We revisit the renormalization group (RG) analysis of the Kohn-Luttinger (KL) mechanism for superconductivity. The KL mechanism leads to superconductivity in a system with a repulsive bare interaction. The key ingredient is the screening effect that renders the induced interaction attractive in channels with nonzero angular momentum $\ensuremath{\ell}\ensuremath{\ne}0$, thereby triggering the Bardeen-Cooper-Schrieffer (BCS) instability. According to the original argument, the resulting gap is exponentially small, with its exponent scaling as $\ensuremath{-}{\ensuremath{\ell}}^{4}$. However, the KL mechanism was originally formulated within perturbation theory, where the series is known to converge poorly in certain cases---most notably, for the
p-wave paring gap induced by a repulsive
s-wave contact interaction. This poor convergence may be attributed to a divergent integrand in a specific class of diagrams containing both the BCS logarithm and the Kohn anomaly, suggesting that one must resum the Kohn anomaly contributions separately from the BCS logarithm. In this work, we incorporate the Kohn anomaly contribution into the $\ensuremath{\beta}$ function of the RG equation governing the BCS instability near the Fermi surface. Our solution shows that the KL gap exponent is then proportional to $\ensuremath{-}\ensuremath{\ell}$, indicating a significant enhancement of the KL mechanism beyond the previously known result. To illustrate this, we study the spin-triplet
p-wave pairing gap arising from a repulsive
s-wave contact interaction and compare our RG-based results with those obtained from the Bethe-Salpeter equation in perturbation theory.
The Kohn-Luttinger (KL) mechanism of pairing, which describes superconductivity emergent from repulsive interactions, typically yields Cooper pairs at high angular-momentum (
ℓ>0) and extremely low transition temperatures (
Tc). Here, we reveal an inter-layer s-wave (
ℓ=0) KL superconductivity with greatly elevated
Tc in a multi-layer Hubbard model, which prototypes stacked two-dimensional (2D) electrons in layered van der Waals materials. By employing determinant quantum Monte Carlo and dynamical mean-field theory simulations, we show that a strong pairing attraction
V∗, without the mediation of collective modes, can emerge between inter-layer electrons in the system. As inter-layer repulsion
U increases,
V∗ evolves from a conventional KL relation of
V∗∝−U2, to a linear strong-coupling scaling of
V∗∝−U, resulting in enhanced superconductivity at large
U. This strong-coupling KL pairing is robust against changes in lattice geometries and dimensionalities, and it can persist, in the presence of a large remnant Coulomb repulsion
U∗ between pairing electrons. Using \textit{ab initio} calculations, we propose a few 2D layered van der Waals materials that can potentially realize and control this unconventional superconductivity.
We address the outstanding problem of electron pairing in the presence of strong Coulomb repulsion at small to moderate values of the Coulomb parameter, ${r}_{s}\ensuremath{\lesssim}2$, and demonstrate that the pseudopotential framework is fundamentally biased and uncontrolled. Instead, one has to break the net result into two distinctively different effects: the Fermi liquid renormalization factor and the change in the effective low-energy coupling. The latter quantity is shown to behave nonmonotonically with an extremum at ${r}_{s}\ensuremath{\approx}0.75$. Within the random-phase approximation, Coulomb interaction starts to enhance the effective pairing coupling at
rs>2, and the suppression of the critical temperature is entirely due to the renormalized Fermi liquid properties. Leading vertex corrections change this picture only quantitatively. Our results call for radical reconsideration of the widely accepted repulsive pseudopotential approach and show the need for precise microscopic treatment of Coulomb interactions in the problem of superconducting instability.
We consider possible superconducting instabilities in a two-dimensional Fermi system with short-ranged repulsive interactions between electrons. The possibility of an unusual superconducting paring due to the Kohn-Luttinger mechanism is examined. The quasiparticle scattering amplitude is shown to possess an attractive harmonic in second-order perturbation theory for finite values of the energy transfer. The corresponding singularity in the pairing vertex leads to a superconducting pairing of the electron excitations with finite energies. We identify the energy transfer in the Cooper channel as the binding energy of the excited pair. At low enough temperatures, the Fermi system is a mixture of normal electron excitations and fluctuating d-wave Cooper pairs possessing a finite gap.
We analyze the effect of the nonvanishing range of electron-electron repulsion on the mechanism of unconventional superconductivity. We present asymptotically exact weak-coupling results for dilute electrons in the continuum and for the 2D extended Hubbard model, as well as density-matrix renormalization group results for the two-leg extended Hubbard model at intermediate couplings, and approximate results for the case of realistically screened Coulomb interactions. We show that
Tc is generally suppressed in some pairing channels as longer range interactions increase in strength, but superconductivity is not destroyed. Our results confirm that electron-electron interaction can lead to unconventional superconductivity under physically realistic circumstances.
Motivation With few exceptions [131] it is widely accepted that the conventional Bardeen–Cooper–Schrieffer (BCS) theory [12] and its intermediate-coupling Eliashberg extension [14] do not suffice to explain high-temperature superconductivity. As shown in the preceding chapters the true origin of high-temperature superconductivity could be found in a proper combination of the bare Coulomb repulsion with the bare electron–phonon interaction in the framework of the many-body polaron theory [51] beyond the BCS–Eliashberg approximation. Nevertheless some researchers have maintained that the repulsive electron–electron interaction alone should provide pairing at high temperatures without phonons despite substantial experimental evidence that the EPI is an important player and needs to be included in the successful theory of high- T c superconductivity. Following the original proposal by P. W. Anderson, many authors [22, 74] argued that the electron–electron interaction in high-temperature superconductors was strong but repulsive providing high T c without phonons via superexchange and/or spin fluctuations in the d -wave pairing channel ( l = 2). A motivation for this concept can be found in the earlier works by Kohn and Luttinger (KL) [132, 133] who showed that the Cooper pairing of fermions with any weak repulsion was possible since the two-particle interaction induced by many-body effects is attractive for pairs with large orbital momenta, l ≫ 1.