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arXiv:2208.04328 · SciPost Physics

Loop current fluctuations and quantum critical transport

Zhengyan Darius Shi, Dominic V. Else, Hart Goldman, T. Senthil

Published May 15, 2023• 37 citations
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Abstract

We study electrical transport at quantum critical points (QCPs) associated with loop current ordering in a metal, focusing specifically on models of the “Hertz-Millis” type. At the infrared (IR) fixed point and in the absence of disorder, the simplest such models have infinite DC conductivity and zero incoherent conductivity at nonzero frequencies. However, we find that a particular deformation, involving N N species of bosons and fermions with random couplings in flavor space, admits a finite incoherent, frequency-dependent conductivity at the IR fixed point, \sigma(\omega>0)\sim\omega^{-2/z} σ ( ω > 0 ) ∼ ω − 2 / z , where z z is the boson dynamical exponent. Leveraging the non-perturbative structure of quantum anomalies, we develop a powerful calculational method for transport. The resulting "anomaly-assisted large N N expansion" allows us to extract the conductivity systematically. Although our results imply that such random-flavor models are problematic as a description of the physical N = 1 N = 1 system, they serve to illustrate some general conditions for quantum critical transport as well as the anomaly-assisted calculational methods. In addition, we revisit an old result that irrelevant operators generate a frequency-dependent conductivity, \sigma(\omega>0) \sim \omega^{-2(z-2)/z} σ ( ω > 0 ) ∼ ω − 2 ( z − 2 ) / z , in problems of this kind. We show explicitly, within the scope of the original calculation, that this result does not hold for any order parameter.

DOI
10.21468/scipostphys.14.5.113

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We study electrical transport at quantum critical points (QCPs) associated with loop current ordering in a metal, focusing specifically on models of the “Hertz-Millis” type. At the infrared (IR) fixed point and in the absence of disorder, the simplest such models have infinite DC conductivity and zero incoherent conductivity at nonzero frequencies. However, we find that a particular deformation, involving N N species of bosons and fermions with random couplings in flavor space, admits a finite incoherent, frequency-dependent conductivity at the IR fixed point, \sigma(\omega>0)\sim\omega^{-2/z} σ ( ω > 0 ) ∼ ω − 2 / z , where z z is the boson dynamical exponent. Leveraging the non-perturbative structure of quantum anomalies, we develop a powerful calculational method for transport. The resulting "anomaly-assisted large N N expansion" allows us to extract the conductivity systematically. Although our results imply that such random-flavor models are problematic as a description of the physical N = 1 N = 1 system, they serve to illustrate some general conditions for quantum critical transport as well as the anomaly-assisted calculational methods. In addition, we revisit an old result that irrelevant operators generate a frequency-dependent conductivity, \sigma(\omega>0) \sim \omega^{-2(z-2)/z} σ ( ω > 0 ) ∼ ω − 2 ( z − 2 ) / z , in problems of this kind. We show explicitly, within the scope of the original calculation, that this result does not hold for any order parameter.
Non-Fermi liquids arise when strong interactions destroy stable fermionic quasiparticles. The simplest models featuring this phenomenon involve a Fermi surface coupled to fluctuating gapless bosonic order parameter fields, broadly referred to as ``Hertz-Millis'' models. We revisit a controlled approach to Hertz-Millis models that combines an expansion in the inverse number (N)(N)(N) of fermion species with an expansion in the deviation of the boson dynamical critical exponent zzz from 2. The structure of the expansion is found to be qualitatively different in the quantum critical regime $\mathrm{\ensuremath{\Omega}}\ensuremath{\ll}q$ and in the transport regime $\mathrm{\ensuremath{\Omega}}\ensuremath{\gg}q$. In particular, correlation functions in the transport regime involve infinitely many diagrams at each order in perturbation theory. We provide an explicit and tractable recipe to classify and resum these diagrams. For the simplest Hertz-Millis models, we show that this recipe is consistent with nonperturbative anomaly arguments and correctly captures the fixed point optical conductivity as well as leading corrections from irrelevant operators. We comment on potential applications of this expansion to transport in more complicated Hertz-Millis models as well as certain beyond-Landau metallic quantum critical points.
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.
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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 Q⃗=0⃗\vec{Q}=\vec{0}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 ΛΛΛ and ΛfΛ_fΛf​, which we fix by analyzing the RG flow of the scale-dependent ordering wave-vector Q⃗Λ\vec{Q}_ΛQ​Λ​. 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 Q⃗=0⃗\vec{Q}=\vec{0}Q​=0 and development of a first-order phase transition or a phase characterized by Q⃗≠0⃗\vec{Q}\neq\vec{0}Q​=0.
Instabilities of quantum critical metals in the limit Nf→0N_f\rightarrow0Nf​→0
We study a model in 2+1 dimensions composed of a Fermi surface of N_f N f flavors of fermions coupled to scalar fluctuations near quantum critical points (QCPs). The N_f\rightarrow0 N f → 0 limit allows us to non-perturbatively calculate the long-range behavior of fermion correlation functions. We use this to calculate charge, spin and pair susceptibilities near different QCPs at zero and finite temperatures, with zero and finite order parameter gaps. While fluctuations smear out the fermionic quasiparticles, we find QCPs where the overall effect of fluctuations leads to enhanced pairing. We also find QCPs where the fluctuations induce spin and charge density wave instabilities for a finite interval of order parameter fluctuation gaps at T=0 T = 0 . We restore a subset of the diagrams suppressed in the N_f\rightarrow0 N f → 0 limit, all diagrams with internal fermion loops with at most 2 vertices, and find that this does not change the long-range behavior of correlators except right at the QCPs.
We use the Kubo response functions to calculate the electrical and thermal conductivity and Seebeck coefficient at low temperatures and frequencies in the quantum-critical region for fermions on a lattice. The theory uses scattering of the fermions with the previously derived collective fluctuations due to topological defects of the quantum XY model coupled to fermions. The microscopic model is applicable to the fluctuations of the loop-current order in cuprates as well as to a class of quasi-two-dimensional heavy-fermion and other metallic antiferromagnets, and proposed recently also for the possible loop-current order in Moiré twisted bi-layer graphene and bilayer WSe2_22​. All these metals have a linear-in-temperature electrical resistivity in the quantum-critical region of their phase diagrams, often termed ``Planckian" resistivity. The solution of the Kubo equation for transport shows that vertex renormalizations to the external fields, beside those caused by Aslamazov-Larkin (A-L) processes, are absent. A-L appears as an Umklapp scattering matrix, which gives a temperature-independent multiplicative factor for the electrical resistivity but does not affect the thermal conductivity. We also show that the mass renormalization which gives a logarithmic enhancement of the marginal Fermi-liquid specific heat does not appear in the electrical resistivity and, more remarkably, in the thermal conductivity. On the other hand the mass renormalization ∝ln⁡ωc/T\propto \ln ω_c/T∝lnωc​/T appears in the Seebeck coefficient. We also discuss in detail the conservation laws which play a crucial role in all transport properties. We calculate exactly, the numerical coefficients of the transport properties for a circular Fermi surface. The leading temperature dependences is shown to remain the same for a general Fermi surface, but it is too messy to calculate the numerical coefficient.