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arXiv:1307.0004 · Physical Review B

Non-Fermi-liquid fixed point in a Wilsonian theory of quantum critical metals

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

Published Sep 10, 2013• 102 citations
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Abstract

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 ($d=3$ 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-$N$ limit. In this limit, we find that the theory has no superconducting instability.

DOI
10.1103/physrevb.88.125116

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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 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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We use renormalization group (RG) analysis and dimensional regularization techniques to study potential superconductivity-inducing four-fermion interactions in systems with critical Fermi surfaces of general dimensions (m)(m)(m) and codimensions $(d\ensuremath{-}m)$, arising as a result of quasiparticle interaction with a gapless Ising-nematic order parameter. These are examples of non-Fermi liquid states in ddd spatial dimensions. Our formalism allows us to treat the corresponding zero-temperature low-energy effective theory in a controlled approximation close to the upper critical dimension d=dc(m)d={d}_{c}(m)d=dc​(m). The fixed points are identified from the RG flow equations, as functions of ddd and mmm. We find that the flow toward the non-Fermi liquid fixed point is preempted by Cooper pair formation for both the physical cases of (d=3,m=2)(d=3,\phantom{\rule{0.16em}{0ex}}m=2)(d=3,m=2) and (d=2,m=1)(d=2,\phantom{\rule{0.16em}{0ex}}m=1)(d=2,m=1). In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point.
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)].
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)].
When a 2D superconductor is subjected to a strong in-plane magnetic field, Zeeman polarization of the Fermi surface can give rise to inhomogeneous FFLO order with a spatially modulated gap. Further increase of the magnetic field eventually drives the system into a normal metal state. Here, we perform a renormalization group analysis of this quantum phase transition, starting from an appropriate low-energy theory recently introduced in Phys. Rev. B 93, 085112 (2016). We compute one-loop flow equations within the controlled dimensional regularization scheme with fixed dimension of Fermi surface, expanding in $\ensuremath{\epsilon}=5/2\ensuremath{-}d$. We find a new stable non-Fermi-liquid fixed point and discuss its critical properties. One of the most interesting aspects of the FFLO non-Fermi-liquid scenario is that the quantum critical point is potentially naked, with the scaling regime observable down to arbitrary low temperatures. In order to study this possibility, we perform a general analysis of competing instabilities, which suggests that only charge density wave order is enhanced in the vicinity of the quantum critical point.