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arXiv:1707.07352 · Physical review. B./Physical review. B

Optical conductivity of a two-dimensional metal near a quantum critical point: The status of the extended Drude formula

Andrey V. Chubukov, Dmitrii L. Maslov

Published Nov 20, 2017• 21 citations
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Abstract

This paper is devoted to the analysis of the frequency dependence of the optical conductivity, ${\ensuremath{\sigma}}^{\ensuremath{'}}(\mathrm{\ensuremath{\Omega}})$, of a metal near a quantum critical point (QCP). According to general reasoning, the frequency dependence of ${\ensuremath{\sigma}}^{\ensuremath{'}}(\mathrm{\ensuremath{\Omega}})$ may come via the scattering time and the effective mass, ${m}^{\ensuremath{\star}}$, but not via the quasiparticle residue $Z$, because the conductivity probes quasiparticles rather than bare electrons. However, in local theories of QCPs, ${m}^{\ensuremath{\star}}/m$, coincides with $1/Z$, and it is not straightforward to separate the two quantities. The authors use a direct diagrammatic approach, in which ${m}^{\ensuremath{\star}}/m$ and $Z$ are treated separately. They compute the optical conductivity ${\ensuremath{\sigma}}^{\ensuremath{'}}(\mathrm{\ensuremath{\Omega}})$ near two-dimensional nematic and spin-density wave (SDW) QCPs. They explicitly demonstrate that the $Z$ factor is canceled out by series of vertex corrections, and ${m}^{\ensuremath{\star}}/m$ does not appear separately from $Z$. As a result, ${\ensuremath{\sigma}}^{\ensuremath{'}}(\mathrm{\ensuremath{\Omega}})$ diverges as $1/{\mathrm{\ensuremath{\Omega}}}^{2/3}$ in the nematic case and as $1/\mathrm{\ensuremath{\Omega}}$ (up to logarithmic factors) in the SDW case.

DOI
10.1103/physrevb.96.205136

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This paper is devoted to the analysis of the frequency dependence of the optical conductivity, \ensuremathσ\ensuremath′(\ensuremathΩ){\ensuremath{\sigma}}^{\ensuremath{'}}(\mathrm{\ensuremath{\Omega}})\ensuremathσ\ensuremath′(\ensuremathΩ), of a metal near a quantum critical point (QCP). According to general reasoning, the frequency dependence of \ensuremathσ\ensuremath′(\ensuremathΩ){\ensuremath{\sigma}}^{\ensuremath{'}}(\mathrm{\ensuremath{\Omega}})\ensuremathσ\ensuremath′(\ensuremathΩ) may come via the scattering time and the effective mass, m\ensuremath⋆{m}^{\ensuremath{\star}}m\ensuremath⋆, but not via the quasiparticle residue ZZZ, because the conductivity probes quasiparticles rather than bare electrons. However, in local theories of QCPs, m\ensuremath⋆/m{m}^{\ensuremath{\star}}/mm\ensuremath⋆/m, coincides with 1/Z1/Z1/Z, and it is not straightforward to separate the two quantities. The authors use a direct diagrammatic approach, in which m\ensuremath⋆/m{m}^{\ensuremath{\star}}/mm\ensuremath⋆/m and ZZZ are treated separately. They compute the optical conductivity \ensuremathσ\ensuremath′(\ensuremathΩ){\ensuremath{\sigma}}^{\ensuremath{'}}(\mathrm{\ensuremath{\Omega}})\ensuremathσ\ensuremath′(\ensuremathΩ) near two-dimensional nematic and spin-density wave (SDW) QCPs. They explicitly demonstrate that the ZZZ factor is canceled out by series of vertex corrections, and m\ensuremath⋆/m{m}^{\ensuremath{\star}}/mm\ensuremath⋆/m does not appear separately from ZZZ. As a result, \ensuremathσ\ensuremath′(\ensuremathΩ){\ensuremath{\sigma}}^{\ensuremath{'}}(\mathrm{\ensuremath{\Omega}})\ensuremathσ\ensuremath′(\ensuremathΩ) diverges as 1/\ensuremathΩ2/31/{\mathrm{\ensuremath{\Omega}}}^{2/3}1/\ensuremathΩ2/3 in the nematic case and as 1/\ensuremathΩ1/\mathrm{\ensuremath{\Omega}}1/\ensuremathΩ (up to logarithmic factors) in the SDW case.
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 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.
We study the frequency dependence of the electron self-energy and the optical conductivity in a recently developed field theory of the spin-density-wave quantum phase transition in two-dimensional metals. We focus on the interplay between the Fermi surface ``hot spots'' and the remainder of the ``cold'' Fermi surface. Scattering of electrons off the fluctuations of the spin-density-wave order parameter, $\ensuremath{\varphi}$, is strongest at the hot spots; we compute the conductivity due to this scattering in a rainbow approximation. We point out the importance of composite operators, built of products of the primary electron or $\ensuremath{\varphi}$ fields: These have important effects also away from the hot spots. The simplest composite operator, ${\ensuremath{\varphi}}^{2}$, leads to deviations from Landau Fermi-liquid behavior on the entire Fermi surface. We also find an intermediate frequency window in which the cold electrons lose their quasiparticle form due to effectively one-dimensional scattering processes. The latter processes are part of umklapp scattering, which leads to singular contributions to the optical conductivity at the lowest frequencies at zero temperature.
The hyperscaling property implies that spatially isotropic critical quantum states in ddd spatial dimensions have a specific heat, which scales with temperature as Td/z{T}^{d/z}Td/z, and an optical conductivity, which scales with frequency as ${\ensuremath{\omega}}^{(d\ensuremath{-}2)/z}$ for $\ensuremath{\omega}\ensuremath{\gg}T$, where zzz is the dynamic critical exponent. We examine the spin density wave critical fixed point of metals in d=2d=2d=2 found by Sur and Lee [Phys. Rev. B 91, 125136 (2015)] in an expansion in $\ensuremath{\epsilon}=3\ensuremath{-}d$. We find that the contributions of the ``hot spots'' on the Fermi surface to the optical conductivity and specific heat obey hyperscaling (up to logarithms), and agree with the results of the large NNN analysis of the optical conductivity by Hartnoll et al. [ Phys. Rev. B 84, 125115 (2011)]. With a small bare velocity of the boson associated with the spin density wave order, there is an intermediate energy regime where hyperscaling is violated with $d\ensuremath{\rightarrow}{d}_{t}$, where dt=1{d}_{t}=1dt​=1 is the number of dimensions transverse to the Fermi surface. We also present a Boltzmann equation analysis which indicates that the hot-spot contribution to the dc conductivity has the same scaling as the optical conductivity, with TTT replacing $\ensuremath{\omega}$.
Recent progress in experimental techniques has made it possible to extract detailed information on dynamics of carriers in a correlated electron material from its optical conductivity, [Formula: see text]. This review consists of three parts, addressing the following three aspects of optical response: (1) the role of momentum relaxation; (2) [Formula: see text] scaling of the optical conductivity of a Fermi-liquid metal, and (3) the optical conductivity of non-Fermi-liquid metals. In the first part (section 2), we analyze the interplay between the contributions to the conductivity from normal and umklapp electron-electron scattering. As a concrete example, we consider a two-band metal and show that although its optical conductivity is finite it does not obey the Drude formula. In the second part (sections 3 and 4), we re-visit the Gurzhi formula for the optical scattering rate, [Formula: see text], and show that a factor of [Formula: see text] is the manifestation of the 'first-Matsubara-frequency rule' for boson response, which states that [Formula: see text] must vanish upon analytic continuation to the first boson Matsubara frequency. However, recent experiments show that the coefficient b in the Gurzhi-like form, [Formula: see text], differs significantly from b = 4 in most of the cases. We suggest that the deviations from Gurzhi scaling may be due to the presence of elastic but energy-dependent scattering, which decreases the value of b below 4, with b = 1 corresponding to purely elastic scattering. In the third part (section 5), we consider the optical conductivity of metals near quantum phase transitions to nematic and spin-density-wave states. In the last case, we focus on 'composite' scattering processes, which give rise to a non-Fermi-liquid behavior of the optical conductivity at T = 0: [Formula: see text] at low frequencies and [Formula: see text] at higher frequencies. We also discuss [Formula: see text] scaling of the conductivity and show that [Formula: see text] in the same model scales in a non-Fermi-liquid way, as [Formula: see text].
In an isotropic strongly interacting quantum liquid without quasiparticles, general scaling arguments imply that the dimensionless ratio $({k}_{B}/\ensuremath{\hbar})\phantom{\rule{0.16em}{0ex}}\ensuremath{\eta}/s$, where $\ensuremath{\eta}$ is the shear viscosity and sss is the entropy density, is a universal number. We compute the shear viscosity of the Ising-nematic critical point of metals in spatial dimension d=2d=2d=2 by an expansion below d=5/2d=5/2d=5/2. The anisotropy associated with directions parallel and normal to the Fermi surface leads to a violation of the scaling expectations: $\ensuremath{\eta}$ scales in the same manner as a chiral conductivity, and the ratio $\ensuremath{\eta}/s$ diverges at low temperature (TTT) as ${T}^{\ensuremath{-}2/z}$, where zzz is the dynamic critical exponent for fermionic excitations dispersing normal to the Fermi surface.