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arXiv:1703.08172 · Annual Review of Condensed Matter Physics

Recent Developments in Non-Fermi Liquid Theory

Sung-Sik Lee

Published Dec 8, 2017• 211 citations
arXivHTMLPDF

Abstract

Non-Fermi liquids are unconventional metals whose physical properties deviate qualitatively from those of noninteracting fermions due to strong quantum fluctuations near Fermi surfaces. They arise when metals are subject to singular interactions mediated by soft collective modes. In the absence of well-defined quasiparticles, universal physics of non-Fermi liquids is captured by interacting field theories which replace Landau Fermi liquid theory. However, it has been difficult to understand their universal low-energy physics due to a lack of theoretical methods that take into account strong quantum fluctuations in the presence of abundant low-energy degrees of freedom. In this review, we discuss two approaches that have been recently developed for non-Fermi liquid theory with emphasis on two space dimensions. The first is a perturbative scheme based on a dimensional regularization, which achieves a controlled access to the low-energy physics by tuning the codimension of Fermi surface. The second is a nonperturbative approach which treats the interaction ahead of the kinetic term through a non-Gaussian scaling called interaction-driven scaling. Examples of strongly coupled non-Fermi liquids amenable to exact treatments through the interaction-driven scaling are discussed.

DOI
10.1146/annurev-conmatphys-031016-025531

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Non-Fermi liquids are unconventional metals whose physical properties deviate qualitatively from those of noninteracting fermions due to strong quantum fluctuations near Fermi surfaces. They arise when metals are subject to singular interactions mediated by soft collective modes. In the absence of well-defined quasiparticles, universal physics of non-Fermi liquids is captured by interacting field theories which replace Landau Fermi liquid theory. However, it has been difficult to understand their universal low-energy physics due to a lack of theoretical methods that take into account strong quantum fluctuations in the presence of abundant low-energy degrees of freedom. In this review, we discuss two approaches that have been recently developed for non-Fermi liquid theory with emphasis on two space dimensions. The first is a perturbative scheme based on a dimensional regularization, which achieves a controlled access to the low-energy physics by tuning the codimension of Fermi surface. The second is a nonperturbative approach which treats the interaction ahead of the kinetic term through a non-Gaussian scaling called interaction-driven scaling. Examples of strongly coupled non-Fermi liquids amenable to exact treatments through the interaction-driven scaling are discussed.
Landau theory of Fermi liquids is the foundational concept behind our understanding of conventional metals. Its basic postulate is that the low-energy physics can be described by weakly interacting quasiparticles --- a dressed but effectively free version of electrons formally represented by poles of the single-particle Green's function. However, this concept notoriously fails for strongly interacting systems such as high-Tc{T}_{c}Tc​ cuprate superconductors, and our understanding of properties of these ``non-Fermi'' quantum liquids is far from complete. One well-known candidate theory for non-Fermi liquids is Hertz-Millis theory: a finite-density fermion system coupled to a massless critical boson. In (2+1) dimensions, this theory is strongly interacting at low energies, but as of yet there is no controlled insight in this regime; conventional perturbation theory fails. In this article, the authors show that in the so-called quenched approximation the fermionic spectrum can be computed exactly to all orders in the coupling constant. They show extensive detailed analysis that, in this quenched limit, the nonperturbative low-energy dynamics are indeed that of a non-Fermi liquid. Instead of coherent quasiparticles one finds the emergence of critical fermionic excitations described by branch cuts in the single particle Green's function.
Understanding non-Fermi liquids in dimensions higher than one remains one of the most formidable challenges in modern condensed matter physics. These systems, characterized by an abundance of gapless degrees of freedom and the absence of well-defined quasiparticles, defy conventional analytical frameworks. Inspired by recent work [Delacretaz et al. Phys. Rev. Res. 4, 033131 (2022)], we present a procedure for bosonizing Fermi surfaces that does not rely on the existence of sharp excitation and is thus directly applicable to non-Fermi liquids. Our method involves parametrizing the generalized fermionic distribution function through a bosonic field that describes frequency-dependent local variations of the chemical potential in momentum space. We propose an effective action that produces the collisionless quantum Boltzmann equation as its equation of motion and can be used for any dimension and Fermi surface of interest. Even at the quadratic order, this action reproduces nontrivial results obtainable only through involved analysis with alternative means. By offering an alternative method directly applicable to studying the low-energy physics of Fermi and non-Fermi liquids, our work potentially stands as an important building block in advancing the comprehension of strange metals and associated phenomena.
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.
In metals, low-energy effective theories are characterized by a set of coupling functions. Among them, the angle-dependent Fermi momentum specifies the size and shape of Fermi surface. Since the Fermi momentum grows incessantly under the renormalization group flow, a metallic fixed point is defined only modulo a rescaling of Fermi momentum. In this paper, we discuss the physical consequences of this projective nature of fixed points for non-Fermi liquids with hot Fermi surfaces. The first is the absence of a unique dynamical critical exponent that dictates the relative scaling between energy and momentum. The second is mismatches between the scaling dimensions of couplings and their relevancy. Nonetheless, each projective fixed point is characterized by a few marginal and relevant coupling functions, and the notion of universality survives. We illustrate our findings by charting the space of projective fixed points and extracting their universal properties for the Ising-nematic quantum critical metal beyond the patch theory. To control the theory, we use the dimensional regularization scheme that tunes the co-dimension of Fermi surface. Near the upper critical dimension, two exactly marginal coupling functions span the space of stable projective fixed points: functions that specify the shape of the Fermi surface and the angle-dependent Fermi velocity. All other coupling functions, including the Landau functions and the universal pairing interaction, are fixed by those two marginal functions. With decreasing dimensions, the forward scattering remains irrelevant while the pairing interaction becomes relevant near two dimensions. In two dimensions, it is expected that the universal superconducting fluctuations lower the symmetry of the non-Fermi liquid realized above the superconducting transition temperatures from the loop U(1) group to a proper subgroup.
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.
An introductory survey of the theoretical ideas and calculations and experimental results which depart from Landau Fermi liquids is presented. The common themes and possible routes to the singularities leading to breakdown of Landau Fermi liquids are categorized following an elementary discussion of the theory. Soluble examples of singular or Non Fermi‐liquids include models of impurities in metals with special symmetries and one‐dimensional interacting fermions. A review of these is followed by a discussion of singular Fermi liquids in a wide variety of experimental situations and theoretical models. These include the effects of low energy collective fluctuations, gauge fields, fluctuations around quantum critical points, the normal state of high temperature superconductors and the two dimensional metallic state.