We study the optical conductivity $\ensuremath{\sigma}(\ensuremath{\Omega})$ of an electron system near a quantum-critical point with finite-wavelength ordering. $\ensuremath{\sigma}(\ensuremath{\Omega})$ vanishes in clean Galilean-invariant systems, unless electrons are coupled to dynamical collective modes, which dissipate the current. This coupling introduces a nonuniversal energy scale. Depending on the parameters of each specific system, a variety of responses arise near criticality: scaling peaks at a temperature- and doping-dependent frequency, peaks at a fixed frequency, or no peaks to be associated with criticality. Therefore the lack of scaling in the far-infrared conductivity in cuprates does not necessarily call for new concepts of quantum criticality.
Optical phonons naturally generate linear-in-temperature ( T T ) resistivity in the equipartition regime, but their finite gap prevents this mechanism from surviving to asymptotically low temperatures. Here we analyze whether proximity to an electronic quantum critical point can remove this obstruction by strongly softening an optical phonon. We first derive a model-independent criterion for such softened phonons to control low-temperature transport. In addition to reducing the renormalized optical gap, the Landau-damped phonon must acquire a dynamical exponent z_p;gt d z p > d , where d d is the spatial dimension of the phonon, so that a sufficiently large thermally occupied phase space survives as T\to 0 T → 0 . We then analyze a concrete mechanism in which the phonon couples nonlinearly to long-wavelength electronic collective modes near a quantum critical point associated with an order parameter carrying zero center-of-mass momentum, and apply it to the Ising-nematic problem. Within a large- N N field theoretic formulation, the phonon softening is enhanced near criticality, but in the clean theory the resulting dynamics lies at or near the boundary for asymptotic T T -linear scattering. Including feedback from the softened phonon back onto the electronic critical sector further weakens the tendency toward robust low-temperature T T -linear transport. Our results sharpen both the promise and the limitations of phonon-based explanations of strange-metal transport near electronic criticality.
We study high-frequency response functions, notably the optical conductivity, in the vicinity of quantum critical points (QCPs) by allowing for both detuning from the critical coupling and finite temperature. We consider general dimensions and dynamical exponents. This leads to a unified understanding of sum rules. In systems with emergent Lorentz invariance, powerful methods from quantum field theory allow us to fix the high-frequency response in terms of universal coefficients. We test our predictions analytically in the large-N O(N) model and using the gauge-gravity duality and numerically via quantum Monte Carlo simulations on a lattice model hosting the interacting superfluid-insulator QCP. In superfluid phases, interacting Goldstone bosons qualitatively change the high-frequency optical conductivity and the corresponding sum rule.
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].
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=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=3 critical theory.
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.