We study a model in 2+1 dimensions composed of a Fermi surface of N_f N f flavors of fermions coupled to scalar fluctuations near quantum critical points (QCPs). The N_f\rightarrow0 N f → 0 limit allows us to non-perturbatively calculate the long-range behavior of fermion correlation functions. We use this to calculate charge, spin and pair susceptibilities near different QCPs at zero and finite temperatures, with zero and finite order parameter gaps. While fluctuations smear out the fermionic quasiparticles, we find QCPs where the overall effect of fluctuations leads to enhanced pairing. We also find QCPs where the fluctuations induce spin and charge density wave instabilities for a finite interval of order parameter fluctuation gaps at T=0 T = 0 . We restore a subset of the diagrams suppressed in the N_f\rightarrow0 N f → 0 limit, all diagrams with internal fermion loops with at most 2 vertices, and find that this does not change the long-range behavior of correlators except right at the QCPs.
Instabilities of quantum critical metals in the limit Nf→0
We study a model in 2+1 dimensions composed of a Fermi surface of N_f N f flavors of fermions coupled to scalar fluctuations near quantum critical points (QCPs). The N_f\rightarrow0 N f → 0 limit allows us to non-perturbatively calculate the long-range behavior of fermion correlation functions. We use this to calculate charge, spin and pair susceptibilities near different QCPs at zero and finite temperatures, with zero and finite order parameter gaps. While fluctuations smear out the fermionic quasiparticles, we find QCPs where the overall effect of fluctuations leads to enhanced pairing. We also find QCPs where the fluctuations induce spin and charge density wave instabilities for a finite interval of order parameter fluctuation gaps at T=0 T = 0 . We restore a subset of the diagrams suppressed in the N_f\rightarrow0 N f → 0 limit, all diagrams with internal fermion loops with at most 2 vertices, and find that this does not change the long-range behavior of correlators except right at the QCPs.
A Framework for Studying a Quantum Critical Metal in the Limit Nf→0
We study a model in 1+2 dimensions composed of a spherical Fermi surface of N_f N f flavors of fermions coupled to a massless scalar. We present a framework to non-perturbatively calculate general fermion n n -point functions of this theory in the limit N_f\rightarrow0 N f → 0 followed by k_F\rightarrow\infty k F → ∞ where k_F k F sets both the size and curvature of the Fermi surface. Using this framework we calculate the zero-temperature fermion density-density correlation function in real space and find an exponential decay of Friedel oscillations.
We analyze quantum fluctuation effects at the onset of charge or spin-density wave order in two-dimensional metals with an incommensurate nesting (2kF) wave vector connecting two pairs of hot spots on the Fermi surface. We first compute the momentum and frequency dependence of the fermion self-energy near the hot spots to leading order in a perturbation expansion (one loop). Non-Fermi liquid behavior with a linear (in energy) quasiparticle decay rate and a logarithmically vanishing quasiparticle weight is obtained. The momentum dependence of the self-energy entails only finite renormalizations of the Fermi velocity and the Fermi-surface curvature at the hot spots. The perturbative one-loop result is not self-consistent and casts doubt on the stability of the 2kF quantum critical point. We construct a self-consistent solution of the one-loop equations with self-energy feedback, where the quantum critical point is stabilized rather than being destroyed by fluctuations, while the non-Fermi liquid behavior as found in the perturbative one-loop calculation is confirmed.
We analyze quantum fluctuation effects at the onset of charge- or spin-density wave order with a 2kF wave vector Q in two-dimensional metals, for the special case where Q connects a pair of hot spots situated at high symmetry points of the Fermi surface with a vanishing Fermi surface curvature. We compute the order parameter susceptibility and the fermion self-energy in one-loop approximation. The susceptibility has a pronounced peak at Q, and the self-energy displays non-Fermi liquid behavior at the hot spots, with a linear frequency dependence of its imaginary part. The real part of the one-loop self-energy exhibits logarithmic divergencies with universal prefactors as a function of both frequency and momentum, which may be interpreted as perturbative signatures of power laws with universal anomalous dimensions. As a result, one obtains a non-Fermi liquid metal with a vanishing quasiparticle weight at the hot spots, and a renormalized dispersion relation with anomalous algebraic momentum dependencies near the hot spots.
We consider pairing of itinerant fermions in a metal near a quantum-critical point (QCP) towards some form of particle-hole order (nematic, spin-density-wave, charge-density-wave, etc).At a QCP, the dominant interaction between fermions comes from exchanging massless fluctuations of a critical order parameter.At low energies, this physics can be described by an effective model with the dynamical electron-electron interaction V (Ω m ) ∝ 1/|Ω m | γ , up to some upper cutoff Λ.The case γ = 0 corresponds to BCS theory, and can be solved by summing up geometric series of Cooper logarithms.We show that for a finite γ, the pairing problem is still marginal (i.e., perturbation series are logarithmic), but one needs to go beyond logarithmic approximation to find the pairing instability.We discuss specifics of the pairing at γ > 0 in some detail and also analyze the marginal case γ = 0+, when V (Ω m ) = λ log (Λ/|Ω m |).We show that in this case the summation of Cooper logarithms does yield the pairing instability at λ log 2 (Λ/T c ) = O(1), but the logarithmic series are not geometrical.We reformulate the pairing problem in terms of a renormalization group (RG) flow of the coupling, and show that the RG equation is different in the cases γ = 0, γ = 0+, and γ > 0.
We analyze quantum fluctuation effects at the onset of charge or spin density wave order in two-dimensional metals with an incommensurate 2kF wave vector connecting a single pair of hot spots on the Fermi surface. We compute the momentum and frequency dependence of the fermion self-energy near the hot spots to leading order in a fluctuation expansion (one loop). Non-Fermi liquid behavior with anomalous frequency scaling and a vanishing quasiparticle weight is obtained. The momentum dependence yields a divergent renormalization of the Fermi velocity and a flattening of the Fermi surface near the hot spots. Going beyond the leading-order calculation, we find that the one-loop result is not self-consistent. We show that any momentum-independent self-energy with a non-Fermi liquid frequency exponent wipes out the peak of the polarization function at the 2kF wave vector, and thus destroys the mechanism favoring 2kF density waves over those with generic wave vectors. However, a 2kF density wave quantum critical point might survive in presence of a sufficiently flat renormalized Fermi surface.
Abstract We consider pairing of itinerant fermions in a metal near a quantum-critical point (QCP) towards some form of particle-hole order (nematic, spin-density-wave, charge-density-wave, etc.). At a QCP, the dominant interaction between fermions comes from exchanging massless fluctuations of a critical order parameter. At low energies, this physics can be described by an effective model with the dynamical electron-electron interaction V(Ωm) ∝ 1/|Ωm|γ, up to some upper cutoff Λ. The case γ = 0 corresponds to BCS theory, and can be solved by summing up geometric series of Cooper logarithms. We show that for a finite γ, the pairing problem is still marginal (i.e., perturbation series are logarithmic), but one needs to go beyond logarithmic approximation to find the pairing instability. We discuss specifics of the pairing at γ > 0 in some detail and also analyze the marginal case γ = 0+, when V(Ωm) = λlog(Λ/|Ωm|). We show that in this case the summation of Cooper logarithms does yield the pairing instability at λlog2(Λ/Tc) = O(1), but the logarithmic series are not geometrical. We reformulate the pairing problem in terms of a renormalization group (RG) flow of the coupling, and show that the RG equation is different in the cases γ = 0, γ = 0+, and γ > 0.