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arXiv:1303.1587 · Physical Review B

Quantum critical metals in d=3+1 dimensions

Raghu Mahajan, David M. Ramirez, Shamit Kachru A, S. Raghu

Published Sep 10, 2013• 50 citations
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Abstract

We study the problem of disorder-free metals in the vicinity of a quantum critical point in $d=3+1$ dimensions. We begin with perturbation theory in the Yukawa coupling between the electrons and undamped bosons (order-parameter fluctuations) and show that the perturbation expansion breaks down below energy scales where the bosons get substantially Landau damped. Above this scale, however, we find a regime in which low-energy fermions obtain an imaginary self-energy that varies linearly with frequency, realizing the ``marginal Fermi liquid'' phenomenology [Varma et al., Phys. Rev. Lett. 63, 1996 (1989)]. We discuss a large-$N$ theory in which the marginal Fermi liquid behavior is enhanced while the role of Landau damping is suppressed, and show that quasiparticles obtain a decay rate parametrically larger than their energy.

DOI
10.1103/physrevb.88.115116

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We study the problem of disorder-free metals in the vicinity of a quantum critical point in d=3+1d=3+1d=3+1 dimensions. We begin with perturbation theory in the Yukawa coupling between the electrons and undamped bosons (order-parameter fluctuations) and show that the perturbation expansion breaks down below energy scales where the bosons get substantially Landau damped. Above this scale, however, we find a regime in which low-energy fermions obtain an imaginary self-energy that varies linearly with frequency, realizing the ``marginal Fermi liquid'' phenomenology [Varma et al., Phys. Rev. Lett. 63, 1996 (1989)]. We discuss a large-NNN theory in which the marginal Fermi liquid behavior is enhanced while the role of Landau damping is suppressed, and show that quasiparticles obtain a decay rate parametrically larger than their energy.
We present a functional renormalization-group calculation of the properties of a quantum critical metal in d=2d=2d=2 spatial dimensions. Our theory describes a general class of Pomeranchuk instabilities with Nb{N}_{b}Nb​ flavors of boson. At small Nb{N}_{b}Nb​ we find a family of fixed points characterized by weakly non-Fermi-liquid behavior of the conduction electrons and $z\ensuremath{\approx}2$ critical dynamics for the order-parameter fluctuations, in agreement with the scaling observed by Schattner et al. [Phys. Rev. X 6, 031028 (2016)] for the Ising-nematic transition. Contrary to recent suggestions that this represents an intermediate regime en route to the scaling limit, our calculation suggests that this behavior may persist all the way to the critical point. As the number of bosons Nb{N}_{b}Nb​ is increased, the model's fixed-point properties cross over to $z\ensuremath{\approx}1$ scaling and non-Fermi-liquid behavior similar to that obtained by Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].
We study the problem of disorder-free metals near a continuous quantum critical point. We depart from the standard paradigm [Hertz, Phys. Rev. B 14, 1165 (1976) and Millis, Phys. Rev. B 48, 7183 (1993)] and treat both fermions and bosons (i.e., order parameter fields) on equal footing. We construct a Wilsonian effective field theory that integrates out only high-energy boson and fermion modes. Below the upper critical dimension of the theory (d=3d=3d=3 spatial dimensions), we find new fixed points in which the bosons are described by the Wilson-Fisher fixed point and are coupled to a non-Fermi-liquid metal. We describe subtleties with the renormalization group flow of four-Fermi interactions, which can be surmounted in a controlled large-NNN limit. In this limit, we find that the theory has no superconducting instability.
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=3z=3z=3 scalar interacting with an NFL. We finally analyze the IR phases of the theory with arbitrary number of flavors Nc{N}_{c}Nc​. When Nc{N}_{c}Nc​ is small, the superconducting dome covers the NFL behavior; strikingly, for moderately large Nc{N}_{c}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{T}_{c}Tc​ materials in a systematic $\ensuremath{\epsilon}$ expansion.
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.
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.
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{N}_{B}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)SU({N}_{B})SU(NB​) flavor symmetry group. The leading large NB{N}_{B}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/NB1/{N}_{B}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{N}_{B}NB​, our results are valid down to d=2d=2d=2. In the limit $\ensuremath{\epsilon}\ensuremath{\ll}1$, the large NB{N}_{B}NB​ solution is consistent with the renormalization group analysis of Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].