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arXiv:1112.1112 · Physical Review B

General pairing theory for condensed and noncondensed Cooper pairs of a superconductor in a high magnetic field

Peter Scherpelz, Dan Wulin, Břetislav Šopík, Kathryn Levin, A. K. Rajagopal

Published Jan 28, 2013• 6 citations
arXivHTMLPDF

Abstract

We extend Gor'kov theory to address superconducting pairing at high magnetic fields and general temperatures with arbitrary attractive interaction strength. This analysis begins with an interpretation of the high-field Gor'kov gap equation which we associate with an instability in a generalized particle-particle ladder series. Importantly, this interpretation of the nonlinear gap equation enables a treatment of pairing which is distinct from condensation. We also show how to consolidate two distinct fermionic pairing schemes in real and momentum space, both corresponding to an Abrikosov lattice. Numerical results for the fermionic local density of states demonstrate that gapless structure in a field is robust and presumably relevant to quantum oscillation experiments. We find that despite their differences, both pairing schemes contain very similar physics. Our formalism is designed to explore a variety of magnetic field effects in the so-called pseudogap phase and throughout the BCS-BEC crossover.

DOI
10.1103/physrevb.87.024516

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We extend Gor'kov theory to address superconducting pairing at high magnetic fields and general temperatures with arbitrary attractive interaction strength. This analysis begins with an interpretation of the high-field Gor'kov gap equation which we associate with an instability in a generalized particle-particle ladder series. Importantly, this interpretation of the nonlinear gap equation enables a treatment of pairing which is distinct from condensation. We also show how to consolidate two distinct fermionic pairing schemes in real and momentum space, both corresponding to an Abrikosov lattice. Numerical results for the fermionic local density of states demonstrate that gapless structure in a field is robust and presumably relevant to quantum oscillation experiments. We find that despite their differences, both pairing schemes contain very similar physics. Our formalism is designed to explore a variety of magnetic field effects in the so-called pseudogap phase and throughout the BCS-BEC crossover.
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We address the important question of how to characterize the normal state and the nature of noncondensed pairs in fermionic superfluids under the influence of a strong effective magnetic field. In ultracold gases, the magnetic field is implemented through rapid rotation or novel artificial field techniques. We consider the near-unitary regime, where noncondensed pairs are likely to be present at temperatures above the condensation temperature Tc{T}_{c}Tc​. We show (based on Gor'kov theory) that these pairs are associated with a precursor of a vortex configuration. Importantly, this nonuniform normal state appears to enable ``Bose condensation'' in a field which is otherwise problematic due to the effective one-dimensionality of Landau-level dispersion.
This paper addresses the transition from the normal to the superfluid state in strongly correlated two dimensional fermionic superconductors and Fermi gases. We arrive at the Berezinskii-Kosterlitz-Thouless (BKT) temperature TBKTT_{\text{BKT}}TBKT​ as a function of \emph{attractive} pairing strength by associating it with the onset of quasi-condensation in the normal phase. Our approach builds on a criterion for determining the BKT transition temperature for atomic gases which is based on a well established Quantum Monte Carlo analysis of the phase space density. This latter quantity, when derived from BCS-BEC crossover theory for fermions, leads to non-monotonic behavior for TBKTT_{\text{BKT}}TBKT​ as a function of the attractive interaction or inverse scattering length. In Fermi gases, this implies a robust superconducting dome followed by a long tail from the flat BEC asymptote, rather similar to what is observed experimentally. For lattice systems we find that TBKTT_{\text{BKT}}TBKT​ has an absolute maximum of the order of 0.1EF0.1 E_F0.1EF​. We discuss how our results compare with those derived from the Nelson Kosterlitz criterion based on the mean field superfluid density and the approach to the transition from below. While there is agreement in the strict mean-field BCS regime at weak coupling, we find that at moderate pairing strength bosonic excitations cause a substantial increase in TBKTT_\text{BKT}TBKT​ followed by an often dramatic decrease before the system enters the BEC regime.
We review Cooper pairing starting from its simplest, original 1956 version of two electrons interacting above the Fermi sea of an ideal Fermi gas (IFG). The two-electron interaction assumed extensively (if not exclusively), is the attractive two-parameter Cooper, and then BCS, model interactions. Hole Cooper pairs (CPs) and electron-hole CPs are then included along with the initial electron-CPs in terms of the single-fermion Green functions implied by the Bethe-Salpeter (BS) integral equation in the ladder approximation. A purely-imaginary CP energy "instability" is recovered that is well-documented in the literature at least since the late 1950's. A novel interpretation of this instability is that an unperturbed Hamiltonian different from the IFG one first used by Cooper suffices to obtain meaningful CPs. Instead of the IFG sea, a BCS-correlated Fermi "sea" used in the BS equation interpreted as the associated unperturbed Hamiltonian leads to real CP energies (with small imaginary terms implying damping). We survey how this has been achieved in 1D, 2D and 3D, and give a more detailed treatment in 2D. A vital distinction is that the original and generalized CPs are true bosons in contrast with BCS pairs that are not ordinary bosons but rather "hard-core bosons" as they do not obey strict Bose commutation rules. Another important common element of the original or generalized CPs (particularly in 2D where ordinary Bose-Einstein condensation (BEC) does not occur) is their linear dispersion relation in leading order in the total (or, center-of-mass) momentum power-series expansion of the CP energy. This theory encompasses, in principle, all empirically known superconductors including quasi-2D superconductors such as cuprates and the ET organic compounds, as well as quasi-1D ones such as the organometallic Bechgaard salts and nanotubes.
We have developed a method for studying superconductivity in the negative-U Hubbard model. It is based on the Hubbard-alloy-analogy approximation applied to the Gor'kov equation for the model. We find that, unlike the case of the BCS solution, the formation of pairs and their condensation into a superconducting ground state take place at two different temperatures. We calculate the condensation temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ for all U and discuss the nature of the normal state which, for large enough U, features a macroscopically large number of pairs. Briefly, we argue that the model has relevance for high-temperature superconductors.
The equation for the gap parameter represents the main equation of the pairing theory of superconductivity. Although it is formally defined through a single-particle property, physically, it reflects the pairing correlations between opposite-spin fermions. Here, we exploit this physical connection and cast the gap equation in an alternative form which explicitly highlights these two-particle correlations by showing that it is equivalent to a Hugenholtz-Pines condition for fermion pairs. At a formal level, a direct connection is established in this way between the treatment of the condensate fraction in condensate systems of fermions and bosons. At a practical level, the use of this alternative form of the gap equation is expected to make easier the inclusion of pairing fluctuations beyond mean field. As a proof-of-concept of the new method, we apply the modified form of the gap equation to the long-pending problem about the inclusion of the Gorkov-Melik-Barkhudarov correction across the whole BCS-BEC crossover, from the BCS limit of strongly overlapping Cooper pairs to the BEC limit of dilute composite bosons, and for all temperatures in the superfluid phase. Our numerical calculations yield excellent agreement with the recently determined experimental values of the gap parameter for an ultracold Fermi gas in the intermediate regime between BCS and BEC, as well as with the available quantum Monte Carlo data in the same regime.