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Soviet Journal of Low Temperature Physics

On the bound states of two weakly interacting quasiparticles with a strongly degenerate dispersion relation

R. A. Minlos, Alexander Mogilner

Published Oct 1, 1988• 0 citations
Publisher

Abstract

The emergence of several bound states of two weakly interacting quasiparticles with a strongly degenerate dispersion relation is considered. The estimates of the number of bound states, their binding energy and effective mass, and the region of their existence in the center-of-mass quasi-momentum space are obtained from an analysis of the Fredholm determinant of the Schrödinger equation.

DOI
10.1063/10.0032037

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The emergence of several bound states of two weakly interacting quasiparticles with a strongly degenerate dispersion relation is considered. The estimates of the number of bound states, their binding energy and effective mass, and the region of their existence in the center-of-mass quasi-momentum space are obtained from an analysis of the Fredholm determinant of the Schrödinger equation.
We investigate one-dimensional three-body systems composed of two identical bosons and one mass-imbalanced atom (impurity) with attractive two-body and three-body zero-range interactions. In the absence of three-body interaction, we give a complete phase diagram of the number of three-body bound states in the whole region of mass ratio and the ratio of intra- and intercomponent interaction strength via direct calculation of Skornyakov-Ter-Martirosyan equations. We demonstrate that other low-lying three-body bound states emerge when the mass of the impurity particle is different from other two identical particles. We obtain the binding energies together with the corresponding wave functions. When the mass of impurity atom is very large, there are at most three three-body bound states. In the presence of three-body zero-range interaction, we reveal that weak three-body interaction will not always induce one more three-body bound state. At some special parameter points, arbitrary small three-body interaction can generate one more three-body bound state. This corresponds to the transition of the number of three-body bound states induced only by two-body attractive interaction.
Cooper pairing in two dimensions is analyzed with a set of renormalized equations to determine its binding energy for any fermion number density and all coupling assuming a generic pairwise residual interfermion interaction. Also considered are Cooper pairs (CP's) with nonzero center-of-mass momentum (CMM) and their binding energy is expanded analytically in powers of the CMM up to quadratic terms. A Fermi-sea-dependent linear term in the CMM dominates the pair excitation energy in weak coupling (also called the BCS regime) while the more familiar quadratic term prevails in strong coupling (the Bose regime). The crossover, though strictly unrelated to BCS theory per se, is studied numerically as it is expected to play a central role in a model of superconductivity as a Bose-Einstein condensation of CPs where the transition temperature vanishes for all dimensionality d< 2d<~2d< 2 for quadratic dispersion, but is nonzero for all d> 1d>~1d> 1 for linear dispersion.