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arXiv:cond-mat/9908162

A model of marginal Fermi liquid superconductor at two dimensions

Yu-Liang Liu, Tai Kai Ng

Published Aug 11, 1999• 0 citations
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Abstract

In this paper we study a model of s-wave marginal Fermi liquid superconductor at two dimensions. Besides the usual Bardeen-Cooper-Schrieffer(BCS) point attractive interaction, the fermions in our system also interact {\em via} a long range repulsive interaction $v(q)=g_{D}/q^2$. We find that although superconductivity is not destroyed by the long-ranged repulsion interaction (demonstrated by the existence of Meissner effect), nevertheless the BCS pairing function $Δ_{BCS}$ is strongly suppressed resulting in power-law correlation $\sim|\vec{x}|^{-2γ}$. This observation is explained within a general picture of marginal Fermi liquid.

DOI
10.48550/arxiv.cond-mat/9908162

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In this paper we study a model of an sss-wave marginal Fermi liquid superconductor in two dimensions. Besides the usual Bardeen-Cooper-Schrieffer (BCS) point attractive interaction, the fermions in our system also interact via a long-range repulsive interaction v(q)=gD/q2v(q){\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}g}_{D}/{q}^{2}v(q)=gD​/q2. We find that although superconductivity is not destroyed by the long-ranged repulsive interaction (demonstrated by the response to electromagnetic magnetic fields), the BCS pairing function ${\ensuremath{\Delta}}_{\mathrm{BCS}}$ is strongly suppressed, resulting in a power-law correlation $〈{\ensuremath{\Delta}}_{\mathrm{BCS}}(\stackrel{\ensuremath{\rightarrow}}{x}){\ensuremath{\Delta}}_{\mathrm{BCS}}(0)〉\ensuremath{\sim}|\stackrel{\ensuremath{\rightarrow}}{x}{|}^{\ensuremath{-}2\ensuremath{\gamma}}$. This observation is explained within a general picture of the marginal Fermi liquid.
In this letter we consider DDD-dimensional interacting Fermi liquids, and demonstrate that an attractive interaction with a finite range RsR_sRs​ that is much greater than the Fermi wavelength λFλ_FλF​ breaks the conventional BCS theory of superconductivity. In contrast to the BCS prediction of a finite superconducting gap for all attractive contact interactions, we show that a finite-range interaction does not induce a superconducting gap. Instead, the pair susceptibility develops a power-law singularity at zero momentum and zero frequency signaling quantum critical behavior without long-range ordering. Starting from this, we show that superconductivity can be stabilized by adding a short-range attractive interaction, which is always present in real electronic systems. As an example, we consider a layered quasi-two-dimensional material with attractive electron-electron interactions mediated by optical phonons. We demonstrate a dome shape of the critical temperature TcT_cTc​ versus doping, strongly suppressed isotope effect, and a weak dependence of the optimal doping and maximal Tc∗∼0.1EFT_c^* \sim 0.1 E_FTc∗​∼0.1EF​ on the interaction range at Rs≫λFR_s \gg λ_FRs​≫λF​, EFE_FEF​ is the Fermi energy. We believe that these results could be relevant to high-temperature superconductors.
In this Letter we consider D-dimensional interacting Fermi liquids, and demonstrate that an attractive interaction with a finite range R_{s} that is much greater than the Fermi wavelength λ_{F} breaks the conventional BCS theory of superconductivity. In contrast to the BCS prediction of a finite superconducting gap for all attractive contact interactions, we show that a finite-range interaction does not induce a superconducting gap. Instead, the pair susceptibility develops a power-law singularity at zero momentum and zero frequency signaling quantum critical behavior without long-range ordering. Starting from this, we show that superconductivity can be stabilized by adding a short-range attractive interaction, which is always present in real electronic systems. As an example, we consider a layered quasi-two-dimensional material with attractive electron-electron interactions mediated by optical phonons. We demonstrate a dome shape of the critical temperature T_{c} versus doping, strongly suppressed isotope effect, and a weak dependence of the optimal doping and maximal T_{c}^{*}∼0.1E_{F} on the interaction range at R_{s}≫λ_{F}, E_{F} is the Fermi energy. We believe that these results could be relevant to high-temperature superconductors.
We study the Kohn-Luttinger effect in a two-dimensional (2D) nested fermion liquid with a repulsive interaction via the renormalization group method and identify the resulting order parameter symmetry. Using the band structure of the 2D Hubbard model close to half-filling as a prototype, we construct an effective low-energy theory. We use multidimensional bosonization to incorporate the zero-sound channel and find marginal Fermi liquid behavior in the absence of any instability. We show an analog of the Landau theorem in nested fermion liquids, which serves as the criterion of the BCS instability. Including repulsive or antiferromagnetic exchange interactions in the low-energy theory, we show that the ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$-wave BCS channel is renormalized to be the most attractive. Below half-filling, when the nesting is not perfect, there is competition between the spin-density-wave (SDW) and the BCS channels; when the SDW coupling is small enough, there occurs a ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$-wave superconducting instability at sufficiently low temperatures.
A free Fermi gas has, famously, a superconducting susceptibility that diverges logarithmically at zero temperature. In this paper we ask whether this is still true for a Fermi liquid and find that the answer is that it does not. From the perspective of the renormalization group for interacting fermions, the question arises because a repulsive interaction in the Cooper channel is a marginally irrelevant operator at the Fermi liquid fixed point and thus is also expected to infect various physical quantities with logarithms. Somewhat surprisingly, at least from the renormalization group viewpoint, the result for the superconducting susceptibility is that two logarithms are not better than one. In the course of this investigation we derive a Callan-Symanzik equation for the repulsive Fermi liquid using the momentum-shell renormalization group, and use it to compute the long-wavelength behavior of the superconducting correlation function in the emergent low-energy theory. We expect this technique to be of broader interest.
We use the concept of generalized (almost localized) Fermi Liquid composed of nonstandard quasiparticles with spin-dependence effective masses and the effective field induced by electron correlations.This Fermi liquid is obtained within the so-called statistically-consistent Gutzwiller approximation (SGA) proposed recently [cf.J. Je ¸drak et al., arXiv: 1008.0021] and describes electronic states of the correlated quantum liquid.Particular emphasis is put on real space pairing driven by the electronic correlations, the Fulde-Ferrell state of the heavy-fermion liquid, and the d-wave superconducting state of high temperature cuprate superconductors in the overdoped limit.The appropriate phase diagrams are discussed showing in particular the limits of stability of the Bardeen-Cooper-Schrieffer (BCS) type of state.