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Physical review. B, Condensed matter

Pairing ground states in fermion systems: A static method

Konstantinos Moulopoulos, Neil William Ashcroft

Published Nov 1, 1990• 11 citations
Publisher

Abstract

We present a static recursive method that can straightforwardly be applied to variational treatments of canonical many-body systems whose ground states generally exhibit pairing or higher clustering correlations. We illustrate the utility of the method by applying it to an electron system with pairing interactions where we obtain, as expected, a smooth transition between a Bose condensate of tightly bound electron pairs and a superconducting ground state. The method also reproduces the mean-field results for the electron-hole liquid problem as well as the antiferromagnetic insulator ground state of the Hubbard model but in a simpler way. We give a nontrivial application to a model of dense molecular hydrogen consisting of condensed generalized molecules. We show that the effective interaction between identical particles is of such a form that the state undergoes a dielectric catastrophe at a certain finite density. The competition between this abrupt transition and a smooth transition of atomic pairs in the spinless and translationally invariant version of the problem is also discussed.

DOI
10.1103/physrevb.42.7855

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International Journal of Modern Physics B•Dec 20, 1999•4 citations
We present a static recursive method that can straightforwardly be applied to variational treatments of canonical many-body systems whose ground states generally exhibit pairing or higher clustering correlations. We illustrate the utility of the method by applying it to an electron system with pairing interactions where we obtain, as expected, a smooth transition between a Bose condensate of tightly bound electron pairs and a superconducting ground state. The method also reproduces the mean-field results for the electron-hole liquid problem as well as the antiferromagnetic insulator ground state of the Hubbard model but in a simpler way. We give a nontrivial application to a model of dense molecular hydrogen consisting of condensed generalized molecules. We show that the effective interaction between identical particles is of such a form that the state undergoes a dielectric catastrophe at a certain finite density. The competition between this abrupt transition and a smooth transition of atomic pairs in the spinless and translationally invariant version of the problem is also discussed.
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.
The two-particle spectrum in a dense Fermion system is treated using a thermodynamic Green function approach. A self-consistent description of possible bound states and a superfluid condensate in a correlated medium is given. A detailed understanding of superfluidity and superconductivity in correlated Fermion systems, especially the transition from the Cooper-paired state (weak-coupling limit) to the Bose condensed state of tightly bound pairs of Fermions (strong coupling limit) [1], is of great interest for very different physical systems. The problem of a unified treatment of Bose–Einstein condensation (BEC) and the Bardeen–Cooper–Schrieffer (BCS) phase arises not only in describing the electron structure of strongly correlated electron superfluids such as superconductors [2], the electronhole system in semiconductors [3], spin-polarized hydrogen and liquid He [4], but also in the theory of nuclear matter [5] and quark–gluon systems [6]. Recently, there have been several new approaches to this stimulating problem. A Monte-Carlo simulation of a finite He system at zero temperature has been performed in Ref. [7]. The microscopic theory of strongly coupled quantum fluids has been treated within the Jastrow approximation (cf. Ref. [8]) to obtain the ground state and low-lying excited states of strongly correlated boson quantum fluids. The crossover from weak to strong coupling superconductivity has been considered using a functional integral representation [2] (see also Ref. [7]).
For a dilute gas of fermions interacting via an arbitrary pair potential in d=2 dimensions, we show that the many-body ground state is unstable to s-wave pairing at T=0 if and only if a two-body bound state exists. We further obtain, within a variational pairing Ansatz, a smooth crossover from a Cooper-paired state (${\ensuremath{\xi}}_{0}$kf{\mathrm{k}}_{\mathrm{f}}kf​\ensuremath{\gg}1) to a Bose condensed state of tightly bound pairs (${\ensuremath{\xi}}_{0}$kf{\mathrm{k}}_{\mathrm{f}}kf​\ensuremath{\ll}1). We briefly discuss non--s-wave superconductors. Insofar as this model is appli- cable to the high-Tc{\mathrm{T}}_{\mathrm{c}}Tc​ materials, they are in the interesting regime with the coherence length ${\ensuremath{\xi}}_{0}$ comparable to the interparticle spacing ${\mathrm{K}}_{\mathrm{f}}^{\mathrm{\ensuremath{-}}1}$.
In order to develop our theoretical understanding of the superconducting state of strongly correlated electron systems, we investigate the many-body ground state of the attractive Hubbard model in infinite dimensions using the Gutzwiller-type projected BCS variational (GBCS) wave function. We first calculate various physical quantities by numerically evaluating analytical equations for systems in the thermodynamic limit. We then show that the GBCS wave function has the lowest energy when the gap parameter has a mixture of s wave and extended s wave pairing symmetry and that the ground state energy of the GBCS is lower than that of either the BCS or Fermi liquid at any electron density and any strength of the attractive interactions in our model. We also show that with infinite strength interactions, each pair of electrons with opposite spins forms a tightly bound pair on a lattice site (BEC-like state). We find a smooth crossover from a BCS-like state with large, overlapping Cooper pairs (for weakly attractive interactions) to a BEC-like state of composite bosons formed out of the tightly bound pairs of fermions (for strongly attractive interactions).
We present a nonstandard Hubbard model applicable to arbitrary single-particle potential profiles and interparticle interactions. Our approach involves a treatment of Wannier functions, free from the ambiguities of conventional methods and applicable to finite systems without periodicity constraints. To ensure the consistent evaluation of Wannier functions, we develop a perturbative approach, utilizing the barrier penetration coefficient as a perturbation parameter. With the defined Wannier functions as a basis, we derive the Hubbard Hamiltonian, revealing the emergence of density-induced and pair tunneling terms alongside standard contributions. Our investigation demonstrates that long-range interparticle interactions can induce a mechanism for repulsive particle pairing. This mechanism relies on the effective suppression of single-particle tunneling due to density-induced tunneling. Contrary to expectations based on the standard Hubbard model, an increase in interparticle interaction does not lead to an insulating state. Instead, our proposed mechanism implies the coherent motion of correlated electron pairs, similar to bound states within a multiwell system, resistant to decay from single-electron tunneling transitions. These findings carry significant implications for various phenomena, including the formation of flat bands, the emergence of superconductivity in twisted bilayer graphene, and the possibility of a metal-insulator transition.
The ground-state properties in one-dimensional Hubbard model with on-site attraction and repulsion of electrons in the presence of magnetic field h are calculated by means of the exact Bethe-ansatz formalism and the generalized self-consistent field (GSCF) approach for general electron concentrations n and arbitrary interaction strength. The ground-state properties, including the energy, the average spin (magnetization) and the kinetic energy are compared over a wide range of parameter space. The GSCF theory is in qualitative and in some cases in good quantitative agreement with the exact results. The GSCF theory at U≤0 (or U≥0) differentiates the spin (or charge) energy gap from the BCS (or antiferromagnetic) order parameter and suggests a smooth crossover from the phase with the itinerant BCS-like behavior to the Bose condensation regime of the local pairs.