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.
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.
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.
We study several models of d-dimensional fermions (d=1,2,3) with an emphasis on the properties of their gapless (metallic) phase. It occurs at T=0 as a continuous transition when zeros of the partition function reach the real range of parameters. Those zeros define the $(d\ensuremath{-}1)$-manifold of quantum criticality (Fermi surface). Its appearance or restructuring correspond to the Lifshitz transition. Such $(d\ensuremath{-}1)$-membrane breaks the symmetry of the momentum space, leading to gapless excitations, a hallmark of metallic phase. To probe quantitatively the gapless phase we introduce the geometric order parameter as d-volume of the Fermi sea. From analysis of the chain, ladder, and free fermions with different spectra, this proposal is shown to be consistent with scaling near the Lifshitz points of other quantities: correlation length, oscillation wavelength, susceptibilities, and entanglement. All the (hyper)scaling relations are satisfied. Two interacting cases of the Tomonaga-Luttinger (d=1) and the Fermi (d=2,3) liquids are analyzed, yielding the same universality classes as free fermions.
The problem of continuous quantum phase transitions in metals involves critical bosons coupled to a Fermi surface. We solve the theory in the limit of a large number, NB, of bosonic flavors, where the bosons transform in the adjoint representation (a matrix representation), while the fermions are in the fundamental representation (a vector representation) of a global SU(NB) flavor symmetry group. The leading large NB solution corresponds to a non-Fermi liquid coupled to Wilson-Fisher bosons. In a certain energy range, the fermion velocity vanishes---resulting in the destruction of the Fermi surface. Subleading 1/NB corrections correspond to a qualitatively different form of Landau damping of the bosonic critical fluctuations. We discuss the model in $d=3\ensuremath{-}\ensuremath{\epsilon}$ but because of the additional control afforded by large NB, our results are valid down to d=2. In the limit $\ensuremath{\epsilon}\ensuremath{\ll}1$, the large NB solution is consistent with the renormalization group analysis of Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].
In this thesis we study the scaling properties of unconventional metals that arise at quantum critical points using low-energy effective field theories. Due to high rate of scatterings between electrons and critical fluctuations of the order parameter associated with spontaneous symmetry breaking, Landau’s Fermi liquid theory breaks down at the critical points. The theories that describe these critical points generally flow into strong coupling regimes at low energy in two space dimensions. Here we develop and utilize renormalization group methods that are suitable for the interacting non-Fermi liquids. We focus on the critical points arising at excitonic, and commensurate spin and charge density wave transitions. By controlled analyses we find stable non-Fermi liquid and marginal Fermi liquid states, and extract the scaling behaviour. The field theories for the non-Fermi liquids are characterized by symmetry groups, local curvature of the Fermi surface, the dispersion of the order parameter fluctuations, and dimensions of space and Fermi surface.
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.
We show that the fermionic and bosonic spectrum of d=2 fermions at finite density coupled to a critical boson can be determined nonperturbatively in the combined limit ${k}_{F}\ensuremath{\rightarrow}\ensuremath{\infty},\phantom{\rule{0.16em}{0ex}}{N}_{f}\ensuremath{\rightarrow}0$ with NfkF fixed. In this double scaling limit, the boson two-point function is corrected but only at one loop. This double scaling limit therefore incorporates the leading effect of Landau damping. The fermion two-point function is determined analytically in real space and numerically in (Euclidean) momentum space. The resulting spectrum is discontinuously connected to the quenched ${N}_{f}\ensuremath{\rightarrow}0$ result. For $\ensuremath{\omega}\ensuremath{\rightarrow}0$ with k fixed the spectrum exhibits the distinct non-Fermi-liquid behavior previously surmised from the RPA approximation. However, the exact answer obtained here shows that the RPA result does not fully capture the IR of the theory.