This review examines the theory of superconductivity in systems with singular dynamical electron-electron interaction and contrasts it with a conventional BCS superconductivity. Examples include metals near a Quantum Critical Point, quantum dots and system near a localization (Mott) transition. We show, that the singular interaction destroys the traditional separation of energy scales, invalidating the significance of Cooper logarithm, and, as the consequence, the whole BCS framework. We explore the universal model with dynamical interaction Γ(Ω) ∝ 1/|Ω|^γ (the γ-model) and analyze the competition/interplay between the tendency towards pairing and towards non-Fermi liquid behavior. We show that superconductivity still develops once the pairing interaction exceeds a certain threshold, but the origin of the pairing is qualitatively different from that in BCS theory. We show that the gap equation at T=0 has an infinite set of topologically distinct solutions. These solution disappear one by one once the pairing interaction becomes non-singular (massive). We review the physics underlying these phenomena and outline future directions.

Recent experiments on graphene multilayers under displacement field have demonstrated a wide variety of electronically ordered phases, including valley and/or spin polarized phases as well as potentially unconventional superconducting phases. In addition, quantum oscillation measurements in Bernal bilayer graphene and rhombohedral trilayer graphene showed the presence of electronic nematic states. Here, we investigate the emergence of nematic order in these systems, with emphasis on Bernal bilayer graphene, within a self-consistent Hartree–Fock framework, using a realistic band structure with trigonal warping and a dual-gate screened Coulomb interaction. We compute the phase diagram as carrier density and displacement field are varied, and find a sequence of isospin (spin and valley) polarized states consistent with experiment, including partially polarized phases with a large Fermi pockets for isospin-majority carriers and smaller Fermi pockets for isospin-minority carriers. Within these partially isospin polarized states, we identify regions with three C_3-symmetric pockets for the minority carriers and regions with only one or two such pockets, implying that the system develops a spontaneous nematic order that breaks C_3 symmetry. We find numerically that nematicity emerges near the boundary between fully and partially polarized phases and is controlled by the strength of the screened interaction. We analytically derive a criterion for the nematic order, which agrees with our numerical results.

We analyze the spectrum of collective modes in a superconductor in which pairing is mediated by long-range nematic fluctuations. Previous experimental and theoretical studies have found that the superconducting gap in such a system is highly anisotropic and, at any finite $T<T_c$, vanishes on four arcs of the Fermi surface, even when the pairing symmetry is $s$ wave ($s^{+-}$ between hole and electron pockets). We derive the expression for the pair susceptibility $χ(\mathbf{q},Ω)$ at finite momentum $\mathbf{q}$ and frequency $Ω$ deep in the superconducting phase. We analyze the spectral function, $\operatorname{Im}χ(\mathbf{q},Ω)$, and its pole structure in the transverse (phase) and longitudinal (amplitude) channels, and compare the results with those of a conventional $s$-wave superconductor. We find that the analytic structure of the pair susceptibility in both channels is qualitatively distinct from that in a BCS superconductor. This gives rise to a highly unconventional dispersion of phase and amplitude collective modes.

The Migdal-Eliashberg theory (MET) describes electrons interacting with phonons in the adiabatic limit when the phonon Debye frequency is much smaller than the Fermi energy. A conventional belief is that MET holds even at strong coupling, when electron self-energy is large, and breaks down only near the point where the dressed phonon spectrum softens to near zero. We analyze numerically and analytically a different option – collapse to a polaronic/bipolaronic ground state. The last scenario has never been analyzed in precise quantitative terms for a generic electron density. Using variational considerations, we establish rigorous upper bounds on the coupling λ, at which a FL state transforms into the bipolaron/polaron state. We show that at small and near-maximum densities, this happens well before a dressed phonon softens. This is true both in 2D and 3D systems; in the latter the upper bound on λ tends to zero in the limit of small or near-full density. We present analytical reasoning for this behavior based on hints extracted from exact diagrammatic treatment of the on-site Holstein model for the spin polarized case and argue that polarons are produced by fermions with energies comparable to the bandwidth; i.e., polaron formation is outside the realm of MET. Closer to half-filling, the leading instability upon increasing λ is towards a charge-density-wave state (CDW), and there exists a strong coupling regime of MET near this instability, while the polaron/bipolaron state develops at larger λ out of a CDW-ordered state and inherits a CDW order over some range of coupling.

We study superconductivity in paramagnetic and ferromagnetically-ordered phases in a two-dimensional electron system with parabolic fermionic dispersion and short-range repulsive interaction. In the paramagnetic phase, we find that a weak momentum dependence of a paramagnon propagator parametrically reduces the onset temperature for the pairing compared to that in phenomenological theories which assume a strong dispersion of a paramagnon and also changes the topology of the gap function. In the ferromagnetic phase, we show that the order instantly polarizes low-energy fermionic excitations. We derive the fully renormalized pairing interaction between low-energy fermions, mediated by two transverse Goldstone modes and show that it is attractive in a spatially-odd channel. The pairing temperature in the ferromagnetic phase is found to be a fraction of the Fermi energy, significantly larger than in the paramagnetic phase near the transition. Our results are relevant for understanding superconductivity in proximity to itinerant ferromagnetism in multi-valley graphene systems, particularly the ones with full valley and spin polarization.

We analyze superconductivity in a ferromagnetically ordered state of multi-layer graphene system in proximity to WSe2. We model this material by a two-valley system of interacting fermions with small pockets and Ising spin-orbit coupling. The magnetically-ordered state is a half-metal with a canted ferromagnetic order. We obtain the magnon spectrum and derive two types of magnon-mediated 4-fermion interactions: spin-flip interactions mediated by a single magnon and spin-preserving interactions mediated by two magnons. We argue that both have to be included into the pairing interaction between fermions from the filled bands. The full magnon-mediated interaction satisfies Adler criterion and is attractive in a spin-triplet, valley-odd and spatially-even pairing channel. The attraction is induced by spin-orbit coupling and is confined to energies which are parametrically smaller than the Fermi energy. We argue that near the onset of a canted ferromagnetic order this attraction wins over a repulsive intra-valley density-density interaction.

Stoner instabilities for fermions with repulsive interaction have been well studied within the mean-field (ladder) approximation. In this study, we consider two-dimensional electronic systems with one or two valleys and discuss a Stoner transition beyond the ladder approximation. At weak coupling, the corrections to the ladder approximation come predominantly from the renormalization of the particle-hole vertex in the particle-particle channel, and the lowest-order corrections are logarithmically singular in the low-density limit. To investigate the problem beyond the lowest order, we apply the diagrammatic Monte Carlo algorithm and treat ladder and non-ladder renormalizations on equal footing. We find that in a one-valley system, a Stoner transition to a ferromagnetism occurs at low density only if there is a cutoff on the momentum transfer carried by the interaction. In a two-valley system, the restriction is less severe. Here we find either a direct Stoner transition into a spin- and valley-polarized state, or a set of two Stoner transitions via an intermediate valley-polarized state. The pathway is controlled by the anisotropy of the fermionic dispersion.

It is widely believed that in an adiabatic limit a Fermi liquid state of an electron-phonon system described by Migdal-Eliashberg theory remains stable before a dressed phonon softens. Using Holstein model as a prototypical example and variational/analytic considerations we demonstrate that in a wide range of fillings both in 3D and 2D, a polaronic/bi-polaronic state emerges before phonon softening; at small filling in 3D this happens already at weak coupling. We show that a polaronic/bi-polaronic state emerges, upon increasing coupling, via an intermediate pseudogap-type mixed state, in which some fermions regain Fermi liquid behavior, yet Luttinger theorem is broken. At even larger couplings the density of states gradually approaches its form in the atomic limit.

Motivated by recent observations of superconductivity in a quarter-metal state of spin- and valley- polarized graphene multilayers, we investigate pairing within a ferromagnetic phase of a single-valley model of itinerant two-dimensional (2D) electrons with Hubbard-type interaction and no artificial high-energy cutoff. In 2D, the Stoner transition is first-order into a fully-polarized state wherein the only gapless collective excitations are transverse magnons. We find that in a spin-SU(2) symmetric model, this magnon-mediated pairing interaction between equal-spin fermions vanishes at T=0. We show that a small easy-plane magnetic anisotropy Ω_0 ≪ E_F, where E_F is the Fermi energy, breaks the SU(2) symmetry and generates an attractive interaction for equal-spin p-wave pairing. We explicitly derive the corresponding coupling constant λ_p as the scaling function of both the relative strength of the easy-plane anisotropy, Ω_0/E_F, and the proximity to the ferromagnetic transition. While λ_p is parametrically small in Ω_0/E_F deep inside the ferromagnetic phase, it becomes enhanced near the ferromagnetic transition, reaching order unity regardless of how small Ω_0/E_F is. This mechanism yields a sizable T_c, peaked near the onset of ferromagnetism.

Recent tunneling measurements on β-PdBi_2 reported a magnetic field induced phase transition from a fully gapped to a nodal superconducting state. We develop a microscopic theory of superconductivity in this multi-band material, taking into account spin-orbit coupling and Zeeman splitting. We show that there are two attractive pairing channels in this system: an s-wave and a p-wave. At zero magnetic field, s-wave superconductivity wins. At a finite field, this channel becomes less favorable because Fermi surfaces split, and the gap symmetry changes to p-wave. We show that at a higher field, the excitation spectrum of a p-wave multi-band superconductor becomes gapless, with nodal points located in between the split Fermi surfaces. We argue that this behavior accounts for the experimentally observed field-induced nodal superconductivity in β-PdBi_2.

In the BCS theory of superconductivity, an instability towards pairing develops at arbitrarily weak dimensionless coupling lambda due to a divergence of logarithmic perturbative series for the pairing susceptibility (Cooper logarithms) at Tc similar to omega 0e-1/lambda, where omega 0 is an energy cutoff. On the contrary, in many models of superconductivity out of a non-Fermi-liquid, the Cooper logarithm is absent, and superconductivity emerges only when lambda exceeds a certain threshold. We argue that there are situations when there is no threshold, and at weak coupling the formula for Tc is BCS-like, yet the origin of the pairing instability is fundamentally different from that in the BCS scenario. As an example, we revisit superconductivity in a metal at the onset of (pi, pi) spin-density-wave order. Earlier studies of this problem found no threshold and a BCS-like expression for Tc at weak coupling. We argue that, despite this, the pairing is not caused by the Cooper logarithm, and in many respects it is qualitatively similar to that in non-Fermi-liquids.

We present the results for the evolution of the Fermi surfaces under variation of number density and displacement field for spin and valley-polarized states in Bernal bilayer graphene (BBG) using a realistic form of the electronic dispersion with trigonal warping terms. Earlier studies without trigonal warping have found discrete half-metal and quarter-metal states with full spin and/or valley polarization and complete depletion of some of the Fermi surfaces. We show that when trigonal warping terms are included in the dispersion, partially polarized states with large but non-complete polarization and with both majority and minority carriers present, emerge at small doping, as seen in the experimental data. We show the results when the intervalley and intravalley interactions are equal as well as when they are unequal.

In the BCS theory of superconductivity, an instability towards pairing develops at arbitrary weak dimensionless coupling $\lambda$ due to a divergence of logarithmic perturbative series for the pairing susceptibility (Cooper logarithms) at $T_c \sim \omega_0 e^{-1/\lambda}$, where $\omega_0$ is an energy cutoff. On the contrary, in many models of superconductivity out of a non-Fermi liquid, Cooper logarithm is absent and superconductivity emerges only when $\lambda$ exceeds a certain threshold. We argue that there are situations when there is no threshold and at weak coupling the formula for $T_c$ is BCS-like, yet the origin of the pairing instability is fundamentally different from that in the BCS scenario. As an example, we revisit superconductivity in a metal at the onset of $(\pi,\pi)$ spin-density-wave order. Earlier studies of this problem found no threshold and a BCS-like expression for $T_c$ at weak coupling. We argue that, despite this, the pairing is not caused by the Cooper logarithm and in many respects is qualitatively similar to that in non-Fermi liquids.

We present the results of a mean-field analysis of the temperature evolution of a ferromagnetic Stoner transition in a two-dimensional (2D) system with an isotropic dispersion ε_k ∝ k^2α, which for α>1 models flat dispersions in various multi-layer graphene systems in a displacement field. This study is an extension to a finite T of previous studies at T=0, which found both first-order and second-order Stoner transitions, depending on the value of α and special behavior at α=1 and α=2. We find that the Stoner transition at a finite T displays new features not seen at T=0. The most interesting one is the reentrant behavior, where the ordered state emerges as temperature is increased. This behavior develops at α>1.4 and the range where it holds increases with α. We conjecture that the reentrant behavior is the fundamental feature of the Stoner transition in 2D, not sensitive to the details of the electronic structure.

We consider a superconductor under external perturbation, which forces Cooper pairs to develop with a finite total momentum q. The condensation energy of such a state decreases with q and vanishes at a critical q_c. We analyze how superconducting order evolves at q ≈ q_c. In 3D, the result is well-known: the pairing susceptibility diverges at q = q_c +0, and the gap amplitude Δ(q) gradually increases as q decreases below q_c and reaches its largest value Δ_0 at q=0. In 2D, we find different behavior. Namely, for a parabolic dispersion, the pairing susceptibility also diverges at q = q_c +0, but at q = q_c -0, the gap amplitude jumps to the maximal Δ_0 and remains equal to it for all q 1 the transition becomes second-order, but the gap evolution is rather sharp, whereas for α<1 it becomes first-order, but Δ(q) is non-monotonic. This is similar, but not identical, to the behavior of magnetization near a Stoner transition in 2D.

We analyze superconductivity in a multiorbital fermionic system near the onset of nematic order, using doped FeSe as an example. We associate nematicity with a spontaneous polarization between dxz and dyz orbitals (a Pomeranchuk-type order) and analyze the pairing mediated by soft nematic fluctuations. Such a pairing gives rise to a highly anisotropic gap function whose structure strongly varies with temperature and leads to strongly non-BCS behavior in thermodynamics, spectroscopy, and transport. We compute the specific heat and its directional variation with a magnetic field, magnetic susceptibility, density of states, tunneling conductance, Raman intensity, superfluid stiffness, and penetration depth, both without and with impurity scattering. For the latter, we also compute optical conductivity and the Tc variation. We find good agreement with existing data for FeSe1-xSx and FeSe1-xTex and suggest new experiments.

We analyze the dynamical pairing susceptibility chi pp(omega m) at T = 0 in a quantum-critical metal, where superconductivity emerges out of a non-Fermi-liquid ground state once the attraction exceeds a certain threshold. We obtain chi pp(omega m) as the ratio of the fully dressed dynamical pairing vertex Phi(omega m) and the bare Phi 0(omega m). For superconductivity out of a Fermi liquid, the pairing susceptibility in the normal state is positive above Tc, diverges at Tc, and becomes negative below it. For superconductivity out of a non-Fermi-liquid, the behavior of chi pp(omega m) is different in two aspects: (i) it diverges at the onset of pairing at T = 0 only for a certain subclass of bare Phi 0(omega m) and remains nonsingular for other Phi 0(omega m), and (ii) below the instability, it becomes a nonunique function of a continuous parameter phi for an arbitrary Phi 0(omega m). The susceptibility is negative in some range of phi and diverges at the boundary of this range. We argue that this behavior of the susceptibility reflects a multicritical nature of a superconducting transition in a quantum-critical metal, where, immediately below the instability, an infinite number of superconducting states emerges simultaneously, exhibiting different amplitudes of the order parameter down to an infinitesimally small one.

We examine the spectral properties of collective excitations with finite angular momentum l for a system of interacting fermions near a Pomeranchuk quantum critical point, both in the Fermi liquid and non-Fermi liquid regimes. Previous studies found that deep in the Fermi liquid regime, the spectral functions for even and odd l behave differently - the latter is suppressed compared to the former because of kinematic constraints on scattering processes. The main focus of our paper is to understand how the spectral functions for even and odd l evolve as the system enters the non-Fermi liquid regime. We obtain the full scaling function for the electron polarization bubble at arbitrary l, which interpolates between the Fermi liquid and non-Fermi liquid regimes. We show that collective excitations for all l remain stable and causal throughout the crossover and right at the quantum critical point.

Inspired by empirical evidence of the existence of pair-density-wave (PDW) order in certain underdoped cuprates, we investigate the collective modes in systems with unidirectional PDW order with momenta ± Q and a d-wave form-factor with special focus on the amplitude (Higgs) modes. In the pure PDW state, there are two overdamped Higgs modes. We show that a phase with co-existing PDW and uniform (d-wave) superconducting (SC) order, PDW/SC, spontaneously breaks time-reversal symmetry—and thus is distinct from a simpler phase, SC/CDW, with coexisting SC and charge-density-wave (CDW) order. The PDW/SC phase exhibits three Higgs modes, one of which is sharply peaked and is predominantly a PDW fluctuation, symmetric between Q and −Q, whose damping rate is strongly reduced by SC. This sharp mode should be visible in Raman experiments.