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Ukrains’kyi Matematychnyi Zhurnal

Модельний гамільтоніан БКШ теорії надпровідності як квадратична форма

D. Ya. Petrina

Published Mar 25, 2004• 0 citations
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Abstract

Bogolyubov proved that the average energies (per unit volume) of the ground states for the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide in the thermodynamic limit. In the present paper, we show that this result is also true for all excited states. We also establish that, in the thermodynamic limit, the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide as quadratic forms.

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Bogolyubov proved that the average energies (per unit volume) of the ground states for the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide in the thermodynamic limit. In the present paper, we show that this result is also true for all excited states. We also establish that, in the thermodynamic limit, the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide as quadratic forms.
We consider the model and approximating Hamiltonians directly in the case of infinite volume. We show that each of these Hamiltonians has two branches of the spectrum and two systems of eigenvectors, which represent excitations of the ground states of the model and approximating Hamiltonians as well as the ground states themselves. On both systems of eigenvectors, the model and approximating Hamiltonians coincide with one another. In both branches of the spectrum, there is a gap between the eigenvalues of the ground and excited states.
We investigate the spectrum of a model Hamiltonian with BCS and mean-field interaction in a finite domain under periodic boundary conditions. The model Hamiltonian is considered on the states of pairs and waves of density charges and their excitations. It is represented as the sum of three operators that describe noninteracting pairs, the interaction between pairs, and the interaction between pairs and waves of density charges. The last two operators tend to zero in the thermodynamic limit, and the spectrum of the model Hamiltonian coincides with the spectrum of noninteracting pairs with chemical potential shifted by mean-field interaction. It is shown that the model and approximating Hamiltonians coincide in the thermodynamic limit on their ground and excited states and both have two branches of eigenvalues and eigenvectors.
The BCS Hamiltonian in a finite cube with periodic boundary condition is considered. The special subspace of pairs of particles with opposite momenta and spin is introduced. It is proved that, in this subspace, the spectrum of the BCS Hamiltonian is defined exactly for one pair, and for n pairs the spectrum is defined through the eigenvalues of one pair and a term that tends to zero as the volume of the cube tends to infinity. On the subspace of pairs, the BCS Hamiltonian can be represented as a sum of two operators. One of them describes the spectra of noninteracting pairs and the other one describes the interaction between pairs that tends to zero as the volume of the cube tends to infinity. It is proved that, on the subspace of pairs, as the volume of the cube tends to infinity, the BCS Hamiltonian tends to the approximating Hamiltonian, which is a quadratic form with respect to the operators of creation and annihilation.