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.
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.
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.
Understanding the influence of quenched random potential is crucial for comprehending the exotic electronic transport of non-Fermi liquid metals near metallic quantum critical points. In this study, we identify a stable fixed point governing the quantum critical behavior of two-dimensional non-Fermi liquid metals in the presence of a random potential disorder. By performing renormalization group analysis on a dimensional-regularized field theory for Ising-nematic quantum critical points, we systematically investigate the interplay between random potential disorder for electrons and Yukawa-type interactions between electrons and bosonic order-parameter fluctuations in a perturbative epsilon expansion. At the one-loop order, the effective field theory lacks stable fixed points, instead exhibiting a runaway flow toward infinite disorder strength. However, at the two-loop order, the effective field theory converges to a stable fixed point characterized by finite disorder strength, termed the “disordered non-Fermi liquid (DNFL) fixed point”. Our investigation reveals that two-loop vertex corrections induced by Yukawa couplings are pivotal in the emergence of the DNFL fixed point, primarily through screening disorder scattering. Additionally, the DNFL fixed point is distinguished by a substantial anomalous scaling dimension of fermion fields, resulting in pseudogap-like behavior in the electron’s density of states. These findings shed light on the quantum critical behavior of disordered non-Fermi liquid metals, emphasizing the indispensable role of higher-order loop corrections in such comprehension.
We present a functional renormalization-group calculation of the properties of a quantum critical metal in d=2 spatial dimensions. Our theory describes a general class of Pomeranchuk instabilities with Nb flavors of boson. At small 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 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 develop a theoretical framework based on the nonperturvative renormalization group (RG) in the one-particle irreducible (Wetterich) formulation to tackle the interplay of coupled fermionic and order-parameter fluctuations at metallic quantum critical points (QCPs) with ordering wavevector Q=0. We consistently treat the dynamical emergence of the Landau damping of the bosonic mode and non-Fermi liquid scaling of fermions upon lowering the cutoff scale. The loop integrals of the present theory involve only contributions from fluctuations above the cutoff scale, which protects the system from developing singular bosonic interactions. We emphasize the importance of the nontrivial relative scaling of the bosonic and fermionic cutoffs Λ and Λf, which we fix by analyzing the RG flow of the scale-dependent ordering wave-vector QΛ. Upon neglecting Fermi self-energy in the loop integrals of the functional RG flow, we identify a non-Fermi liquid RG fixed point and recover the features obtained earlier within RPA-type approaches. In a subsequent step, we self-consistently include the scaling of the self-energy and the Yukawa coupling. We find a generic instability of the non-Fermi liquid RG fixed point. This implies, at least at this truncation level, absence of the QCP with Q=0 and development of a first-order phase transition or a phase characterized by Q=0.
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) 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 d 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). The fixed points are identified from the RG flow equations, as functions of d and m. 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) and (d=2,m=1). In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point.