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arXiv:2207.08841 · Physical review. B./Physical review. B

Large- N theory of critical Fermi surfaces. II. Conductivity

Haoyu Guo, Aavishkar A. Patel, Ilya Esterlis, Subir Sachdev A

Published Sep 30, 2022• 71 citations
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Abstract

A Fermi surface coupled to a scalar field can be described in a $1/N$ expansion by choosing the fermion-scalar Yukawa coupling to be random in the $N$-dimensional flavor space, but invariant under translations. We compute the conductivity of such a theory in two spatial dimensions for a critical scalar. We find a Drude contribution, and verify that the proposed $1/{\ensuremath{\omega}}^{2/3}$ contribution to the optical conductivity at frequency $\ensuremath{\omega}$ has vanishing coefficient for a convex Fermi surface. We also describe the influence of impurity scattering of the fermions, and find that while the self-energy resembles a marginal Fermi liquid, the resistivity and optical conductivity behave like a Fermi liquid.

DOI
10.1103/physrevb.106.115151

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A Fermi surface coupled to a scalar field can be described in a 1/N1/N1/N expansion by choosing the fermion-scalar Yukawa coupling to be random in the NNN-dimensional flavor space, but invariant under translations. We compute the conductivity of such a theory in two spatial dimensions for a critical scalar. We find a Drude contribution, and verify that the proposed 1/\ensuremathω2/31/{\ensuremath{\omega}}^{2/3}1/\ensuremathω2/3 contribution to the optical conductivity at frequency \ensuremathω\ensuremath{\omega}\ensuremathω has vanishing coefficient for a convex Fermi surface. We also describe the influence of impurity scattering of the fermions, and find that while the self-energy resembles a marginal Fermi liquid, the resistivity and optical conductivity behave like a Fermi liquid.
We analyze optical conductivity of a clean two-dimensional electron system in a Fermi liquid regime near a T=0T=0T=0 Ising-nematic quantum critical point (QCP) and extrapolate the results to a QCP. We employ direct perturbation theory up to the two-loop order to elucidate how the Fermi surface's geometry (convex vs concave) and fermionic dispersion (parabolic vs nonparabolic) affect the scaling of the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ with frequency $\ensuremath{\omega}$ and correlation length $\ensuremath{\xi}$. We find that for a convex Fermi surface the leading terms in the optical conductivity cancel out, leaving a subleading contribution $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}{\ensuremath{\xi}}^{4}\mathcal{L}$, where L=const\mathcal{L}=\mathrm{const}L=const for a parabolic dispersion and $\mathcal{L}\ensuremath{\propto}ln\ensuremath{\omega}{\ensuremath{\xi}}^{3}$ in a generic case. For a concave Fermi surface, the leading terms do not cancel, and $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\xi}}^{2}$. We extrapolate these results to a QCP and obtain $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2/3}$ for a convex Fermi surface and $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}1/{\ensuremath{\omega}}^{2/3}$ for a concave Fermi surface.
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We study the optical conductivity of a pristine two-dimensional electron system near an Ising-nematic quantum critical point. We discuss the relation between the frequency scaling of the conductivity and the shape of the Fermi surface, namely, whether it is isotropic, convex, or concave. We confirm the cancellation of the leading order terms in the optical conductivity for the cases of isotropic and convex Fermi surfaces and show that the remaining contribution scales as ${|\ensuremath{\omega}|}^{2/3}$ at T=0T=0T=0. On the contrary, the leading term, $\ensuremath{\propto}{|\ensuremath{\omega}|}^{\ensuremath{-}2/3}$, survives for a concave FS. We also address the frequency dependence of the optical conductivity near the convex-to-concave transition. Explicit calculations are carried out for the Fermi-liquid regime using the modified (but equivalent to the original) version of the Kubo formula, while the quantum-critical regime is accessed by employing the space-time scaling of the Z=3Z=3Z=3 critical theory.
We study the fermion-boson coupled system in d=3−ϵ space dimensions near the quantum phase transition; infinite many boson modes locating on a sphere become critical simultaneously, which is dubbed “critical boson surface” (CBS). The fermions on the Fermi surface can be scattered to nearby points locating on a boson ring in the low-energy limit. The large number of the boson scattering channels N renders the well-known Landau damping effect largely suppressed. We propose an effective theory for a single point on the Fermi surface and the associated critical boson ring induced by the boson scattering channels. Based on the effective model, one-loop renormalization group analysis is performed with asymptotic ϵ-expansion. The fermion self-energy and Yukawa interaction vertex are dressed with ϵ poles and are largely enhanced due to the presence of critical boson ring. The imaginary part of the self-energy exhibits a linear-in frequency dependence and the real part gives a vanishing quasiparticle weight in the low-energy limit, which signatures the celebrated “marginal Fermi liquid” behavior. We find a novel non-Fermi liquid fixed point, at which the critical properties show features associated with the CBS.
We study the low-energy effective theory for a non-Fermi-liquid state in 2+12+12+1 dimensions, where a transverse U(1) gauge field is coupled with a patch of Fermi surface with NNN flavors of fermion in the large NNN limit. In the low-energy limit, quantum corrections are classified according to the genus of the two-dimensional surface on which Feynman diagrams can be drawn without a crossing in a double line representation and all planar diagrams are important in the leading order. The emerging theory has the similar structure to the four-dimensional SU(N) gauge theory in the large NNN limit. Because of strong quantum fluctuations caused by the abundant low-energy excitations near the Fermi surface, low-energy fermions remain strongly coupled even in the large NNN limit. As a result, there are infinitely many quantum corrections that contribute to the leading frequency dependence of the Green's function of fermion on the Fermi surface. On the contrary, the boson self-energy is not modified beyond the one-loop level and the theory is stable in the large NNN limit. The nonperturbative nature of the theory also shows up in correlation functions of gauge-invariant operators.