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arXiv:quant-ph/0105130 · Journal of Physics A Mathematical and General

Testing Hall-Post inequalities with exactly solvable N -body problems

Avinash Khare, J. M. Richard

Published Aug 10, 2001• 5 citations
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Abstract

The Hall-Post inequalities provide lower bounds on N -body energies in terms of N '-body energies with N ' < N . They are rewritten and generalized to be tested with exactly solvable models of Calogero-Sutherland type in one and higher dimensions. The bound for N spinless fermions in one dimension is better saturated at large coupling than for noninteracting fermions in an oscillator potential.

DOI
10.1088/0305-4470/34/33/102

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Lower bound energy formulae for the N-fermion problem of the type used by Hall (1967) and Post (1956) are considered in terms of a reduced density operator for the interparticle separation. In bounds of this type the N-body energy is bounded below by a sum of eigen-energies of states of relative motion weighted by occupations for the relative states. It is shown that the wavefunctions of relative motion are subject to constraints and in particular must be zero when the particle separation is zero. This modifies the energy spectrum of the reduced Hamiltonian of relative motion and improves the lower bound due to Hall. The improvement is demonstrated for some simple model problems. Furthermore it is shown that the bounds of this type currently known do not exhibit saturation regardless of the interaction details. This leads to some speculation on improved lower bounds.
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.
Alternative lower bounds for the binding energy of a quantum-mechanical system of interacting particles are presented. These bounds are expressed in terms of two-particle quantities and improve the conventional bounds of the Hall-Post type. They are constructed by considering not only the energy in the two-particle system, but also the structure of the pair wave function. We apply the formal results to various numerical examples, and show that in some cases dramatic improvement over the existing bounds is reached.