Novel Correlations in Arbitrary Dimensions

We present a new three dimensional many-body Hamiltonian with threebody and five-body interactions. We obtain the exact ground state as well as some excited states of this Hamiltonian for arbitrary number of particles. These exact wave-functions describe a novel correlations. Finally, we generalize these three dimensional results to arbitrary higher dimensions. PACS numbers: 03.65.Ge, 05.30.-d Typeset using REVTEX ∗ E-mail: [email protected] 1 Recently, there has been a renewed interest in the study of many-body quantum mechanical systems, like the Calogero-Sutherland Model (CSM) and its variants in one dimension [1–6]. This is primarily because these models are relevant to many diverse branches in physics [7]. Though, such one dimensional many-body systems have been studied extensively in the recent literature, nothing much is known about the appropriate generalization of these models to higher dimensions. As a promising step towards this, it was pointed out recently [8] that in two dimensions there exists novel correlations other than the one used in constructing the Laughlin’s trial wave function [9]. The exact ground state as well as some excited states were also obtained for a model many-body Hamiltonian, where this novel correlations can be realized. The purpose of this letter is to show that this type of correlations can be appropriately generalized to arbitrary higher dimensions. In particular, we construct a new three dimensional many-body Hamiltonian with three-body and five-body interactions. We obtain the exact ground state as well as some excited states of this Hamiltonian. The exact wave-functions of this model describe a novel correlations, which are an appropriate generalization of the two dimensional correlations introduced in [8]. These correlations can be realized by spinless bosons as well as fermions. Finally, all these three dimensional results are generalized to arbitrary higher dimensions. The CSM is described by N identical particles confined in a one-body harmonic oscillator potential or on the rim of a circle, and interacting through each other via a two-body inverse square potential. The wave function of this model contains a Jastrowtype factor of the form Jij = (xi − xj) | xi − xj | , where xi and xj denote the particle positions of the ith and the jth particle, respectively. The parameters α and λ are related to the strength of the inverse square two-body potential. The wave functions of CSM are highly correlated and the nature of correlations is encoded in the Jastrow-type