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arXiv:2502.11624

High-Temperature Superconductivity from Finite-Range Attractive Interaction

Dmitry Miserev, Joel Hutchinson, Herbert Schoeller, Jelena Klinovaja, Daniel Loss

Published Feb 17, 2025• 0 citations
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Abstract

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 $T_c^* \sim 0.1 E_F$ on the interaction range at $R_s \gg λ_F$, $E_F$ is the Fermi energy. We believe that these results could be relevant to high-temperature superconductors.

DOI
10.48550/arxiv.2502.11624

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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 present a simple phenomenological scaling theory for the pairing instability of a quantum critical metal. It can be viewed as a minimal generalization of the classical Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity (SC) for normal Fermi-liquid metals. We assume that attractive interactions are induced in the fermion system by an external ``bosonic glue'' that is strongly retarded. Resting on the small Migdal parameter, all the required information from the fermion system needed to address the superconductivity enters through the pairing susceptibility. Asserting that the normal state is a strongly interacting quantum critical state of fermions, the form of this susceptibility is governed by conformal invariance and one only has the scaling dimension of the pair operator as free parameter. Within this scaling framework, conventional BCS theory appears as the ``marginal'' case but it is now easily generalized to the (ir)relevant scaling regimes. In the relevant regime an algebraic singularity takes over from the BCS logarithm with the obvious effect that the pairing instability becomes stronger. However, it is more surprising that this effect is strongest for small couplings and small Migdal parameters, highlighting an unanticipated important role of retardation. Using exact forms for the finite-temperature pair susceptibility from 1+1D1+1\text{D}1+1D conformal field theory as models, we study the transition temperatures, finding that the gap to transition temperature ratios is generically large compared to the BCS case, showing, however, an opposite trend as a function of the coupling strength compared to the conventional Migdal-Eliashberg theory. We show that our scaling theory naturally produces the superconducting ``domes'' surrounding the quantum critical points (QCPs), even when the coupling to the glue itself is not changing at all. We argue that hidden relations will exist between the location of the crossover lines to the Fermi liquids away from the quantum critical points and the detailed form of the dome when the glue strength is independent of the zero-temperature control parameter. Finally, we discuss the behavior of the orbital-limited upper critical magnetic field as a function of the zero-temperature coupling constant. Compared to the variation in the transition temperature, the critical field might show a much stronger variation pending the value of the dynamical critical exponent.
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.
Domes of TcT_cTc​ in superconductors with finite-range attractive interactions
The rise and fall of the superconducting transition temperature TcT_cTc​ upon tuning carrier density or external parameters, such as pressure or magnetic field, is ubiquitously observed in a wide range of quantum materials. In order to investigate such domes of TcT_cTc​, we go beyond the prototypical Hubbard model, and consider a lattice model of electrons coupled via instantaneous, spatially extended, attractive interactions. Using mean-field theory and functional renormalization group (FRG) methods, we find that for a characteristic interaction range ℓ\ellℓ, there exists a dome in TcT_cTc​ around kFℓ∼O(1)k_F \ell \sim \mathcal{O}(1)kF​ℓ∼O(1). This result can intuitively be understood from the geometric relation between the Fermi surface and the interaction range. We show that our results generally hold for single-band as well as multi-band superconductors, in both two and three dimensions. Our model may be relevant for domes of TcT_cTc​ in dilute weakly coupled superconductors or in engineered cold atom systems.
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.
Relation between the binding energy of a pair of electrons in Cooper's configuration and the magnitude of the attractive interaction between them is investigated theoretically for a two-dimensional electron gas when the electron pair has (a) zero and (b) finite center of mass momentum (CMM) first (i) in the background of a passive Fermi sea and (ii) with inclusion of the effect of Pauli exclusion principle in the presence of an active Fermi sea, at zero temperature. In the Cooper's approach (Cooper 1956 Phys. Rev. 104 1189) two pairing electrons are considered above the Fermi sea. Due to Pauli exclusion principle, these two electrons are forbidden to go into the states whose energies are less than the Fermi energy. Consequently, the role of the electrons which form the Fermi sea is passive in formation of electron pair (Fujita and Godoy 2001 Theory of High Temperature Superconductivity (Kluwer Academic Publishers)). However, in the case of an active Fermi sea, implying possible presence of multiple pairs of electrons coming out of the Fermi sea, the two electrons will have less available region in the momentum space for pair formation due to the Pauli blocking effect. In both the situations, threshold values of attractive interaction are found to exist to bind the electrons having finite CMM even with respect to the Fermi energy while with arbitrarily small value of attractive interaction they can be bound only in the case of zero CMM and that too in the presence of a passive Fermi sea. However, the presence of an active Fermi sea strongly opposes the pair formation having lower CMM. Furthermore, raising the range of the attractive interaction below and above the Fermi surface makes the pair formation more difficult due to enhanced Pauli blocking in this case. In the case of a passive Fermi sea however, with the increase in the range of the attractive interaction it becomes easier for the electron pairs to be in both Cooper pair state (CP) and true bound state (TB). These results are totally distinct from those obtained by Randeria et al (1989 Phys. Rev. Lett. 62 981, 1990 Phys. Rev. B 41 327) that a fermionic bound state in true vacuum is a necessary and sufficient condition for the Cooper pairing instability in the presence of a passive Fermi sea, independent of attractive interaction strength implying that for a potential which is attractive everywhere in two dimension, a two-body bound state exists for an arbitrarily weak attraction. Our calculations show, in the mean field approximation, that co-existence of TB state and CP state with various CMMs can occur only beyond a critical attractive interaction strength. This could lead to the formation of incoherent pairs even under strong coupling situation, which may hint onto the pseudo gap formation leading to some of the exotic features in the anomalous normal state seen in some of the exotic superconductors.