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arXiv:2112.03490 · Physical review. B./Physical review. B

Effect of repulsion on superconductivity at low density

Dan Phan, Andrey V. Chubukov

Published Feb 28, 2022• 7 citations
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Abstract

We examine the effect of repulsion on superconductivity in a three-dimensional system with a Bardeen-Pines-like interaction in the low-density limit, where the chemical potential $\ensuremath{\mu}$ is much smaller than the phonon frequency ${\ensuremath{\omega}}_{L}$. We parametrize the strength of the repulsion by a dimensionless parameter $f$, and find that the superconducting transition temperature ${T}_{c}$ approaches a nonzero value in the $\ensuremath{\mu}=0$ limit as long as $f$ is below a certain threshold ${f}^{*}$. In this limit, we find that ${T}_{c}$ goes to zero as a power of ${f}^{*}\ensuremath{-}f$, in contrast to the high density limit, where ${T}_{c}$ goes to zero exponentially quickly as $f$ approaches ${f}^{*}$. For all nonzero $f$, the gap function $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})$ changes sign along the Matsubara axis, which allows the system to partially overcome the repulsion at high frequencies. We trace the position of the gap node with $f$ and show that it approaches zero frequency as $f$ approaches ${f}^{*}$. To investigate the robustness of our conclusions, we then go beyond the Bardeen-Pines model and include full dynamical screening of the interaction, finding that ${T}_{c}$ still saturates to a nonzero value at $\ensuremath{\mu}=0$ when $f<{f}^{*}$.

DOI
10.1103/physrevb.105.064518

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We examine the effect of repulsion on superconductivity in a three-dimensional system with a Bardeen-Pines-like interaction in the low-density limit, where the chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ is much smaller than the phonon frequency \ensuremathωL{\ensuremath{\omega}}_{L}\ensuremathωL​. We parametrize the strength of the repulsion by a dimensionless parameter fff, and find that the superconducting transition temperature Tc{T}_{c}Tc​ approaches a nonzero value in the \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=0 limit as long as fff is below a certain threshold f∗{f}^{*}f∗. In this limit, we find that Tc{T}_{c}Tc​ goes to zero as a power of f∗\ensuremath−f{f}^{*}\ensuremath{-}ff∗\ensuremath−f, in contrast to the high density limit, where Tc{T}_{c}Tc​ goes to zero exponentially quickly as fff approaches f∗{f}^{*}f∗. For all nonzero fff, the gap function \ensuremathΔ(\ensuremathωm)\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})\ensuremathΔ(\ensuremathωm​) changes sign along the Matsubara axis, which allows the system to partially overcome the repulsion at high frequencies. We trace the position of the gap node with fff and show that it approaches zero frequency as fff approaches f∗{f}^{*}f∗. To investigate the robustness of our conclusions, we then go beyond the Bardeen-Pines model and include full dynamical screening of the interaction, finding that Tc{T}_{c}Tc​ still saturates to a nonzero value at \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=0 when f<f∗f<{f}^{*}f<f∗.
Motivated by the observation of superconductivity in SrTiO3{\mathrm{SrTiO}}_{3}SrTiO3​ and Bi, we analyze phonon-mediated superconductivity in three-dimensional systems at low carrier density, when the chemical potential $\ensuremath{\mu}$ is comparable to or even smaller than the characteristic phonon frequency ${\ensuremath{\omega}}_{L}$. We consider the attractive part of the Bardeen-Pines pairing interaction, in which the frequency-dependent electron-phonon interaction is dressed by the Coulomb potential. This dressing endows the pairing interaction with momentum dependence. We argue that the conventional Migdal-Eliashberg approximation becomes invalid when $\ensuremath{\mu}\ensuremath{\le}{\ensuremath{\omega}}_{L}$, chiefly because the dominant contribution to pairing comes from electronic states away from the Fermi surface. We obtain the pairing onset temperature, which is equal to Tc{T}_{c}Tc​ in the absence of phase fluctuations, as a function of $\ensuremath{\mu}/{\ensuremath{\omega}}_{L}$. We find both analytically and numerically that Tc{T}_{c}Tc​ increases as the ratio $\ensuremath{\mu}/{\ensuremath{\omega}}_{L}$ becomes smaller. In particular, in the dilute regime, $\ensuremath{\mu}\ensuremath{\rightarrow}0$, it holds that ${T}_{c}\ensuremath{\propto}{\ensuremath{\omega}}_{L}{(\frac{Ry}{{\ensuremath{\omega}}_{L}})}^{\ensuremath{\eta}}$, where Ry is the Rydberg constant and $\ensuremath{\eta}\ensuremath{\sim}0.2$.
Abstract We consider a model of electrons at zero temperature, with a repulsive interaction which is a function of the energy transfer. Such an interaction can arise from the combination of electron–electron repulsion at high energies and the weaker electron–phonon attraction at low energies. As shown in previous works, superconductivity can develop despite the overall repulsion due to the energy dependence of the interaction, but the gap Δ(ω) must change sign at some (imaginary) frequency ω0 to counteract the repulsion. However, when the constant repulsive part of the interaction is increased, a quantum phase transition towards the normal state occurs. We show that, as the phase transition is approached, Δ and ω0 must vanish in a correlated way such that 1/∣log⁡[Δ(0)]∣∼ω021/| \log [{{\Delta }}(0)]| \sim {\omega }_{0}^{2}1/∣log[Δ(0)]∣∼ω02​ 1 / ∣ log [ Δ ( 0 ) ] ∣ ~ ω 0 2 . We discuss the behavior of phase fluctuations near this transition and show that the correlation between Δ(0) and ω0 locks the phase stiffness to a non-zero value.
The two-dimensional Hubbard model is studied for small values of the interaction strength (UUU of the order of the hopping amplitude ttt), using a variational ansatz well suited for this regime. The wave function, a refined Gutzwiller ansatz, has a BCS mean-field state with ddd-wave symmetry as its reference state. Superconducting order is found for densities n<1n<1n<1 (but not for n=1n=1n=1). This resolves a discrepancy between results obtained with the functional renormalization group, which do predict superconducting order for small values of UUU, and numerical simulations, which did not find superconductivity for $U\ensuremath{\lesssim}4t$. Both the gap parameter and the order parameter have a domelike shape as a function of nnn with a maximum for $n\ensuremath{\approx}0.8$. Expectation values for the energy, the particle number, and the superconducting order parameter are calculated using a linked-cluster expansion up to second order in UUU. In this way large systems (millions of sites) can be readily treated and well converged results are obtained. A big size is indeed required to see that the gap becomes very small at half filling and probably tends to zero in the thermodynamic limit, whereas away from half filling a finite asymptotic limit is reached. For a lattice of a given size the order parameter becomes finite only above a minimal coupling strength Uc{U}_{c}Uc​. This threshold value decreases steadily with increasing system size, which indicates that superconductivity exists for arbitrarily small UUU for an infinite system. For moderately large systems the size dependence is quite irregular, due to variations in level spacings at the Fermi energy. The fluctuations die out if the gap parameter spans several level spacings.
In this paper we continue the analysis of the interplay between non-Fermi liquid and superconductivity for quantum-critical systems, the low-energy physics of which is described by an effective model with dynamical electron-electron interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\propto}1/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). In paper I [A. Abanov and A. V. Chubukov, Phys. Rev. B 102, 024524 (2020)], two of us analyzed the $\ensuremath{\gamma}$ model at T=0T=0T=0 for $0<\ensuremath{\gamma}<1$ and argued that there exists a discrete, infinite set of topologically distinct solutions for the superconducting gap, all with the same spatial symmetry. The gap function ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ for the nthn\mathrm{th}nth solution changes sign nnn times as the function of Matsubara frequency. In this paper we analyze the linearized gap equation at a finite TTT. We show that there exists an infinite set of pairing instability temperatures, Tp,n{T}_{p,n}Tp,n​, and the eigenfunction ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes sign nnn times as a function of a Matsubara number mmm. We argue that ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ retains its functional form below Tp,n{T}_{p,n}Tp,n​ and at T=0T=0T=0 coincides with the nthn\mathrm{th}nth solution of the nonlinear gap equation. Like in paper I, we extend the model to the case when the interaction in the pairing channel has an additional factor 1/N1/N1/N compared to that in the particle-hole channel. We show that Tp,0{T}_{p,0}Tp,0​ remains finite at large NNN due to special properties of fermions with Matsubara frequencies $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$, but all other Tp,n{T}_{p,n}Tp,n​ terminate at Ncr=O(1){N}_{\text{cr}}=O(1)Ncr​=O(1). The gap function vanishes at $T\ensuremath{\rightarrow}0$ for N>NcrN>{N}_{\text{cr}}N>Ncr​ and remains finite for N<NcrN<{N}_{\text{cr}}N<Ncr​. This is consistent with the T=0T=0T=0 analysis.
Near a quantum-critical point in a metal a strong fermion-fermion interaction, mediated by a soft boson, destroys fermionic coherence and also gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other, and in a class of large NNN models, where the tendency to incoherence is parametrically stronger, one would naively expect an incoherent (non-Fermi liquid) normal state behavior to persist down to T=0T=0T=0. However, this is not the case for quantum-critical systems, described by Eliashberg theory. In such systems, the part of the fermionic self-energy $\mathrm{\ensuremath{\Sigma}}({\ensuremath{\omega}}_{m})$, relevant for spin-singlet pairing, is large for a generic Matsubara frequency ${\ensuremath{\omega}}_{m}=\ensuremath{\pi}T(2m+1)$ but vanishes for fermions with ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$, while the pairing interaction between fermions with these two frequencies remains strong. It has been shown [Y. Wang et al. Phys. Rev. Lett. 117, 157001 (2016)] that due to this peculiarity, the onset temperature for the pairing Tp{T}_{p}Tp​ is finite even at large NNN, when the scaling analysis predicts a non-Fermi liquid normal state. We consider the system behavior below Tp{T}_{p}Tp​ and contrast the conventional case, when ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$ are not special, and the case when the pairing is induced by fermions with ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$. We obtain the solution of the nonlinear gap equations in Matsubara frequencies and then convert to real frequency axis and obtain the spectral function $A(k,\ensuremath{\omega})$ and the density of states $N(\ensuremath{\omega})$. In a conventional BCS-type superconductor, $A(k,\ensuremath{\omega})$ and $N(\ensuremath{\omega})$ are peaked at the gap value $\mathrm{\ensuremath{\Delta}}(T)$, and the peak position shifts to a smaller $\ensuremath{\omega}$ as temperature increases towards Tp{T}_{p}Tp​, i.e., the gap ``closes in.'' We show that, when the pairing is induced by fermions with ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$, the situation is qualitatively different from the standard BCS result. Namely, the peak in $N(\ensuremath{\omega})$ remains at a finite frequency even at $T={T}_{p}\ensuremath{-}0$, the gap just ``fills in'' near this TTT. The spectral function $A(k,\ensuremath{\omega})$ either shows almost the same ``gap filling'' behavior as the density of states, or its peak position shifts to zero frequency already at a finite $\mathrm{\ensuremath{\Delta}}$ (``emergent Fermi arc'' behavior), depending on the position of k\mathbf{k}k on the Fermi surface. As an example, we compare our results with the data for the cuprates and argue that ``gap filling'' behavior holds in the antinodal region, while the ``emergent Fermi arc'' behavior holds in the nodal region.
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