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arXiv:cond-mat/0004012 · Journal of Physics A Mathematical and General

A class ofN-body problems with nearest- and next-to-nearest-neighbour interactions

Guy Auberson, Sudhir R. Jain, Avinash Khare

Published Jan 18, 2001• 49 citations
arXivHTMLPDF

Abstract

We obtain the exact ground state and part of the excitation spectrum in one dimension on a line and the exact ground state on a circle in the case where the N particles are interacting via nearest- and next-to-nearest-neighbour interactions. Furthermore, using the exact ground state, we establish a mapping between these N -body problems and the short-range Dyson models introduced recently to model intermediate spectral statistics. Using this mapping we compute the one- and two-point functions of a related many-body theory in the thermodynamic limit and show the absence of long-range order. However, quite remarkably, we prove the existence of an off-diagonal long-range order in the symmetrized version of the related many-body theory. Generalization of the models to other root systems is also considered. Besides, we also generalize the model on the full line to higher dimensions. Finally, we consider a model in two dimensions in which all the states exhibit novel correlations.

DOI
10.1088/0305-4470/34/4/302

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We obtain the exact ground state and part of the excitation spectrum in one dimension on a line and the exact ground state on a circle in the case where the N particles are interacting via nearest- and next-to-nearest-neighbour interactions. Furthermore, using the exact ground state, we establish a mapping between these N -body problems and the short-range Dyson models introduced recently to model intermediate spectral statistics. Using this mapping we compute the one- and two-point functions of a related many-body theory in the thermodynamic limit and show the absence of long-range order. However, quite remarkably, we prove the existence of an off-diagonal long-range order in the symmetrized version of the related many-body theory. Generalization of the models to other root systems is also considered. Besides, we also generalize the model on the full line to higher dimensions. Finally, we consider a model in two dimensions in which all the states exhibit novel correlations.
We obtain the exact ground state of N-particles in one dimension both on a line and on a circle in case the N particles are interacting via nearest and next-to-nearest neighbour interactions. Further, we establish a mapping between these N-body problems and short range Dyson models introduced recently to model intermediate spectral statistics. Using this mapping we compute one and two point functions of a related many-body theory in the thermodynamic limit and show the absence of long range order. Generalization of the model on the line to higher dimensions as well as to other root systems is also considered. Finally, we also consider a variant of the above model in two dimensions in which all the states exhibit novel correlations.
The bound and scattering states of the many-body systems, related to the short-range Dyson model, are studied. The Hamiltonian for the full-line problem is connected to decoupled oscillators. The analytically obtainable eigenstates are smaller in number as compared to the Calogero-Sutherland family, indicating the quasiexactly solvable nature of these models. The exactly found scattering states, a smaller set as compared to the Calogero case, can be realized as coherent states. The relation of the scattering Hamiltonian to free particles is also established algebraically. We analyze both ${A}_{N\ensuremath{-}1}$ and BCNB{C}_{N}BCN​ models on a circle and construct a part of the excitation spectrum by making use of the symmetry arguments.
The bound and scattering states of the many-body systems, related to the short-range Dyson model, are studied. The Hamiltonian for the full-line problem is mapped to decoupled oscillators; using this connection we establish an equivalent of the Stanley-Macdonald conjecture in the present case. Further, we show that the scattering states can be realized as coherent states. As in the Calogero case, the scattering scenario in our case yields that the initial momenta, {ki}, go over to the final momenta, {k ′ i = kN+1−i}, with energy independent phase shift. We also establish the unitary equivalence of the scattering Hamiltonian to free particles. Finally, we analyse both the AN−1 and BCN models on circle and construct a number of eigenfunctions by making
The scattering and bound states of the many-body systems, related to the short-range Dyson model, are studied. First, we show that the scattering states can be realized as coherent states and the scattering Hamiltonian can be connected to a free system. Unlike the closely related Calogero-Moser model, only a part of the partial waves acquire energy independent phase shifts, after scattering. The cause of the same is traced to the reduction in the degeneracies. The bound state Hamiltonian for the full-line problem is also studied and the relationship of its Hilbert space with that of the decoupled oscillators elucidated. Finally, we analyze the related models on circle and construct a part of the excitation spectrum through symmetry arguments.
We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction g = 2. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order. Calogero-Sutherland type models are a class of exactly solvable models. Starting from the inception [1], till today, these models continue to be of interest, especially due to their exact solvability [2]. Moreover, these models are related to (1 + 1)-dimensional conformal field theory, random matrices and a host of other things [3]. Some variants of the models have attracted attention in very recent past [3–5]. The original model of Calogero [1] described N particles in one dimension, interacting through an inverse square law and a two-body harmonic interaction. The latter interaction was added to the effect of complete discretization of the spectrum. This model gave in to an analytic treatment yielding an exact solution. Sutherland [6] considered a variant of this problem where the harmonic interaction was replaced by a harmonic well containing the N particles. It was shown that one can exactly solve for the ground state of the system. The norm of the ground state wave-function was shown to be of a form that coincides with the joint probability density function of the eigenvalues of random matrices from an orthogonal, unitary or symplectic Gaussian ensemble, provided, the
Abstract A 1D cluster model with next-nearest-neighbor interactions and two additional composite interactions is solved; the free energy is obtained and a correlation function is derived exactly. The model is diagonalized by a transformation obtained automatically from its interactions, which is an algebraic generalization of the Jordan–Wigner transformation. The gapless condition is expressed as a condition on the roots of a cubic equation, and the phase diagram is obtained exactly. We find that the distribution of roots for this algebraic equation determines the existence of long-range order, and we again obtain the ground-state phase diagram. We also derive the central charges of the corresponding conformal field theory. Finally, we note that our results are universally valid for an infinite number of solvable spin chains whose interactions obey the same algebraic relations.