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arXiv:math-ph/0603061 · Reviews in Mathematical Physics

PROOF OF THE VARIATIONAL PRINCIPLE FOR A PAIR HAMILTONIAN BOSON MODEL

Joseph V. Pulé, Valentin Anatol'evich Zagrebnov

Published Mar 1, 2007• 5 citations
arXivHTMLPDF

Abstract

We give a two-parameter variational formula for the grand-canonical pressure of the Pair Boson Hamiltonian model. By using the Approximating Hamiltonian Method we provide a rigorous proof of this variational principle.
DOI
10.1142/s0129055x07002924

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The many-boson system with repulsive interactions is treated by a variational method based on a variational trial state of an exponential pair-excitation type obtained by a generalization of that of Bogoliubov; the treatment is closely related to an intermediate-coupling approximation with respect to pairs. The nonlinear integral equation which determines the variational ground state is derived, and various properties of this ground state and the associated excited states are examined. The resultant low-lying spectrum lies below that of Bogoliubov by an amount proportional to the total number of particles. The variational principle is shown to produce rigorous energy eigenvalue differences for the pair part of the Hamiltonian. The variational states, however, still exhibit unphysical features characteristic of pair-excitation states: The pair correlation function does not go to zero at zero particle separation and the phonon spectrum exhibits a gap above the ground state. It is suggested that these features can be removed by using states which take into account excitation of momentum-conserving groups of more than two particles.
A novel variational approach is outlined for treating electron correlation in many-particle systems and is applied to the one-band Hubbard model. The authors' treatment is based on the concept of fermion pairs and their representation in terms of ideal-boson operators. In this representation approximate eigen-states of the Hamiltonian are obtained in the form of an antisymmetrised product of pair orbitals. For the ground state of the Hubbard model these orbitals are then determined by the use of the variational principle. Because of the complexity of the resulting equation a simplified ansatz is used for the orbitals, which leads to the same energy expression as is obtained by the traditional Bogolubov method. The trial states used in both approaches are, however, quite different. They find reasonable qualitative agreement with well established results for the Hubbard model in one dimension even in those cases where the Bogolubov or related approaches fail. They are also able to show that in the atomic limit the Bogolubov energy bound is the best upper bound that can be obtained by using a single determinant of pair orbitals. Therefore, the only way to improve the ground-state energy in the atomic limit is by means of configuration interaction.