We study the interplay between superconductivity and non-Fermi liquid behavior of a Fermi surface coupled to a massless
SU(N) matrix boson near the quantum critical point. The presence of thermal infrared singularities in both the fermionic self-energy and the gap equation invalidates the Eliashberg approximation, and makes the quantum-critical pairing problem qualitatively different from that at zero temperature. Taking the large
N limit, we solve the gap equation beyond the Eliashberg approximation, and obtain the superconducting temperature
Tc as a function of
N. Our results show an anomalous scaling between the zero-temperature gap and
Tc. For
N greater than a critical value, we find that
Tc vanishes with a Berezinskii-Kosterlitz-Thouless scaling behavior, and the system retains non-Fermi liquid behavior down to zero temperature. This confirms and extends previous renormalization-group analyses done at
T=0, and provides a controlled example of a naked quantum critical point. We discuss the crucial role of thermal fluctuations in relating our results with earlier work where superconductivity always develops due to the special role of the first Matsubara frequency.
Non-Fermi liquids in
d=2 spatial dimensions can arise from coupling a Fermi surface to a gapless boson. At finite temperature, however, the perturbative quantum field theory description breaks down due to infrared divergences. These are caused by virtual static bosonic modes and affect both fermionic and bosonic correlators. We show how these divergences are resolved by self-consistent boson and fermion self-energies that resum an infinite class of diagrams and correct the standard Eliashberg equations. Extending a previous approach in $d=3\ensuremath{-}\ensuremath{\epsilon}$ dimensions, we find a new ``thermal non-Fermi liquid'' regime that violates the scaling laws of the zero-temperature fixed point and dominates over a wide range of scales. We conclude that basic properties of quantum phase transitions and quantum-classical crossovers at finite temperature are modified in crucial ways in systems with soft bosonic fluctuations, and we begin a study of some of the phenomenological consequences.
We study the quantum theory of a Fermi surface coupled to a gapless boson scalar in $D=4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions as a simple model for non-Fermi liquids (NFL) near a quantum phase transition. Our analysis takes into account the full backreaction from Landau damping of the boson, and obtains an RG flow that proceeds through three distinct stages. Above the scale of Landau damping, the Fermi velocity flows to zero, while the coupling evolves according to its classical dimension. Once damping becomes important, its backreaction leads to a crossover regime where dynamic and static damping effects compete and the fermion self-energy does not respect scaling. Below this crossover and having tuned the boson to criticality, the theory flows to a
z=3 scalar interacting with an NFL. We finally analyze the IR phases of the theory with arbitrary number of flavors
Nc. When
Nc is small, the superconducting dome covers the NFL behavior; strikingly, for moderately large
Nc, we find that NFL effects become important first, before the onset of superconductivity. A generic prediction of the theory is that the Fermi velocity and quasiparticle residue vanish with a power law ${\ensuremath{\omega}}^{\ensuremath{\epsilon}}$ as the fixed point is approached. These features may be useful for understanding some of the phenomenology of high-
Tc materials in a systematic $\ensuremath{\epsilon}$ expansion.
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.
Near a quantum critical point in a metal, a strong fermion-fermion interaction, mediated by a soft boson, acts in two different directions: it destroys fermionic coherence and it gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other. We analyze a class of quantum critical models, in which momentum integration and the selection of a particular pairing symmetry can be done explicitly, and the competition between non-Fermi liquid and pairing can be analyzed within 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 this paper, the first in the series, we consider the case
T=0 and $0<\ensuremath{\gamma}<1$. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that a superconducting state is highly nontrivial as there exists a discrete set of topologically distinct solutions for the pairing gap ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ ($n=0,1,2,...,\ensuremath{\infty}$). All solutions have the same spatial pairing symmetry, but differ in the time domain: ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes sign
n times as a function of Matsubara frequency ${\ensuremath{\omega}}_{m}$. The
n=0 solution ${\mathrm{\ensuremath{\Delta}}}_{0}({\ensuremath{\omega}}_{m})$ is sign preserving and tends to a finite value at ${\ensuremath{\omega}}_{m}=0$, like in BCS theory. The $n=\ensuremath{\infty}$ solution corresponds to an infinitesimally small $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})$, which oscillates down to the lowest frequencies as $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})\ensuremath{\propto}|{\ensuremath{\omega}}_{m}{|}^{\ensuremath{\gamma}/2}cos[2\ensuremath{\beta}\phantom{\rule{0.16em}{0ex}}log(|{\ensuremath{\omega}}_{m}|/{\ensuremath{\omega}}_{0})]$, where $\ensuremath{\beta}=O(1)$ and ${\ensuremath{\omega}}_{0}$ is of order of fermion-boson coupling. As a proof, we obtain the exact solution of the linearized gap equation at
T=0 on the entire frequency axis for all $0<\ensuremath{\gamma}<1$, and an approximate solution of the nonlinear gap equation. We argue that the presence of an infinite set of solutions opens up a new channel of gap fluctuations. We extend the analysis to the case where the pairing component of the interaction has additional factor
1/N and show that there exists a critical
Ncr>1, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid. We show that all solutions develop simultaneously once
N gets smaller than
Ncr.
We study the fermion-boson coupled system in d=3−ϵ space dimensions near the quantum phase transition; infinite many boson modes locating on a sphere become critical simultaneously, which is dubbed “critical boson surface” (CBS). The fermions on the Fermi surface can be scattered to nearby points locating on a boson ring in the low-energy limit. The large number of the boson scattering channels N renders the well-known Landau damping effect largely suppressed. We propose an effective theory for a single point on the Fermi surface and the associated critical boson ring induced by the boson scattering channels. Based on the effective model, one-loop renormalization group analysis is performed with asymptotic ϵ-expansion. The fermion self-energy and Yukawa interaction vertex are dressed with ϵ poles and are largely enhanced due to the presence of critical boson ring. The imaginary part of the self-energy exhibits a linear-in frequency dependence and the real part gives a vanishing quasiparticle weight in the low-energy limit, which signatures the celebrated “marginal Fermi liquid” behavior. We find a novel non-Fermi liquid fixed point, at which the critical properties show features associated with the CBS.