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arXiv:2610.25713

Categorical classification of non-Fermi liquids in 2+1 dimensions

Shubham Kukreja, Dawson M Willerton, Sungsik Lee

Published Oct 19, 2026• 0 citations
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Abstract

A defining feature of non-Fermi liquids is a lack of the quasiparticle peak on the Fermi surface in the spectral function, which is usually realized by coupling a finite density of fermions with critical collective fluctuations. However, these features of the two- and three-point vertex functions are qualitatively the same across many non-Fermi liquids, which has made it difficult to classify them beyond the definition by negation. We point out that the four-point function in the pairing channel contains rich information that positively differentiates distinct types of non-Fermi liquids. Using the generalized notion of fixed points that captures scale-dependent universal pairing interaction, we provide a non-perturbative classification of 2+1-dimensional non-Fermi liquids. In the large Fermi momentum limit with fixed dimensionless couplings, metals coupled with long-wavelength critical fluctuations are classified into seven superuniversality classes. Each of them exhibits a unique phase diagram in the space of temperature and bare four-fermion coupling. The present work generalizes the earlier perturbative classification to strongly interacting non-Fermi liquids by leveraging kinematic constraints.

DOI
10.48550/arxiv.2610.25713

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A defining feature of non-Fermi liquids is a lack of the quasiparticle peak on the Fermi surface in the spectral function, which is usually realized by coupling a finite density of fermions with critical collective fluctuations. However, these features of the two- and three-point vertex functions are qualitatively the same across many non-Fermi liquids, which has made it difficult to classify them beyond the definition by negation. We point out that the four-point function in the pairing channel contains rich information that positively differentiates distinct types of non-Fermi liquids. Using the generalized notion of fixed points that captures scale-dependent universal pairing interaction, we provide a non-perturbative classification of 2+1-dimensional non-Fermi liquids. In the large Fermi momentum limit with fixed dimensionless couplings, metals coupled with long-wavelength critical fluctuations are classified into seven superuniversality classes. Each of them exhibits a unique phase diagram in the space of temperature and bare four-fermion coupling. The present work generalizes the earlier perturbative classification to strongly interacting non-Fermi liquids by leveraging kinematic constraints.
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.
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.
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.
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.
Fixed points of the renormalization group flow play a crucial role in classifying phases of matter. In metals, these fixed points are defined only projectively due to the indefinitely increasing Fermi momentum under scale transformations. Here, the authors study the physical implications of the projective nature of metallic fixed points and chart the space of projective fixed points for Ising-nematic quantum critical metals. Their universality classes are characterized by a nontunable universal forward interaction and intrinsic superconducting fluctuations.
We develop a general theory of fermion liquids in spatial dimensions greater than 1. The principal method, bosonization, is applied to the cases of short- and long-range longitudinal interactions and to transverse gauge interactions. All the correlation funtions of the system may be obtained with the use of a generating functional. Short-range and Coulomb interactions do not destroy the Landau-Fermi fixed point. Non-Fermi liquid fixed points are found, however, in the cases of a super-long-range longitudinal interaction in two dimensions and transverse gauge interactions in two and three spatial dimensions. We consider in some detail the (2+1)-dimensional problem of a Chern-Simons gauge action combined with a longitudinal two-body interaction V(q)\ensuremath{\propto}\ensuremath{\Vert}q${\mathrm{\ensuremath{\Vert}}}^{\mathit{y}\mathrm{\ensuremath{-}}1}$, which controls the density, and hence gauge, fluctuations. For y0 we find that the gauge interaction is irrelevant and the Landau fixed point is stable, while for y>0 the interaction is relevant and the fixed point cannot be accessed by bosonization. Of special importance is the case y=0 (Coulomb interaction), which describes the Halperin-Lee-Read theory of the half-filled Landau level. We obtain the full quasiparticle propagator, which is of a marginal Fermi-liquid form. Using Ward identities, we show that neither the inclusion of nonlinear terms in the fermion dispersion nor vertex corrections alters our results: the fixed point is accessible by bosonization. As the two-point fermion Green's function is not gauge invariant, we also invetigate the gauge-invariant density response function. Near momentum Q=2kF{\mathit{k}}_{\mathit{F}}kF​, in addition to the Kohn anomaly we find other nonanalytic behavior. In the appendies we present a numerical calculation of the spectral function for a Fermi liquid with Landau parameter f0{\mathit{f}}_{0}f0​\ensuremath{\ne}0. We also show how Kohn's theorem is satisfied within the bosonization framework.