Exactly Solvable Models in Arbitrary Dimensions

Ghosh, R K, Rao, S

Published May 21, 1997• 9 citations

Abstract

We construct a new class of quasi-exactly solvable many-body Hamiltonians in arbitrary dimensions, whose ground states can have any correlations we choose. Some of the known correlations in one dimension and some recent novel correlations in two and higher dimensions are reproduced as special cases. As specific interesting examples, we also write down some new models in two and higher dimensions with novel correlations.

Related papers

Novel Correlations in Two Dimensions: Some Exact Solutions

M. V. N. Murthy, R. K. Bhaduri, Diptiman Sen

We construct a new many-body Hamiltonian with two- and three-body interactions in two space dimensions and obtain its exact many-body ground state for an arbitrary number of particles. This ground state has a novel pairwise correlation. A class of exact solutions for the excited states is also found. These excited states display an energy spectrum similar to the Calogero-Sutherland model in one dimension. The model reduces to an analog of the well-known trigonometric Sutherland model when projected on to a circular ring.

arXiv:cond-mat/9603155Physical Review LettersMay 27, 199631 citations

Novel Correlations in Arbitrary Dimensions

Pijush K. Ghosh

We present 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 for arbitrary number of particles. These exact wave-functions describe a novel correlations. Finally, we generalize these three dimensional results to arbitrary higher dimensions.

arXiv:cond-mat/9607009Jul 2, 19962 citations

Exactly solvable models for 2D interacting fermions

Edwin Langmann A

I discuss many-body models for correlated fermions in two space dimensions which can be solved exactly using group theory. The simplest example is a model of a quantum Hall system: two-dimensional (2D) fermions in a constant magnetic field and a particular non-local four-point interaction. It is exactly solvable due to a dynamical symmetry corresponding to the Lie algebra gl ∞ ⊕ gl ∞ . There is an algorithm to construct all energy eigenvalues and eigenfunctions of this model. The latter are, in general, many-body states with spatial correlations. The model also has a non-trivial zero temperature phase diagram. I point out that this QH model can be obtained from a more realistic one using a truncation procedure generalizing a similar one leading to mean field theory. Applying this truncation procedure to other 2D fermion models I obtain various simplified models of increasing complexity which generalize mean field theory by taking into account non-trivial correlations but nevertheless are treatable by exact methods.

arXiv:cond-mat/0206045Journal of Physics A Mathematical and GeneralDec 15, 20035 citations

Hubbard model with next-nearest-neighbour interaction terms in higher dimensions: New, exactly solvable cases

Péter Gurin, Zsolt Gulácsi

The exact ground-state wavefunction and energy of an extended Hubbard model are presented containing also next-nearest-neighbour hopping, bond-charge and density interaction terms besides the nearest-neighbour and on-site contributions. Our results are valid at half-filling for a wide range of parameters, on simple hypercubic lattices in arbitrary D > I dimensions.

Philosophical Magazine BNov 1, 19972 citations

A class of exactly solvable models in quantum mechanics

Francisco Marcelo Fernández

We consider exactly solvable models in quantum mechanics with potential-energy functions that exhibit exponential dependence on the coordinate. As particular examples we choose a spherically symmetric well to illustrate the concept of critical strengths, and a barrier in one dimension to discuss tunnelling.

European Journal of PhysicsJul 1, 20011 citation

Novel correlations in two dimensions: Two-body problem

R. K. Bhaduri, Avinash Khare, J. Law +2 more

We discuss a many-body Hamiltonian with two- and three-body interactions in two dimensions introduced recently by Murthy, Bhaduri and Sen. Apart from an analysis of some exact solutions in the many-body system, we analyse in detail the two-body problem which is completely solvable. We show that the solution of the two-body problem reduces to solving a known differential equation due to Heun. We show that the two-body spectrum becomes remarkably simple for large interaction strengths and the level structure resembles that of the Landau levels. We also clarify the 'ultraviolet' regularization which is needed to define an inverse-square potential properly and discuss its implications for our model.

Apr 7, 19979 citations