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arXiv:cond-mat/0610676 · Physical Review B

Optical conductivity near finite-wavelength quantum criticality

S. Caprara, M. Grilli, C. Di Castro, Tilman Enss

Published Apr 13, 2007• 19 citations
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Abstract

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.

DOI
10.1103/physrevb.75.140505

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We study the optical conductivity \ensuremathσ(\ensuremathΩ)\ensuremath{\sigma}(\ensuremath{\Omega})\ensuremathσ(\ensuremathΩ) of an electron system near a quantum-critical point with finite-wavelength ordering. \ensuremathσ(\ensuremathΩ)\ensuremath{\sigma}(\ensuremath{\Omega})\ensuremathσ(\ensuremathΩ) 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.
We calculate the optical conductivity for a system of quasiparticles coupled to collective modes, which are (nearly) critical at a finite wavevector in the proximity of a quantum critical point. We show that a conserving approximation scheme is crucial to obtain correct results, because the contributions arising from the exchange of critical collective modes are all of the same order. We also show that a non-universal energy scale associated to non-electronic degrees of freedom, responsible for electron momentum dissipation, rules the low-energy response and may mask the effects of the collective modes even when these become fully critical.
Understanding optical conductivity data in the optimally doped cuprates in the framework of quantum criticality requires a strongly coupled quantum critical metal which violates hyperscaling. In the simplest scaling framework, hyperscaling violation can be characterized by a single nonzero exponent $\ensuremath{\theta}$, so that in a spatially isotropic state in ddd spatial dimensions, the specific heat scales with temperature as ${T}^{(d\ensuremath{-}\ensuremath{\theta})/z}$, and the optical conductivity scales with frequency as ${\ensuremath{\omega}}^{(d\ensuremath{-}\ensuremath{\theta}\ensuremath{-}2)/z}$ for $\ensuremath{\omega}\ensuremath{\gg}T$, where zzz is the dynamic critical exponent defined by the scaling of the fermion response function transverse to the Fermi surface. We study the Ising-nematic critical point, using the controlled dimensional regularization method proposed by Dalidovich and Lee [Phys. Rev. B 88, 245106 (2013)]. We find that hyperscaling is violated, with $\ensuremath{\theta}=1$ in d=2d=2d=2. We expect that similar results apply to Fermi surfaces coupled to gauge fields in d=2d=2d=2.
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.
We demonstrate that the optical conductivity of a Fermi liquid (FL) in the absence of umklapp scattering is dramatically affected by the topology of the Fermi surface (FS). Specifically, electron-electron (ee\mathit{ee}ee) scattering leads to rapid current relaxation in systems with multiple, or multiply connected, FSs, provided that the valleys have different effective masses. This effect results from intervalley drag. We microscopically derive the optical conductivity of a two-valley system, both within the FL regime and near a quantum critical point (QCP) of the Ising-nematic type. In the FL regime, intervalley drag restores the Gurzhi-like scaling of the conductivity, $\mathrm{Re}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{0}$. This dependence contrasts sharply with the previously identified subleading contribution to the conductivity of a two-dimensional FL with a single convex FS, where $\mathrm{Re}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{2}ln|\ensuremath{\omega}|$. The vanishing of the leading term in the optical conductivity is a signature of geometric constraints on ee\mathit{ee}ee scattering channels, which are lifted for a multiply connected FS. A large differential response, $d\mathrm{Re}\ensuremath{\sigma}/d\ensuremath{\mu}$ with $\ensuremath{\mu}$ being the chemical potential, is predicted at the Lifshitz transition from a single-valley to a multivalley FS, which should be observable within the experimentally accessible frequency range. Near a QCP, intervalley drag leads to a ${|\ensuremath{\omega}|}^{\ensuremath{-}2/3}$ scaling of $\mathrm{Re}\ensuremath{\sigma}(\ensuremath{\omega})$ in two dimensions, thus providing a specific current-relaxing process for this long-standing conjecture.
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 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.