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Progress of Theoretical Physics

Two-Body Correlation of Interacting Fermions

Ki-ichi Nakamura

Published May 1, 1959• 36 citations
Publisher

Abstract

The electron-pair model, proposed for superconductivity by Bardeen, Cooper and Schrieffer, is discussed in the configuration space. The wave function for the system of electron-pairs is taken as follows,
DOI
10.1143/ptp.21.713

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Abstract One of the important steps leading to the successful formulation of the microscopic theory of superconductivity was the model calculation performed by Cooper (1956) which showed that two electrons, interacting above a filled Fermi sea, had a bound state in the presence of an arbitrarily weak attractive interaction. This chapter first examines Cooper’s calculation. The success of the Cooper model calculation in producing a bound state, with its implication that a proper many-body theory would contain a gap in the excitation spectrum, strongly suggests that the ground state wave function should be constructed from pairs of electrons. However, it is difficult to perform calculations using the many-body pair wave function from Cooper’s model. The chapter then introduces the wave function proposed by Bardeen, Cooper, and Schrieffer (BCS) as an alternative.
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Topological pairing of composite fermions has led to remarkable ideas, such as excitations obeying non-Abelian braid statistics and topological quantum computation. We construct a ppp-wave paired Bardeen-Cooper-Schrieffer (BCS) wave function for composite fermions in the torus geometry, which is a convenient geometry for formulating momentum space pairing as well as for revealing the underlying composite-fermion Fermi sea. Following the standard BCS approach, we minimize the Coulomb interaction energy at half filling in the lowest and the second Landau levels, which correspond to filling factors $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=5/2$ in GaAs quantum wells, by optimizing two variational parameters that are analogous to the gap and the Debye cutoff energy of the BCS theory. Our results show no evidence for pairing at $\ensuremath{\nu}=1/2$ but a clear evidence for pairing at $\ensuremath{\nu}=5/2$. To a good approximation, the highest overlap between the exact Coulomb ground state at $\ensuremath{\nu}=5/2$ and the BCS state is obtained for parameters that minimize the energy of the latter, thereby providing support for the physics of composite-fermion pairing as the mechanism for the 5/25/25/2 fractional quantum Hall effect. We discuss the issue of modular covariance of the composite-fermion BCS wave function, and calculate its Hall viscosity and pair correlation function. By similar methods, we look for but do not find an instability to sss-wave pairing for a spin-singlet composite-fermion Fermi sea at half-filled lowest Landau level in a system where the Zeeman splitting has been set to zero.