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arXiv:1902.02337

A mathematical derivation of zero-temperature 2D superconductivity from microscopic Bardeen-Cooper-Schrieffer model

Magnen, J., Jérémie M. Unterberger

Published Feb 6, 2019• 0 citations
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Abstract

Starting from H. Fröhlich's second-quantized Hamiltonian for a $d$-dimensional electron gas in interaction with lattice phonons describing the quantum vibrations of a metal, we present a rigorous mathematical derivation of the superconducting state, following the principles laid out originally in 1957 by J. Bardeen, L. Cooper and J. Schrieffer. As in the series of papers written on the subject in the 90es, of which the present paper is a continuation, the representation of ions as a uniform charge background allows for a $(1+d)$-dimensional fermionic quantum-field theoretic reformulation of the model at equilibrium. For simplicity, we restrict in this article to $d=2$ dimensions and zero temperature, and disregard effects due to electromagnetic interactions. Under these assumptions, we prove transition from a Fermi liquid state to a superconducting state made up of Cooper pairs of electrons at an energy level $Γ_ϕ\sim \hbarω_D e^{-π/mλ}$ equal to the mass gap, expressed in terms of the Debye frequency $ω_D$, electron mass $m$ and coupling constant $λ$. The dynamical $U(1)$-symmetry breaking produces at energies lower than the energy gap $Γ_ϕ$ a Goldstone boson, a non-massive particle described by an effective $(2+1)$-dimensional non-linear sigma-model, whose parameters and correlations are computed. The proof relies on a mixture of general concepts and tools (multi-scale cluster expansions, Ward identities), adapted to this quantum many-body problem with its extended infra-red singularity located on the Fermi circle, and a specific $1/N$-expansion giving the leading diagrams at intermediate energies. Ladder diagrams are proved to provide the leading behavior in the infra-red limit, in agreement with mean-field theory predictions.

DOI
10.48550/arxiv.1902.02337

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We review efforts to unify both the Bardeen, Cooper and Schrieffer (BCS) and Bose-Einstein condensation (BEC) pictures of superconductivity. We have finally achieved this in terms of a "\textit{complete} boson-fermion (BF) model" (CBFM) that reduces in special cases to all the main continuum (as opposed to "spin") statistical theories of superconductivity. Our BF model is "complete" in the sense that not only two-electron (2e) but also two-hole (2h) Cooper pairs (CPs) are allowed in arbitrary proportions. In contrast, BCS-Bogoliubov theory--which can also be considered as the theory of a mixture of kinematically independent electrons, 2e- and 2h-CPs--allows only equal, 50%-50%, mixtures of the two kinds of CPs. This is obvious from the perfect symmetry about μμμ, the electron chemical potential, of the well-known Bogoliubov v2(ε)v^{2}(ε)v2(ε) and u2(ε)u^{2}(ε)u2(ε) coefficients, where εεε is the electron energy. The CBFM is then applied to see: a) whether the BCS model interaction for the electron-phonon dynamical mechanism is sufficient to predict the unusually high values of TcT_{c}Tc​ (in units of the Fermi temperature) of ≃0.01−0.1\simeq 0.01-0.1≃0.01−0.1 exhibited by the so-called ``exotic'' superconductors \cite{Brandow} in both 2D and 3D--relative to the low values of ≲10−3\lesssim 10^{-3}≲10−3 more or less correctly predicted by BCS theory for conventional, elemental superconductors; and b) whether it can at least suggest, if not explain, why "hole superconductors" have higher TcT_{c}Tc​'s.
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.
The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the sss-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or XY\mathit{XY}XY universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple sss-wave picture.
Abstract As charge carrier of the macroscopic superconductivity, the Cooper pair is a composite particle of two paired electrons, which has both center-of-mass and inner-pair degrees of freedom. In most cases, these two different degrees of freedom can be well described by the macroscopic Ginzburg-Landau theory and the microscopic Bardeen-Cooper-Schrieffer (BCS) theory, respectively. Near the superconducting phase transition where the Cooper pair is fragile and unstable because of the small binding energy, there are non-trivial couplings between these two different degrees of freedom due to such as finite energy and/or momentum transfer. The non-trivial couplings make the original derivation of the Ginzburg-Landau theory from the BCS theory fail in principle as where these two different degrees of freedom should not be decoupled. In this article, we will present a renormalization formalism for an extended Ginzburg-Landau action for the superconducting phase transition where there is finite energy transfer between the center-of-mass and the inner-pair degrees of freedom of Cooper pairs. This renormalization formalism will provide a theoretical tool to study the unusual dynamical effects of the inner-pair time-retarded physics on the superconducting phase transition.