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Physical Review Letters

Model of a Two-Dimensional Marginal Fermi Liquid Superconductor

Yu-Liang Liu, Tai Kai Ng

Published Dec 27, 1999• 9 citations
Publisher

Abstract

In this paper we study a model of an $s$-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 $v(q){\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}g}_{D}/{q}^{2}$. 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.

DOI
10.1103/physrevlett.83.5539

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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 \ensuremathΔBCS{\ensuremath{\Delta}}_{\mathrm{BCS}}\ensuremathΔBCS​ is strongly suppressed, resulting in a power-law correlation 〈\ensuremathΔBCS(x\ensuremath→)\ensuremathΔBCS(0)〉\ensuremath∼∣x\ensuremath→∣\ensuremath−2\ensuremathγ〈{\ensuremath{\Delta}}_{\mathrm{BCS}}(\stackrel{\ensuremath{\rightarrow}}{x}){\ensuremath{\Delta}}_{\mathrm{BCS}}(0)〉\ensuremath{\sim}|\stackrel{\ensuremath{\rightarrow}}{x}{|}^{\ensuremath{-}2\ensuremath{\gamma}}〈\ensuremathΔBCS​(x\ensuremath→)\ensuremathΔBCS​(0)〉\ensuremath∼∣x\ensuremath→∣\ensuremath−2\ensuremathγ. This observation is explained within a general picture of the marginal Fermi liquid.
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)=gD/q2v(q)=g_{D}/q^2v(q)=gD​/q2. 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 ΔBCSΔ_{BCS}ΔBCS​ is strongly suppressed resulting in power-law correlation ∼∣x⃗∣−2γ\sim|\vec{x}|^{-2γ}∼∣x∣−2γ. This observation is explained within a general picture of 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.
We develop a general theory of fermion liquids in spatial dimensions greater than 1. The principal method, bosonization, is applied to the cases of short- and long-range longitudinal interactions and to transverse gauge interactions. All the correlation funtions of the system may be obtained with the use of a generating functional. Short-range and Coulomb interactions do not destroy the Landau-Fermi fixed point. Non-Fermi liquid fixed points are found, however, in the cases of a super-long-range longitudinal interaction in two dimensions and transverse gauge interactions in two and three spatial dimensions. We consider in some detail the (2+1)-dimensional problem of a Chern-Simons gauge action combined with a longitudinal two-body interaction V(q)\ensuremath{\propto}\ensuremath{\Vert}q${\mathrm{\ensuremath{\Vert}}}^{\mathit{y}\mathrm{\ensuremath{-}}1}$, which controls the density, and hence gauge, fluctuations. For y0 we find that the gauge interaction is irrelevant and the Landau fixed point is stable, while for y>0 the interaction is relevant and the fixed point cannot be accessed by bosonization. Of special importance is the case y=0 (Coulomb interaction), which describes the Halperin-Lee-Read theory of the half-filled Landau level. We obtain the full quasiparticle propagator, which is of a marginal Fermi-liquid form. Using Ward identities, we show that neither the inclusion of nonlinear terms in the fermion dispersion nor vertex corrections alters our results: the fixed point is accessible by bosonization. As the two-point fermion Green's function is not gauge invariant, we also invetigate the gauge-invariant density response function. Near momentum Q=2kF{\mathit{k}}_{\mathit{F}}kF​, in addition to the Kohn anomaly we find other nonanalytic behavior. In the appendies we present a numerical calculation of the spectral function for a Fermi liquid with Landau parameter f0{\mathit{f}}_{0}f0​\ensuremath{\ne}0. We also show how Kohn's theorem is satisfied within the bosonization framework.
(1) The temperature dependence of the specific heat for a marginal Fermi liquid has been calculated. (2) We calculated the self-energy at T=0 for a two dimensional fermionic system with hyperbolic dispersion. The existence of the saddle points in the energy gives rise to a marginal behavior. (3) We showed that the two-dimensional fermionic system with the energy εk=kxkyε_{\mathbf{k}}=k_xk_yεk​=kx​ky​ has a non-Fermi behavior. (4) The electronic self-energy due to the electron-spin interaction is calculated for a two dimensional system. (5) We study the influence of the amplitude fluctuations of a non-Fermi superconductor on the energy spectrum of the two-dimensional Anderson non-Fermi system. (6) Using the field-theoretical methods we studied the evolution from BCS theory to Bose-Einstein condensation (BEC) for a non-Fermi system. (7) Using the renormalization group approach proposed by Millis we calculated the specific heat coefficient γ(T)γ(T)γ(T) for the magnetic fluctuations with susceptibility χ−1∼δα+∣ω∣α+f(q)χ^{-1}\sim δ^α+| ω| ^α+f(q)χ−1∼δα+∣ω∣α+f(q) near a Lifshitz point.