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Physical Review Letters

Bound states, Cooper pairing, and Bose condensation in two dimensions

Mohit Randeria, Ji-Min Duan, Lih-Yir Shieh

Published Feb 27, 1989• 435 citations
Publisher

Abstract

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}$${\mathrm{k}}_{\mathrm{f}}$\ensuremath{\gg}1) to a Bose condensed state of tightly bound pairs (${\ensuremath{\xi}}_{0}$${\mathrm{k}}_{\mathrm{f}}$\ensuremath{\ll}1). We briefly discuss non--s-wave superconductors. Insofar as this model is appli- cable to the high-${\mathrm{T}}_{\mathrm{c}}$ materials, they are in the interesting regime with the coherence length ${\ensuremath{\xi}}_{0}$ comparable to the interparticle spacing ${\mathrm{K}}_{\mathrm{f}}^{\mathrm{\ensuremath{-}}1}$.

DOI
10.1103/physrevlett.62.981

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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}$${\mathrm{k}}_{\mathrm{f}}\ensuremath{\gg}1) to a Bose condensed state of tightly bound pairs ({\ensuremath{\xi}}_{0}$${\mathrm{k}}_{\mathrm{f}}\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 \ensuremathξ0{\ensuremath{\xi}}_{0}\ensuremathξ0​ comparable to the interparticle spacing Kf\ensuremath−1{\mathrm{K}}_{\mathrm{f}}^{\mathrm{\ensuremath{-}}1}Kf\ensuremath−1​.
We study a two-dimensional continuum model of a dilute gas of fermions at T=0 interacting via a given two-body potential, with an aim to investigate superconductors with coherence length of the order of the interparticle spacing [kF{k}_{F}kF​${\ensuremath{\xi}}_{0}$\ensuremath{\sim}O(1)], a striking feature of the high-Tc{T}_{c}Tc​ materials. We find that a two-body bound state in vacuum is a necessary and sufficient condition for an s-wave pairing instability. We also find that the existence of such a bound state in a higher-angular-momentum (l>0) channel is not a necessary condition for an l-wave pairing instability. We further investigate using a variational ansatz the evolution from a state with large overlapping Cooper pairs (kF{k}_{F}kF​${\ensuremath{\xi}}_{0}$\ensuremath{\gg}1) to one with Bose condensation of composite bosons (kF{k}_{F}kF​${\ensuremath{\xi}}_{0}$\ensuremath{\ll}1). For the s-wave case an exact solution of the variational equations shows a smooth crossover from one regime to the other at T=0. The p-wave solution has a weak singularity when the chemical potential goes through zero, which is the bottom of the band. We show, quite generally, independent of the dimensionality and of model details, that the gap--to--single-particle excitations is nodeless, even if the anisotropic pair wave function has nodes, when the coupling is strong enough that the chemical potential is negative.
The crossover between the Cooper-pair condensation and the Bose-Einstein condensation of ``di-electronic'' molecules in two-dimensional superconductors is investigated in detail on the basis of the Nozi\`eres and Schmitt-Rink formalism. It is shown that temperature dependence of the chemical potential \ensuremath{\mu} so calculated is classified into two classes as decreasing temperatures; i.e., class (a) where \ensuremath{\mu} approaches the point of Bose-Einstein condensation of two-dimensional ideal Bose gas of ``di-electronic'' molecules, and class (b) where \ensuremath{\mu} diverges positively along the line of BCS-type mean-field pair condensation. This feature is rather universal irrespective of strength V of the attractive interaction of the s-wave type. While the former class (a) has been found by Schmitt-Rink, Varma, and Ruckenstein, existence of the latter class (b) is recognized here. In the case where V is fixed, class (a) is realized for electron number density N smaller than Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​, which is an increasing function of V, and class (b) is realized for N larger than Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​. If N\ensuremath{\gg}Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​ in particular, there exists a regime, where the Fermi-liquid-like description is valid, between the BCS-type mean-field transition temperature and the Fermi temperature. In the situation where V is changed with N being fixed, low-temperature states for the strong-coupling case belong to class (a) while those for the weak-coupling case belong to class (b). Therefore, with decreasing V, the chemical potential \ensuremath{\mu}(T), at temperatures far below the Fermi temperature, shows a discontinuous jump at V=Vcr{\mathit{V}}_{\mathrm{cr}}Vcr​(N) corresponding to the transition from class (a) to (b). However, this is in contradiction to a physical picture that the chemical potential should smoothly cross over between the above two limits unless the liquid-gas transition occurs. This shows in turn a necessity of improving the Nozi\`eres and Schmitt-Rink formalism itself especially in two dimensions. A preliminary approach beyond their formalism is briefly discussed.
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We report results of quantum Monte Carlo calculations of the ground state of dilute Fermi gases with attractive short-range two-body interactions. The strength of the interaction is varied to study different pairing regimes which are characterized by the product of the sss-wave scattering length and the Fermi wave vector, akFa{k}_{F}akF​. We report results for the ground-state energy, the pairing gap $\ensuremath{\Delta}$, and the quasiparticle spectrum. In the weak-coupling regime, $1∕a{k}_{F}<\ensuremath{-}1$, we obtain Bardeen-Cooper-Schrieffer (BCS) superfluid and the energy gap $\ensuremath{\Delta}$ is much smaller than the Fermi gas energy EFG{E}_{\mathrm{FG}}EFG​. When a>0a>0a>0, the interaction is strong enough to form bound molecules with energy Emol{E}_{\mathrm{mol}}Emol​. For $1∕a{k}_{F}\ensuremath{\gtrsim}0.5$, we find that weakly interacting composite bosons are formed in the superfluid gas with $\ensuremath{\Delta}$ and gas energy per particle approaching $\ensuremath{\mid}{E}_{\mathrm{mol}}\ensuremath{\mid}∕2$. In this region, we seem to have Bose-Einstein condensation (BEC) of molecules. The behavior of the energy and the gap in the BCS-to-BEC transition region, $\ensuremath{-}0.5<1∕a{k}_{F}<0.5$, is discussed.
The spectrum of collective (pair) excitations in the ground state of a dilute twodimensional (2D) attractive Fermi-gas is studied within the functional integral formalism. The linearized equations for the fluctuations about the non-trivial saddle point are analyzed for all coupling regimes, which are characterized by the ratio ε 0 /ε F , where ε 0 is the two-fermion binding energy and ε F is the Fermi energy. The approximation takes into account propagation of the fluctuations and their interaction with the condensate. In the strong-coupling, or ‘;Bosegas’, regime (ε 0 /ε F > 1) the spectrum is continuous and has the Bogolubov form, but in the weak-coupling limit (ε 0 /ε F «1) there are two types of excitations (different from the two-fermion scattering states): (i) long-wavelength sound-like excitations with the cut-off at momentum q c ≃ l/ξ 0 (where ξ 0 is the Cooper pair size), and (ii) pair excitations with q ≃ (8 m μ) ½ , where μ is the chemical potential and m is the fermion mass. The crossover between weakand strong-coupling behavior of the excitation spectrum is found to occur at the value of the coupling parameter ε 0 /ε F ≃¼. Introduction Since the discovery of high- T c superconductivity there has been growing interest in studying 2D Fermi gases with attractive interaction, especially in the crossover regime, when the pair size ξ 0 is of the order of the interparticle distance. The importance of such a model, which can be regarded as a semiphenomenological model of a 2D superconductor, is highlighted by experimental evidence that the high- T c superconductors, most of which have a layered structure, have a short coherence length, so that k F ξ 0 ≃1, where k F is the Fermi momentum.