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Physical Review Letters

Pairing in two dimensions

Stefan Schmitt-Rink, Chandra Mohan Varma, Andrei E. Ruckenstein

Published Jul 24, 1989• 158 citations
Publisher

Abstract

A two-dimensional Fermi gas with an attractive s-wave interaction is shown to display a finite number of bound fermion pairs at all densities and temperatures. We study the chemical equilibrium between bound and ionized fermion pairs by including single-pair fluctuations in the free energy. While this turns out to be sufficient to drive the critical temperature for the onset of long-range order to zero, higher-order processes must be invoked to describe correctly the low-temperature regime.

DOI
10.1103/physrevlett.63.445

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Journal of Physics Communications•Oct 16, 2017•3 citations
A two-dimensional Fermi gas with an attractive s-wave interaction is shown to display a finite number of bound fermion pairs at all densities and temperatures. We study the chemical equilibrium between bound and ionized fermion pairs by including single-pair fluctuations in the free energy. While this turns out to be sufficient to drive the critical temperature for the onset of long-range order to zero, higher-order processes must be invoked to describe correctly the low-temperature regime.
It is demonstrated that the interaction between fluctuations of the pairing field in a two-dimensional (2D) dilute attractive Fermi gas stabilizes the low-temperature phase of the system. This result, which is valid for all values of the "coupling parameter" $\frac{{\ensuremath{\varepsilon}}_{0}}{{\ensuremath{\varepsilon}}_{F}}$ (where ${\ensuremath{\varepsilon}}_{0}$ is the two-fermion binding energy and ${\ensuremath{\varepsilon}}_{F}$ is the Fermi energy), then allows one to conclude that a 2D dilute attractive Fermi gas is superfluid at small but finite temperature.
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 nature of the normal phase of strongly correlated fermionic systems is an outstanding question in quantum many-body physics. We used spatially resolved radio-frequency spectroscopy to measure pairing energy of fermions across a wide range of temperatures and interaction strengths in a two-dimensional gas of ultracold fermionic atoms. We observed many-body pairing at temperatures far above the critical temperature for superfluidity. In the strongly interacting regime, the pairing energy in the normal phase considerably exceeds the intrinsic two-body binding energy of the system and shows a clear dependence on local density. This implies that pairing in this regime is driven by many-body correlations, rather than two-body physics. Our findings show that pairing correlations in strongly interacting two-dimensional fermionic systems are remarkably robust against thermal fluctuations.
The Hubbard model of attractively interacting fermions provides a paradigmatic setting for fermion pairing. It features a crossover between Bose-Einstein condensation of tightly bound pairs and Bardeen-Cooper-Schrieffer superfluidity of long-range Cooper pairs, and a "pseudo-gap" region where pairs form above the superfluid critical temperature. We directly observe the nonlocal nature of fermion pairing in a Hubbard lattice gas, using spin- and density-resolved imaging of [Formula: see text]1000 fermionic potassium-40 atoms under a bilayer microscope. Complete fermion pairing is revealed by the vanishing of global spin fluctuations with increasing attraction. In the strongly correlated regime, the fermion pair size is found to be on the order of the average interparticle spacing. Our study informs theories of pseudo-gap behavior in strongly correlated fermion systems.
We study pairing of an interacting three-component Fermi gas in two dimensions. By using a mean-field theory to decouple the interactions between different pairs of Fermi components, we study the free energy landscapes as a function of various system parameters including chemical potentials, binding energies, and temperature. We find that the sss-wave pairing channel is determined by both chemical potentials and the interaction strengths between the three available channels. We find a second-order thermal phase transition and a series of first-order quantum phase transitions for a homogenous system as we change the parameters. In particular, for symmetric parameters, we find the simultaneous existence of three superfluid orders as well as re-entrant quantum phase transitions as we tune the parameters.
Relation between the binding energy of a pair of electrons in Cooper's configuration and the magnitude of the attractive interaction between them is investigated theoretically for a two-dimensional electron gas when the electron pair has (a) zero and (b) finite center of mass momentum (CMM) first (i) in the background of a passive Fermi sea and (ii) with inclusion of the effect of Pauli exclusion principle in the presence of an active Fermi sea, at zero temperature. In the Cooper's approach (Cooper 1956 Phys. Rev. 104 1189) two pairing electrons are considered above the Fermi sea. Due to Pauli exclusion principle, these two electrons are forbidden to go into the states whose energies are less than the Fermi energy. Consequently, the role of the electrons which form the Fermi sea is passive in formation of electron pair (Fujita and Godoy 2001 Theory of High Temperature Superconductivity (Kluwer Academic Publishers)). However, in the case of an active Fermi sea, implying possible presence of multiple pairs of electrons coming out of the Fermi sea, the two electrons will have less available region in the momentum space for pair formation due to the Pauli blocking effect. In both the situations, threshold values of attractive interaction are found to exist to bind the electrons having finite CMM even with respect to the Fermi energy while with arbitrarily small value of attractive interaction they can be bound only in the case of zero CMM and that too in the presence of a passive Fermi sea. However, the presence of an active Fermi sea strongly opposes the pair formation having lower CMM. Furthermore, raising the range of the attractive interaction below and above the Fermi surface makes the pair formation more difficult due to enhanced Pauli blocking in this case. In the case of a passive Fermi sea however, with the increase in the range of the attractive interaction it becomes easier for the electron pairs to be in both Cooper pair state (CP) and true bound state (TB). These results are totally distinct from those obtained by Randeria et al (1989 Phys. Rev. Lett. 62 981, 1990 Phys. Rev. B 41 327) that a fermionic bound state in true vacuum is a necessary and sufficient condition for the Cooper pairing instability in the presence of a passive Fermi sea, independent of attractive interaction strength implying that for a potential which is attractive everywhere in two dimension, a two-body bound state exists for an arbitrarily weak attraction. Our calculations show, in the mean field approximation, that co-existence of TB state and CP state with various CMMs can occur only beyond a critical attractive interaction strength. This could lead to the formation of incoherent pairs even under strong coupling situation, which may hint onto the pseudo gap formation leading to some of the exotic features in the anomalous normal state seen in some of the exotic superconductors.