completely integrable potential of BCN type due to Olshanetsky and Perelomov. As an illustration of our method, we exactly compute several energy levels and their corresponding wavefunctions of an elliptic quasi-exactly solvable potential for two and three particles of spin 1/2.

Haldane-Shastry spin chains of BCN type

We introduce four types of SU(2M + 1) spin chains which can be regarded as the BCN 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.

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

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.

Exactly solvable DN-type quantum spin models with long-range interaction

Abstract We derive the spectra of the D N -type Calogero (rational) su ( m ) spin model, including the degeneracy factors of all energy levels. By taking the strong coupling limit of this model, in which its spin and dynamical degrees of freedom decouple, we compute the exact partition function of the su ( m ) Polychronakos–Frahm spin chain of D N type. In particular, we show that this partition function cannot be obtained as a limiting case of its BC N counterpart. With the help of the partition function we study several statistical properties of the chains spectrum, such as the density of energy levels and the distribution of spacings between consecutive levels.

The Lamé equation in parametric resonance after inflation

We show that the most general inflaton potential in Minkowski spacetime for which the differential equation for the Fourier modes of the matter fields reduces to Lames equation is of the form V(phi)=lambda phi^4/4+Kphi^2/2+mu/(2 phi^2)+V_0. As an application, we study the preheating phase after inflation for the above potential with K=0 and arbitrary lambda,mu >0. For certain values of the coupling constant between the inflaton and the matter fields, we calculate the instability intervals and the characteristic exponents in closed form.

Solvable scalar and spin models with near-neighbors interactions

We construct new solvable rational and trigonometric spin models with near-neighbors interactions by an extension of the Dunkl operator formalism. In the trigonometric case we obtain a finite number of energy levels in the center of mass frame, while the rational models are shown to possess an equally spaced infinite algebraic spectrum. For the trigonometric and one of the rational models, the corresponding eigenfunctions are explicitly computed. We also study the scalar reductions of the models, some of which had already appeared in the literature, and compute their algebraic eigenfunctions in closed form. In the rational cases, for which only partial results were available, we give concise expressions of the eigenfunctions in terms of generalized Laguerre and Jacobi polynomials.

Inozemtsev's hyperbolic spin model and its related spin chain

In this paper we study Inozemtsevs su(m) quantum spin model with hyperbolic interactions and the associated spin chain of Haldane-Shastry type introduced by Frahm and Inozemtsev. We compute the spectrum of Inozemtsevs model, and use this result and the freezing trick to derive a simple analytic expression for the partition function of the Frahm-Inozemtsev chain. We show that the energy levels of the latter chain can be written in terms of the usual motifs for the Haldane-Shastry chain, although with a different dispersion relation. The formula for the partition function is used to analyze the behavior of the level density and the distribution of spacings between consecutive unfolded levels. We discuss the relevance of our results in connection with two well-known conjectures in quantum chaos.

A novel quasi-exactly solvable spin chain with nearest-neighbors interactions

In this paper we study a novel spin chain with nearest-neighbors interactions depending on the sites coordinates, which in some sense is intermediate between the Heisenberg chain and the spin chains of Haldane–Shastry type. We show that when the number of spins is sufficiently large both the density of sites and the strength of the interaction between consecutive spins follow the Gaussian law. We develop an extension of the standard freezing trick argument that enables us to exactly compute a certain number of eigenvalues and their corresponding eigenfunctions. The eigenvalues thus computed are all integers, and in fact our numerical studies evidence that these are the only integer eigenvalues of the chain under consideration. This fact suggests that this chain can be regarded as a finite-dimensional analog of +

The spin Sutherland model of DN type and its associated spin chain

Abstract 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 Polychronakoss 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 N-dimensional 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.