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arXiv:cond-mat/0007005

A Study of Quantum Many-Body Systems With Nearest And Next-to-Nearest Neighbour Long-Range Interactions

Meripeni Ezung, Nakka Gurappa, Avinash Khare, Prasanta Kumar Panigrahi

Published Jul 1, 2000• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0007005

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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.
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
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.
To clarify the relation of energy shifts to scattering phase shifts in one-body and many-body problems, we examine their relation in a number of different situations. We derive, for a particle in a container of arbitrary shape with a short-range scattering center, a general result for the energy eigenvalues in terms of the s-wave scattering phase shift and the eigenstates in the absence of the scatterer. We show that, while the energy shifts for a spherical container are proportional to the phase shift over large ranges, those for a cubic container have a more complicated behavior. We connect our result to the description of energy shifts in terms of the scattering T-matrix. The general relation is extended to problems of particles in traps with smoothly varying potentials, including, e.g., the interaction of a small neutral atom with a Rydberg atom. We then consider the many-body problem for particles with a two-body interaction and show that the free energy change due to the interaction is proportional to an average of a generalized phase shift that includes the effects of the medium. Finally, we discuss why, even though individual energy levels are very sensitive to boundary conditions, the energy of a many-body system is not.