Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions

Abstract

We completely solve the problem of classifying all one-dimensional quantum potentials with nearest-and next-to-nearest-neighbors interactions whose ground state is Jastrowlike, i.e., of Jastrow type but depending only on differences of consecutive particles. In particular, we show that these models must necessarily contain a three-body interaction term, as was the case with all previously known examples. We discuss several particular instances of the general solution, including a new hyperbolic potential and a model with elliptic interactions which reduces to the known rational and trigonometric ones in appropriate limits.Here x ≡ (x 1 , . . . , x N ) andfor the (harmonic) Calogero model, whilefor the Sutherland model (with ω > 0 and a > −1/2). As first noted by Sutherland [2, 23], this property makes it possible to compute in closed form certain correlation functions of the latter models by exploiting their connection with random matrix theory. Indeed, for Calogero's model ψ 2 coincides with the joint probability density of the eigenvalues of the Gaussian orthogonal, unitary and symplectic ensembles respectively for 2ω = 2a = 1, 2, 4, while the same relation holds for the ground state of the Sutherland model and Dyson's circular unitary ensembles (with eigenvalues parametrized as e 2ix k ) [24]. Later on, Dyson [25] showed how to construct analogues of the Gaussian ensembles with eigenvalues distributed according to ψ 2 in Eq. (1.1), with essentially arbitrary ρ and χ(x) = |x| β/2 (as usual, β will be assumed to take the values 1, 2, 4 for the orthogonal, unitary and symplectic ensembles, respectively). In view of these results, it is natural to look for the most general quantum Hamiltonian of Calogero-Sutherland (CS) type (i.e, with one-and two-body long-range interactions) whose ground state is of the form (1.1). A restricted version of this problem (with ρ = 1) was already formulated by Sutherland himself [2], who later found a solution thereof with an elliptic two-body interaction potential [26,27]. Shortly afterwards, Calogero [28] showed that this is in fact the most general solution of this restricted problem. The general problem (with ρ not necessarily equal to 1) was tackled by Inozemtsev and Meshcheryakov [29], who claimed to have found a complete solution. A decade later, however, Forrester [30] found a model of CS type whose ground state, which exhibits longrange crystalline order in the thermodynamic limit, is of the factorized form (1.1) and yet did not appear in the classification of Ref. [29]. The latter classification was finally completed several years later by Koprucki and Wagner [31], who obtained Forrester's model as a particular case.The probability distribution p β (s) of the (normalized) spacing s between two consecutive eigenvalues of the Gaussian β-ensembles is approximately given by Wigner's surmise p β (s) = A β s β e −c β s 2 , where the positive parameters A β , c β are fixed by normalization and the condition that the mean spacing be equal to 1 (see, e.g., Refs. [32,33]). By contra...