The stability of a Fermi liquid is analyzed by summing series of diagrams with an interaction mediated by a system close to quantum criticality. The critical temperature and the gap are derived in terms of an effective coupling constant and do not depend on the density of states at the Fermi surface. The forward scattering process is identified as the main pairing mechanism for the case of low, or vanishing, levels of doping.
The stability of a Fermi liquid is analyzed by summing series of diagrams with an interaction mediated by a system close to quantum criticality. The critical temperature and the gap are derived in terms of an effective coupling constant and do not depend on the density of states at the Fermi surface. The forward scattering process is identified as the main pairing mechanism for the case of low, or vanishing, levels of doping.
Landau Fermi liquid theory is a fixed-point theory of metals that includes the forward-scattering amplitudes as exact marginal couplings. However, the fixed-point theory that only includes the strict forward scatterings is nonlocal in real space. In this paper, we revisit the Fermi liquid theory for charge-neutral fermions using the field-theoretic functional renormalization group formalism and show how the scale-invariant fixed point emerges as a local theory, which includes not only the forward scatterings but also nonforward scatterings with small but nonzero momentum transfers. In the low-energy limit, the near-forward scattering and pairing interactions take scale-invariant forms if the momentum transfer and the center-of-mass momentum of Cooper pairs, respectively, are comparable to the energy. The coupling functions fully capture the universal low-energy dynamics of the collective modes and instabilities of Fermi liquids. A runaway flow of the coupling function in the particle-hole channel beyond a critical interaction suggests an instability toward an ordered phase with a wave vector that depends on the interaction strength. At the critical interaction, the instability corresponds to the uniform Pomeranchuk or Stoner instability, but the momentum of the leading instability becomes nonzero for stronger attractive interaction. In the particle-particle channel, the coupling function reveals the dynamics of the unstable mode associated with the BCS instability. When an unstable normal metal evolves into the superconducting state, there exists a period in which a superconducting state with spatially nonuniform phase appears due to the presence of unstable Cooperon modes with nonzero momenta.
Interactions in Fermi systems can generate a Pomeranchuk instability leading to orientational symmetry breaking, that is, nematic order. In a metallic system close to such an instability the Fermi surface is easily deformed by anisotropic perturbations and exhibits enhanced collective fluctuations. We analyze fluctuation effects in the quantum critical regime near a d-wave Pomeranchuk instability in two dimensions. Density correlations with a d-wave form factor and the dynamical forward scattering interaction diverge near the instability. The singular forward scattering leads to large self-energy contributions, which destroy Fermi liquid behavior over the whole Fermi surface except at “cold spots” on the Brillouin zone diagonal. The decay rate of single-particle excitations, which is related to the width of the peaks in the spectral function, exceeds the excitation energy in the low-energy limit. The dispersion of the spectra flattens strongly near those portions of the Fermi surface which are remote from the zone diagonal. The decay rate for DC transport is linear in temperature except at the cold spots. We discuss the possible relevance of d-wave Fermi surface fluctuations for the “strange metal” behavior observed in the normal phase of cuprates.
In this paper, the sixth in series, we continue our analysis of the interplay between non-Fermi liquid and pairing in the effective low-energy model of fermions with singular dynamical interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})={\overline{g}}^{\ensuremath{\gamma}}/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). The model describes low-energy physics of various quantum-critical metallic systems at the verge of an instability towards density or spin order, pairing of fermions at the half-filled Landau level, color superconductivity, and pairing in SYK-type models. In previous papers I--V, we analyzed the $\ensuremath{\gamma}$ model for $\ensuremath{\gamma}\ensuremath{\le}2$ and argued that the ground state is an ordinary superconductor, but there is an infinite number of local minima of the condensation energy. We further argued that the condensation energy spectrum becomes continuous and gapless, and superconducting Tc vanishes due to critical longitudinal gap fluctuations. In this paper, we consider larger $2<\ensuremath{\gamma}<3$. We show that the system moves away from criticality in that the condensation energy spectrum again becomes discrete and gapless, and Tc becomes finite. Yet, we show that the gap functions for $\ensuremath{\gamma}>2$ and $\ensuremath{\gamma}<2$ are topologically different as they live on different sheets of the Riemann surface. This makes $\ensuremath{\gamma}=2$ a topological quantum-critical point. We further show that the fermionic excitation spectrum for $\ensuremath{\gamma}>2$ acquires a new feature---a bound state at the edge of a continuum, with a macroscopic degeneracy, which is a fraction of the total number of states in the system. We obtain the phase diagram on the $({\ensuremath{\omega}}_{D},\ensuremath{\gamma})$ plane, where ${\ensuremath{\omega}}_{D}$ is a mass of a pairing boson, and on the $(T,\ensuremath{\gamma})$ plane. The latter consists of two distinct superconducting phases at $\ensuremath{\gamma}<2$ and $\ensuremath{\gamma}>2$ and the intermediate pseudogap state of preformed pairs in between.
Forward scattering and backscattering play an exceptional role in the physics of two-dimensional interacting fermions. In a Fermi liquid, both give rise to a nonanalytic ${\ensuremath{\omega}}^{2}ln(\ensuremath{\omega})$ form of the fermionic scattering rate at second order in the interaction. Here we argue that higher powers of $ln(\ensuremath{\omega})$ appear in the backscattering contribution at higher orders. We show that these terms come from ``planar'' processes, which are effectively one-dimensional. This is explicitly demonstrated by extending a Fermi liquid to the limit of $N\ensuremath{\gg}1$ fermionic flavors, when only planar processes survive. We sum the leading logarithms for the case of a two-dimensional Fermi liquid near a nematic transition, and we obtain an expression for the scattering rate at T=0 to all orders in the interaction. For a repulsive interaction, the resulting rate is logarithmically suppressed, and the result is valid down to $\ensuremath{\omega}=0$. For an attractive interaction, the ground state is an s-wave superconductor with a gap ${\mathrm{\ensuremath{\Delta}}}_{0}$. We show that in this case the scattering rate increases as $\ensuremath{\omega}$ is reduced toward ${\mathrm{\ensuremath{\Delta}}}_{0}$. At $\ensuremath{\omega}\ensuremath{\ge}{\mathrm{\ensuremath{\Delta}}}_{0}$, the behavior of the scattering rate is rather unconventional as many pairing channels compete near a nematic critical point, and the s-wave wins only by a narrow margin. We take superconductivity into consideration and obtain the scattering rate also at smaller $\ensuremath{\omega}\ensuremath{\simeq}{\mathrm{\ensuremath{\Delta}}}_{0}$.
Superconductivity results from an instability of the Fermi surface---contour of poles of the single-particle propagator---to an infinitesimally small attraction between electrons. Here, we instead discuss the analogous problem on a model Luttinger surface, or contour of zeros of the Green function. At zero temperature $(\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty})$ and a critical interaction strength $({u}_{c\ensuremath{\infty}})$ characterized by the residue of self-energy pole, we find that the pair susceptibility diverges leading to a superconducting instability. We evaluate the pair fluctuation partition function and find that the spectral density in the normal state has an interaction-driven, power-law $\frac{1}{\sqrt{\ensuremath{\omega}}}$ type, van-Hove singularity (vHS), indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit $(\ensuremath{\beta}u\ensuremath{\gg}1)$, the leading-order fluctuation free-energy terms in the normal state of this NFL-SC transition resembles the equivalent [O(1)] terms of the Sachdev-Ye-Kitaev (SYK) model. This free-energy contribution takes a simple form $\ensuremath{-}\ensuremath{\beta}F=\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\ensuremath{-}\ensuremath{\gamma}\phantom{\rule{0.16em}{0ex}}\text{ln}\left(\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\right),$ where $\ensuremath{\gamma}$ is a constant equal to 21. Weak impurity scattering $(\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\beta}}^{\ensuremath{-}1})$ leaves the low-energy spectral density unaffected but leads to an interaction-driven enhancement of superconductivity. Our results shed light on the role played by order-parameter fluctuations in providing the key missing link between Mott physics and strongly coupled toy models exhibiting gravity duals.
We review the theory of interacting Fermi systems whose low-energy physics is dominated by forward scattering, that is scattering processes generated by effective interactions with small momentum transfers. These systems include Fermi liquids as well as several important non-Fermi-liquid phases: one-dimensional Luttinger liquids, systems with long-range interactions, and fermions coupled to a gauge field. We report results for the critical dimensions separating different 'universality classes' and discuss the behaviour of physical quantities such as the momentum distribution function, the single-particle propagator and low-energy response functions in each class. The renormalization group for Fermi systems will be reviewed and applied as a link between microscopic models and effective lowenergy theories. Particular attention is paid to conservation laws, which constrain any effective low-energy theory of interacting Fermi systems. In scattering processes with small momentum transfers the velocity of each scattering particle is (almost) conserved. This asymptotic conservation law leads to non-trivial cancellations of Feynman diagrams and other simplifications, making thus possible a non-perturbative treatment of forward scattering via Ward identities or bosonization techniques.