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Collisionless transport close to a fermionic quantum critical point

Bitan Roy, Vladimir Juričić

Published Jan 10, 2018• 2 citations
OpenAlex

Abstract

We here report the scaling of optical conductivity (OC) in the \emph{collisionless} regime ($\hbar \omega \gg k_B T$) in the vicinity of a relativistic quantum critical point, separating two-dimensional massless Dirac fermions from a fully gapped insulator. Inside the critical regime gapless fermionic and bosonic excitations are strongly coupled, leading to a \emph{universal} suppression of the inter-band OC according to $\sigma^{\rm IB}_0 [1-N_b/(2N_f)]$ for $N_f \gg 1$ as $T \to 0$ to the leading order in $\epsilon$-expansion, where $N_f (N_b)$ is the fermionic(bosonic) flavor number, $\sigma^{\rm IB}_0$ is the OC of a non-interacting Dirac fluid, and $\epsilon=3-d$. Furthermore, the weight of the Drude peak inside the critical regime also gets suppressed following a universal scaling function, while maintaining its delta function profile. Correction to the OC at such a non-Gaussian critical point due to the long-range Coulomb interaction and generalization of these scenarios to a strongly interacting three-dimensional Dirac or Weyl liquid are also presented.

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We here report the scaling of optical conductivity (OC) in the \emph{collisionless} regime (ℏω≫kBT\hbar \omega \gg k_B Tℏω≫kB​T) in the vicinity of a relativistic quantum critical point, separating two-dimensional massless Dirac fermions from a fully gapped insulator. Inside the critical regime gapless fermionic and bosonic excitations are strongly coupled, leading to a \emph{universal} suppression of the inter-band OC according to σ0IB[1−Nb/(2Nf)]\sigma^{\rm IB}_0 [1-N_b/(2N_f)]σ0IB​[1−Nb​/(2Nf​)] for Nf≫1N_f \gg 1Nf​≫1 as T→0T \to 0T→0 to the leading order in ϵ\epsilonϵ-expansion, where Nf(Nb)N_f (N_b)Nf​(Nb​) is the fermionic(bosonic) flavor number, σ0IB\sigma^{\rm IB}_0σ0IB​ is the OC of a non-interacting Dirac fluid, and ϵ=3−d\epsilon=3-dϵ=3−d. Furthermore, the weight of the Drude peak inside the critical regime also gets suppressed following a universal scaling function, while maintaining its delta function profile. Correction to the OC at such a non-Gaussian critical point due to the long-range Coulomb interaction and generalization of these scenarios to a strongly interacting three-dimensional Dirac or Weyl liquid are also presented.
Quantum transport close to a critical point is a fundamental, but enigmatic problem due to fluctuations, persisting at all length scales. We report the scaling of optical conductivity (OC) in the collisionless regime (ℏω≫k_{B}T) in the vicinity of a relativistic quantum critical point, separating two-dimensional (d=2) massless Dirac fermions from a fully gapped insulator or superconductor. Close to such a critical point, gapless fermionic and bosonic excitations are strongly coupled, leading to a universal suppression of the interband OC as well as of the Drude peak (while maintaining its delta function profile) inside the critical regime, which we compute to the leading order in 1/N_{f}- and ε expansions, where N_{f} counts the fermion flavor number and ε=3-d. Correction to the OC at such a non-Gaussian critical point due to the long-range Coulomb interaction and generalizations of these scenarios to a strongly interacting three-dimensional Dirac or Weyl liquid are also presented, which can be tested numerically and possibly from nonperturbative gauge-gravity duality, for example.
We compute universal electrical transport near continuous transitions between a composite Fermi liquid (CFL) and a metallic phase in moire Chern bands, focusing on fillings ν=−1/2ν=-1/2ν=−1/2 and ν=−3/4ν=-3/4ν=−3/4. The critical theory represents a novel QED-Chern-Simons framework: a charged sector at a bosonic Laughlin-superfluid critical point is coupled, via emergent gauge fields and Chern-Simons mixing, to a neutral spinon Fermi surface. Integrating out matter fields to quadratic order yields an explicit Ioffe-Larkin composition rule for the full resistivity tensor, showing how longitudinal channels add in series while Chern-Simons terms generate Hall response. To obtain the DC limit in the quantum critical fan, we develop a controlled large-N expansion where both fermion flavors and Chern-Simons levels scale with NNN, and solve a quantum Boltzmann equation at leading nontrivial order 1/N1/N1/N. Gauge-mediated inelastic scattering removes the collisionless Drude singularity and produces a universal scaling function Σ(ω/T)Σ(ω/T)Σ(ω/T) and finite DC conductivities σ(0)≈0.033e2/ℏσ(0) \approx 0.033 e^2/\hbarσ(0)≈0.033e2/ℏ (ν=−1/2ν=-1/2ν=−1/2) and 0.047e2/ℏ0.047 e^2/\hbar0.047e2/ℏ (ν=−3/4ν=-3/4ν=−3/4). We also identify a Chern-Simons "filtering" mechanism that suppresses transmission of Landau damping from the spinon Fermi surface to the critical gauge mode. Our approach provides concrete transport diagnostics for detecting quantum criticality in moire superlattices.
We analyze emergent quantum multicriticality for strongly interacting, massless Dirac fermions in two spatial dimensions (d=2d=2d=2) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break O(S1)\text{O}({S}_{1})O(S1​) and O(S2)\text{O}({S}_{2})O(S2​) symmetries in the ordered phase. Performing a renormalization-group analysis by using the $\ensuremath{\epsilon}=(3\ensuremath{-}d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an enlarged O(S1+S2)\text{O}({S}_{1}+{S}_{2})O(S1​+S2​) chiral symmetry. Such a fixed point acts as an exotic quantum multicritical point (MCP), governing the continuous semimetal-insulator as well as insulator-insulator (for example, antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either O(S1)\text{O}({S}_{1})O(S1​) or O(S2)\text{O}({S}_{2})O(S2​) chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power-law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher-dimensional Dirac fermions is also outlined.
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.
Four-component massive and massless Dirac fermions in the presence of long range Coulomb interaction and chemical potential disorder exhibit striking fermionic quantum criticality. For an odd number of flavors of Dirac fermions, the sign of the Dirac mass distinguishes the topological and the trivial band insulator phases, and the gapless semimetallic phase corresponds to the quantum critical point that separates the two. Up to a critical strength of disorder, the semimetallic phase remains stable, and the universality class of the direct phase transition between two insulating phases is unchanged. Beyond the critical strength of disorder the semimetallic phase undergoes a phase transition into a disorder controlled diffusive metallic phase, and there is no longer a direct phase transition between the two types of insulating phases.
We study electric transport near the Mott metal-insulator transition in a triangular-lattice Hubbard model at half filling. We calculate optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ based on a cellular dynamical mean-field theory including vertex corrections inside the cluster. Near the Mott critical end point, a Drude analysis in the metallic region suggests that the change in the Drude weight is important rather than that in the transport scattering rate for the Mott transition. In the insulating region, there emerges an ``in-gap'' peak in $\ensuremath{\sigma}(\ensuremath{\omega})$ at low $\ensuremath{\omega}$ near the Mott transition, and this smoothly connects to the Drude peak in the metallic region with decreasing Coulomb repulsion. We find that the weight of these peaks exhibits a power-law behavior upon controlling Coulomb repulsion at the critical temperature. The obtained critical exponent suggests that conductivity does not correspond to magnetization or energy density of the Ising universality class in contrast to several previous works.