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arXiv:2405.09450

Space of non-Fermi liquids

Shubham Kukreja, Afshin Besharat, Sung-Sik Lee

Published May 15, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2405.09450

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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.
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 devise a dimensional regularization scheme for quantum field theories with a Fermi surface to study scaling behavior of non-Fermi liquid states in a controlled approximation. Starting from two space dimensions, the co-dimension of the Fermi surface is extended to a general value while its dimension is fixed. When the Fermi surface is coupled with a critical boson centered at zero momentum, the interaction becomes marginal at a critical space dimension dc=5/2{d}_{c}=5/2dc​=5/2. A deviation from the critical dimension is used as a small parameter for a systematic expansion. We apply this method to the theory where two patches of the Fermi surface are coupled with a critical boson, and show that the Ising-nematic critical point is described by a stable non-Fermi liquid state slightly below the critical dimension. Critical exponents are computed up to the two-loop order.
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.
Landau Fermi liquid theory is a fixed-point theory of metals that includes the forward-scattering amplitudes as exact marginal couplings. However, the fixed-point theory that only includes the strict forward scatterings is nonlocal in real space. In this paper, we revisit the Fermi liquid theory for charge-neutral fermions using the field-theoretic functional renormalization group formalism and show how the scale-invariant fixed point emerges as a local theory, which includes not only the forward scatterings but also nonforward scatterings with small but nonzero momentum transfers. In the low-energy limit, the near-forward scattering and pairing interactions take scale-invariant forms if the momentum transfer and the center-of-mass momentum of Cooper pairs, respectively, are comparable to the energy. The coupling functions fully capture the universal low-energy dynamics of the collective modes and instabilities of Fermi liquids. A runaway flow of the coupling function in the particle-hole channel beyond a critical interaction suggests an instability toward an ordered phase with a wave vector that depends on the interaction strength. At the critical interaction, the instability corresponds to the uniform Pomeranchuk or Stoner instability, but the momentum of the leading instability becomes nonzero for stronger attractive interaction. In the particle-particle channel, the coupling function reveals the dynamics of the unstable mode associated with the BCS instability. When an unstable normal metal evolves into the superconducting state, there exists a period in which a superconducting state with spatially nonuniform phase appears due to the presence of unstable Cooperon modes with nonzero momenta.
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.
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.