We study the Cooper instability in jellium model in the controlled regime of small to intermediate values of the Coulomb parameter $r_s \leq 2$. We confirm that superconductivity naturally emerges from purely repulsive interactions described by the Kukkonen-Overhauser vertex function. By employing the implicit renormalization approach we reveal that even in the small-$r_s$ limit, the dominant mechanism behind Cooper instability is based on dynamic screening of the Coulomb interaction--accurately captured by the random phase approximation, whereas the Kohn-Luttinger contribution is negligibly small and, thus, not relevant.
The problem of Cooper instability is studied by implementing the Kohn-Luttinger mechanism in the framework of the Shubin-Vonsowsky model taking into account the intersite Coulomb interactions within the first and second coordination spheres. It is shown that the Kohn-Luttinger renormalization for the effective interaction in the second-order terms of perturbation theory, as well as the inclusion of intersite hoppings in the second and third coordination spheres, significantly affects the interplay between the superconducting phases with $d_{xy}$-, $p$-, $s$-, and $d_{x^2-y^2}$-wave symmetries …
The effect of the long-range Coulomb interaction on the realization of the Kohn-Luttinger superconductivity in idealized monolayer doped graphene is studied. It is shown that the allowance for the Kohn-Luttinger renormalizations up to the second order in perturbation theory in the on-site Hubbard interaction inclusively, as well as the intersite Coulomb interaction significantly affects the competition between the superconducting phases with the $f$-wave, $p+ip$-wave and $d + id$-wave symmetries of the order parameter. It is shown that the account for the Coulomb repulsion of electrons located…
Idealized graphene monolayer is considered neglecting the van der Waals potential of the substrate and the role of the nonmagnetic impurities. The effect of the long-range Coulomb repulsion in an ensemble of Dirac fermions on the formation of the superconducting pairing in a monolayer is studied in the framework of the Kohn-Luttinger mechanism. The electronic structure of graphene is described in the strong coupling Wannier representation on the hexagonal lattice. We use the Shubin-Vonsowsky model which takes into account the intra- and intersite Coulomb repulsions of electrons. The Cooper ins…
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 effect of the intersite and interplane Coulomb interactions between the Dirac fermions on the formation of the Kohn-Luttinger superconductivity in bilayer doped graphene is studied disregarding the effects of the van der Waals potential of the substrate and both magnetic and non-magnetic impurities. The phase diagram determining the boundaries of superconductive domains with different types of symmetry of the order parameter is built using the extended Hubbard model in the Born weak-coupling approximation with allowance for the intratomic, interatomic, and interlayer Coulomb interactions b…
Kohn and Luttinger showed that a many body system of fermions interacting via short range forces becomes superfluid even if the interaction is repulsive in all partial waves. In gauge theories such as QCD the interaction between fermions is long range and the assumptions of Kohn and Luttinger are not satisfied. We show that in a U(1) gauge theory the Kohn-Luttinger phenomenon does not take place. In QCD attractive channels always exist, but there are cases in which the primary pairing channel leaves some fermions ungapped. As an example we consider the unpaired fermion in the 2SC phase of QCD …
Weak coupling theory predicts the critical temperature of a phonon superconductor to be $T_c = 1.13 e^{-3/2} ω_D e^{-{1/λ}}$, where $ω_D$ is the Debye frequency, $λ$ is the dimensionless electron-phonon coupling constant, and the factor $e^{-3/2}$ comes from fermionic self-energy and frequency dependence of the interaction. Other corrections are small either in $ω_D/E_F$, by Migdal's theorem, or in $λ$. However, this formula assumes that $ω_D \ll E_F$, where $E_F$ is the Fermi energy. We obtain $T_c$ in the dilute regime, when the Fermi energy is smaller than $ω_D$. We argue that in this situa…
The effect of Coulomb interaction in an ensemble of Dirac fermions on the formation of superconducting pairing in monolayer and bilayer doped graphene is studied using the Kohn-Luttinger mechanism disregarding the Van der Waals potential of the substrate and impurities. The electronic structure of graphene is described using the Shubin-Vonsovsky model taking into account the intratomic, interatomic, and interlayer (in the case of bilayer graphene) Coulomb interactions between electrons. The Cooper instability is determined by solving the Bethe-Saltpeter integral equation. The renormalized scat…
Synthesizing Strong-Coupling Kohn-Luttinger Superconductivity in 2D Van der Waals materials
Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices（中山大学物理学院）；School of Physics, Sun Yat-sen University（中山大学（珠海校区）物理与天文学院）；School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus)
We present a review of theoretical investigations into the Kohn-Luttinger nonphonon superconductivity mechanism in various 3D and 2D repulsive electron systems described by the Fermi-gas, Hubbard, and Shubin-Vonsovsky models. Phase diagrams of the superconducting state are considered, including regions of anomalous $s$-, $p$-, and $d$-wave pairing. The possibility of a strong increase in the superconducting transition temperature $T_c$ even for a low electron density is demonstrated by analyzing the spin-polarized case or the two-band situation. The Kohn-Luttinger theory explains or predicts s…
We investigate the development of superconductivity in graphene when the Fermi level becomes close to one of the Van Hove singularities of the electron system. The origin of the pairing instability lies in the strong anisotropy of the e-e scattering at the Van Hove filling, which leads to a channel with attractive coupling when making the projection of the BCS vertex on the symmetry modes with nontrivial angular dependence along the Fermi line. We show that the scale of the superconducting instability may be pushed up to temperatures larger than 10 K, depending on the ability to tune the syste…
We show that the recently observed superconductivity in twisted bilayer graphene (TBG) can be explained as a consequence of the Kohn-Luttinger (KL) instability which leads to an effective attraction between electrons with originally repulsive interaction. Usually, the KL instability takes place at extremely low energy scales, but in TBG, a doubling and subsequent strong coupling of the van Hove singularities (vHS) in the electronic spectrum occurs as the magic angle is approached, leading to extended saddle points in the highest valence band (VB) with almost perfect nesting between states belo…
The Kohn-Luttinger mechanism for unconventional superconductivity (SC) driven by weak repulsive electron-electron interactions on a periodic lattice is generalized to the quasicrystal (QC) via a real-space perturbative approach. The repulsive Hubbard model on the Penrose lattice is studied as an example, on which a classification of the pairing symmetries is performed and a pairing phase diagram is obtained. Two remarkable properties of these pairing states are revealed, due to the combination of the presence of the point-group symmetry and the lack of translation symmetry on this lattice. Fir…
Motivated by experiments on twisted bilayer graphene, we study the emergence of superconductivity from $\textit{weak}$ repulsive interactions in the Hubbard model on a honeycomb lattice, with both spin and orbital degeneracies, and with the filling treated as a tunable control parameter. The attraction is generated through the Kohn-Luttinger mechanism. We find, similar to old studies of single layer graphene, that the leading superconducting instability is in a $d$-wave pairing channel close to Van Hove filling, and is in an $f$-wave pairing channel away from Van Hove filling. The $d$-wave pai…
We study the emergence of superconductivity in rhombohedral trilayer graphene due purely to the long-range Coulomb repulsion. This repulsive-interaction-driven phase in rhombohedral trilayer graphene is significantly different from those found in twisted bilayer and trilayer graphenes. In the latter case, the nontrivial momentum-space geometry of the Bloch wavefunctions leads to an effective attractive electron-electron interaction; this allows for less modulated order parameters and for spin-singlet pairing. In rhombohedral trilayer graphene, we instead find spin-triplet superconductivity wit…
Kohn--Luttinger Superconductivity in Flat Chern Bands
University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校）；Max Planck Institute for Chemical Physics of Solids（马克斯·普朗克固体化学物理研究所）；University of Nevada, Reno（内华达大学里诺分校）；Min H. Kao Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校闵浩高电气与计算机工程系）
We study how the Rashba spin-orbit interaction influences unconventional superconductivity in a two dimensional electron gas partially spin-polarized by a magnetic field. Somewhat surprisingly, we find that for all field orientations, only the larger Fermi surface is superconducting. When the magnetic field is oriented out-of-plane the system realizes a topological $p+ip$ pairing state. When the field is rotated in-plane the order parameter develops nodes along the field direction and finite center-of-mass-momentum pairing is realized. We demonstrate that the pairing symmetry of the system can…
We analyze the pairing instabilities for fermions on hexagonal lattices (both honeycomb and triangular ones) in a wide range of fermionic densities. We argue that for a generic doping in this range, superconductivity at weak coupling is of Kohn-Luttinger type, and, due to the presence of electronic interactions beyond on-site repulsion, is a threshold phenomenon, with superconductivity emerging only if the attraction generated by the Kohn-Luttinger mechanism exceeds the bare repulsion in some channel. For disconnected Fermi pockets, we predict that Kohn-Luttinger superconductivity, if it occur…
Pairing around a Single Dirac Point: A Unifying View of Kohn-Luttinger Superconductivity in Chern Bands, Quarter Metals, and Topological Surface States
Department of Physics and Astronomy, University of California, Irvine（加州大学尔湾分校物理与天文系）
Motivated by recent experiments on ABC-stacked rhombohedral trilayer graphene (RTG) which observed spin-valley symmetry-breaking and superconductivity, we study instabilities of the RTG metallic state to symmetry breaking orders. We find that interactions select the inter-valley coherent order (IVC) as the preferred ordering channel over a wide range, whose theoretically determined phase boundaries agree well with experiments on both the hole and electron doped sides. The Fermi surfaces near van Hove singularities admit partial nesting between valleys, which promotes both inter-valley supercon…
We demonstrate the instability of the normal state of purely repulsive fermionic systems towards the transition to the Kohn-Luttinger superconducting state. We construct the superconducting phase diagrams of these systems in the framework of the Hubbard and Shubin-Vonsovsky models on the square and hexagonal lattices. We show that an account for the long-range Coulomb interactions, as well as the Kohn- Luttinger renormalizations lead to an increase in the critical superconducting temperatures in various materials, such as high-temperature superconductors, idealized monolayer and bilayer of dop…
The BCS theory of superconductivity named electron-phonon interaction as a glue that overcomes Coulomb repulsion and binds fermions into pairs which then condense and superconduct. We review recent and not so recent works aiming to understand whether a nominally repulsive Coulomb interaction can by itself give rise to a superconductivity. We first discuss a generic scenario of the pairing by electron-electron interaction, put forward by Kohn and Luttinger back in 1965, and then turn to modern studies of the electronic mechanism of superconductivity in the lattice models for the cuprates, the F…
We study the superconducting order in a two-dimensional square lattice Hubbard model with weak repulsive interactions, subject to a Zeeman field and weak Rashba spin-orbit interactions. Diagonalizing the non-interacting Hamiltonian leads to two separate bands, and by deriving an effective low-energy interaction we find the mean field gap equations for the superconducting order parameter on the bands. Solving the gap equations just below the critical temperature, we find that superconductivity is caused by Kohn-Luttinger type interaction, while the pairing symmetry of the bands are indirectly a…
We study superconductivity driven by screened Coulomb repulsion in three-dimensional Luttinger semimetals with a quadratic band touching and strong spin-orbit coupling. In these semimetals, the Cooper pairs are formed by spin-3/2 fermions with non-trivial wavefunctions. We numerically solve the linear Eliashberg equation to obtain the critical temperature of a singlet s-wave gap function as a function of doping, with account of spin-orbit and self-energy corrections. In order to understand the underlying mechanism of superconductivity, we compute the sensitivity of the critical temperature to …
Friedel oscillations and chiral superconductivity in monolayer NbSe$_2$
University of Regensburg（雷根斯堡大学）；Donostia International Physics Center（圣塞巴斯蒂安国际物理中心）；University of Silesia in Katowice（卡托维兹西里西亚大学）；Centro de Física de Materiales（材料物理中心）；Ikerbasque, Basque Foundation for Science（伊克拉巴斯科，巴斯克科学基金会）；Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）
We consider Kohn-Luttinger mechanism for superconductivity in a two-dimensional electron gas confined to a narrow well between two metallic planes with two occupied subbands with Fermi momenta $k_{FL} > k_{FS}$. On the basis of a perturbative analysis, we conclude that non-s-wave superconductivity emerges even when the bands are parabolic. We analyze the conditions that maximize $T_c$ as a function of the distance to the metallic planes, the ratio $k_{FL}/k_{FS}$, and $r_s$, which measures the strength of Coulomb correlations. The largest attraction is in p-wave and d-wave channels, of which p…
The intraorbital repulsive Hubbard interaction cannot lead to attractive superconducting pairing states, except through the Kohn-Luttinger mechanism. This situation may change when we include additional local interactions such as the interorbital repulsion $U^\prime$ and Hund's interactions $J$. Adding these local interactions, we study the nature of the superconducting pairs in systems with tetragonal crystal symmetry including the $d_{xz}$ and $d_{yz}$ orbitals, and in octahedral systems including all three of $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals. In the tetragonal case, spin-orbit inte…
We discuss the mechanisms of unconventional superconductivity and superfluidity in 3D and 2D fermionic systems with purely repulsive interaction at low densities. We construct phase diagrams of these systems and find the areas of the superconducting state in free space, as well as on the lattice in the framework of the Fermi-gas model with hard-core repulsion, the Hubbard model, the Shubin-Vonsovsky model, and the $t-J$ model. We demonstrate that the critical superconducting temperature can be greatly increased in the spin-polarized case or in a two-band situation already at low densities. The…
The Kohn-Luttinger mechanism for superconductivity is investigated in a model for the electron doped cuprates. The symmetry of the order parameter of the superconducting phase is determined as a function of the geometry of the Fermi surface together with the structure of the electron-hole susceptibility. It is found to remain d_{x^2-y^2} wave within a large doping range. The shape of the gap anisotropy evolves with doping, with the maximum gap moving away from (pi ,0), in good agreement with recent experiments. As the shift of the maximum increases, a crossover to d_{xy}-symmetry is found.
In a short review-article we first discuss the results, which are mainly devoted to the generalizations of the famous Kohn-Luttinger mechanism of superconductivity in purely repulsive fermion systems at low electron densities. In the context of repulsive-U Hubbard model and Shubin-Vonsovsky model we consider briefly the superconducting phase diagrams and the symmetries of the order parameter in novel strongly correlated electron systems including idealized monolayer and bilayer graphene. We stress that purely repulsive fermion systems are mainly the subject of unconventional low-temperature su…
We show that in a two-dimensional electron gas with an annular Fermi surface, long-range Coulomb interactions can lead to unconventional superconductivity by the Kohn-Luttinger mechanism. Superconductivity is strongly enhanced when the inner and outer Fermi surfaces are close to each other. The most prevalent state has chiral $p$-wave symmetry, but $d$-wave and extended $s$-wave pairing are also possible. We discuss these results in the context of rhombohedral trilayer graphene, where superconductivity was recently discovered in regimes where the normal state has an annular Fermi surface. Usin…
We investigate the interplay of phonons and correlations in superconducting pairing by introducing a model Hamiltonian with on-site repulsion and couplings to several vibration branches having the Cu-O plane of the cuprates as a paradigm. We express the electron-phonon coupling (EP) through two force constants for O-Cu and O-O bond stretchings. Without phonons, this reduces to the Hubbard Model, and allows purely electronic W=0 pairing. A W=0 pair is a two-body singlet eigenstate of the Hubbard Hamiltonian, with no double occupancy, which gets bound from interactions with background particles.…
Interacting Fermi gas provides an ideal model system to understand unconventional pairing and intertwined orders relevant to a large class of quantum materials. Rydberg-dressed Fermi gas is a recent experimental system where the sign, strength, and range of the interaction can be controlled. The interaction in momentum space has a negative minimum at $q_c$ inversely proportional to the characteristic length-scale in real space, the soft-core radius $r_c$. We show theoretically that single-component (spinless) Rydberg-dressed Fermi gas in two dimensions has a rich phase diagram with novel super…
Motivated by the recent discoveries of superconductivity in bilayer and trilayer graphene, we theoretically investigate superconductivity and other interaction-driven phases in multilayer graphene stacks. To this end, we study the density of states of multilayer graphene with up to four layers at the single-particle band structure level in the presence of a transverse electric field. Among the considered structures, tetralayer graphene with rhombohedral (ABCA) stacking reaches the highest density of states. We study the phases that can arise in ABCA graphene by tuning the carrier density and t…
Quantum Geometry and Pair Breaking in Intra- and Intervalley Superconductivity of Rhombohedral Graphene
Theoretical Division, Los Alamos National Lab（洛斯阿拉莫斯国家实验室理论部）；Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA（田纳西大学诺克斯维尔分校物理与天文系）；Min H. Kao Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA（田纳西大学诺克斯维尔分校明浩高电气与计算机科学系）；Theoretical Division, T-4, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA（洛斯阿拉莫斯国家实验室T-4理论部）；Center for Integrated Nanotechnologies (CINT), Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA（洛斯阿拉莫斯国家实验室集成纳米技术中心）
