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arXiv:cond-mat/9710240 · Journal of Physics Condensed Matter

Universal scaling in Bardeen-Cooper-Schrieffer superconductivity in two dimensions in non-s waves

Sadhan K. Adhikari, Angsula Ghosh

Published Jan 12, 1998• 11 citations
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Abstract

The solutions of a renormalized BCS model are studied in two space dimensions for s, p and d waves for finite-range separable potentials. The gap parameter, the critical temperature , the coherence length and the jump in specific heat at as a function of the zero-temperature condensation energy exhibit universal scalings. In the weak-coupling limit, the present model yields a small and large , appropriate for high- cuprates. The specific heat, penetration depth and thermal conductivity as functions of temperature show universal scaling for p and d waves.

DOI
10.1088/0953-8984/10/1/015

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The solutions of a renormalized BCS model are studied in two space dimensions for s, p and d waves for finite-range separable potentials. The gap parameter, the critical temperature , the coherence length and the jump in specific heat at as a function of the zero-temperature condensation energy exhibit universal scalings. In the weak-coupling limit, the present model yields a small and large , appropriate for high- cuprates. The specific heat, penetration depth and thermal conductivity as functions of temperature show universal scaling for p and d waves.
The BCS superconductivity to Bose condensation crossover problem is studied in two dimensions in S, P, and D waves, for a simple anisotropic pairing, with a finite-range separable potential at zero temperature. The gap parameter and the chemical potential as a function of Cooper-pair binding Bc{\mathrm{B}}_{\mathrm{c}}Bc​ exhibit universal scaling. In the BCS limit the results for coherence length \ensuremath{\xi} and the critical temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ are appropriate for high-Tc{\mathrm{T}}_{\mathrm{c}}Tc​ cuprate superconductors and also exhibit universal scaling as a function of Bc{\mathrm{B}}_{\mathrm{c}}Bc​.
In this paper, we investigate the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity formulated on a three dimensional Riemannian manifold. We derive the gap equation in the path integral picture by employing a Hubbard-Stratonovich transformation followed by a saddle point approximation. Under the assumption that the length scale set by the metric is much larger than the healing length, we solve the gap equation to find a slowly varying gap function that exhibits the effects of curvature. To rigorously address the ultraviolet divergences inherent in the gap equation, we utilize heat kernel expansion techniques, which provide a systematic and mathematically transparent renormalization scheme. We explicitly evaluate the renormalized gap equation at both zero and finite temperatures, computing the relevant integrals asymptotically in the weak-coupling limit (Δ≪μΔ\ll μΔ≪μ). Furthermore, we establish a universal, regularization-independent relation connecting the finite-temperature gap to the zero-temperature gap and the temperature itself. Based on this fundamental relation, we analytically prove the monotonic decrease of the energy gap with increasing temperature (∂Δ/∂T<0\partial Δ/ \partial T < 0∂Δ/∂T<0), successfully recovering the standard universal BCS behavior near the critical temperature within a curved space framework.
Abstract. The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential. 1.
The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.
Analytic expressions for the Bardeen–Cooper–Schrieffer (BCS) gap of a many-electron system with the BCS model interaction are obtained in one, two, and three dimensions in the weak coupling limit, but for arbitrary interaction width ν≡ℏωD/EF, 0<ν<∞. Here ℏωD is the maximum energy of a force-mediating boson, and EF is the Fermi energy (which is fixed by the electronic density). Our results address both phononic (ν≪1) as well as nonphononic (e.g., exciton, magnon, plasmon, etc.) pairing mechanisms where the mediating boson energies are not small compared with EF, provided weak electron–boson coupling prevails. The essential singularity in coupling, sometimes erroneously attributed to the two-dimensional character of the BCS model interaction with ν≪1, is shown to appear in one, two, and three dimensions before the limit ν→0 is taken.
We consider the evolution of superfluid properties of a three-dimensional p-wave Fermi gas from a weak coupling Bardeen-Cooper-Schrieffer (BCS) to strong coupling Bose-Einstein condensation (BEC) limit as a function of scattering volume. At zero temperature, we show that a quantum phase transition occurs for p-wave systems, unlike the s-wave case where the BCS to BEC evolution is just a crossover. Near the critical temperature, we derive a time-dependent Ginzburg-Landau (GL) theory and show that the GL coherence length is generally anisotropic due to the p-wave nature of the order parameter, and becomes isotropic only in the BEC limit.