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arXiv:1312.3321 · Physical Review B

Non-Fermi-liquid behavior of large- NB quantum critical metals

A. Liam Fitzpatrick, Shamit Kachru A, Jared Kaplan A, S. Raghu

Published Apr 11, 2014• 93 citations
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Abstract

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, ${N}_{B}$, 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 $SU({N}_{B})$ flavor symmetry group. The leading large ${N}_{B}$ 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 $1/{N}_{B}$ 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 ${N}_{B}$, our results are valid down to $d=2$. In the limit $\ensuremath{\epsilon}\ensuremath{\ll}1$, the large ${N}_{B}$ solution is consistent with the renormalization group analysis of Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].

DOI
10.1103/physrevb.89.165114

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Significant effort has been devoted to the study of "non-Fermi-liquid" (NFL) metals: gapless conducting systems that lack a quasiparticle description. One class of NFL metals involves a finite density of fermions interacting with soft order parameter fluctuations near a quantum critical point. The problem has been extensively studied in a large-N limit (N corresponding to the number of fermion flavors) where universal behavior can be obtained by solving a set of coupled saddle-point equations. However, a remarkable study by Lee revealed the breakdown of such approximations in two spatial dimensions. We show that an alternate approach, in which the fermions belong to the fundamental representation of a global SU(N) flavor symmetry, while the order parameter fields transform under the adjoint representation (a "matrix large-N" theory), yields a tractable large N limit. At low energies, the system consists of an overdamped boson with dynamical exponent z=3 coupled to a non-Fermi-liquid with self-energy Σ(ω)∼ω^{2/3}, consistent with previous studies.
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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 low-energy behavior of metals coupled to gapless bosons. This problem arises in several contexts in modern condensed matter physics; we focus on the theory of metals near continuous quantum phase transitions (where the boson is the order parameter). In the vicinity of d=3 spatial dimensions, the upper critical dimension of the theory, the ratio of fermion and boson speeds, v/c, acts as an additional control parameter, enabling us to access IR fixed points where this ratio vanishes. This limit corresponds to a non-Fermi liquid coupled to bosons with critical exponents governed by the Wilson-Fisher fixed point.
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The destruction of Fermi-liquid behavior when a gapless Fermi surface is coupled to a fluctuating gapless boson field is studied theoretically. This problem arises in a number of different contexts in quantum many-body physics. Examples include fermions coupled to a fluctuating transverse gauge field pertinent to quantum spin-liquid Mott insulators, and quantum critical metals near a Pomeranchuk transition. We develop a controlled theoretical approach to determine the low-energy physics. Our approach relies on combining an expansion in the inverse number (N)(N)(N) of fermion species with a further expansion in the parameter $ϵ={z}_{b}\ensuremath{-}2$, where zb{z}_{b}zb​ is the dynamical critical exponent of the boson field. We show how this limit allows a systematic calculation of the universal low-energy physics of these problems. The method is illustrated by studying spinon Fermi-surface spin liquids, and a quantum critical metal at a second-order electronic nematic phase transition. We calculate the low-energy single-particle spectra, and various interesting two-particle correlation functions. In some cases, deviations from the popular random-phase approximation results are found. Some of the same universal singularities are also calculated to leading nonvanishing order using a perturbative renormalization-group calculation at small NNN extending previous results of Nayak and Wilczek. Implications for quantum spin liquids and for Pomeranchuk transitions are discussed. For quantum critical metals at a nematic transition, we show that the tunneling density of states has a power-law suppression at low energies.