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arXiv:2401.17392 · Physical review. B./Physical review. B

Fermi surface geometry and optical conductivity of a two-dimensional electron gas near an Ising-nematic quantum critical point

Yasha Gindikin, Andrey V. Chubukov

Published Mar 27, 2024• 12 citations
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Abstract

We analyze optical conductivity of a clean two-dimensional electron system in a Fermi liquid regime near a $T=0$ 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 $\mathcal{L}=\mathrm{const}$ 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.

DOI
10.1103/physrevb.109.115156

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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 \ensuremathσ(\ensuremathω)\ensuremath{\sigma}(\ensuremath{\omega})\ensuremathσ(\ensuremathω) with frequency \ensuremathω\ensuremath{\omega}\ensuremathω and correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ. We find that for a convex Fermi surface the leading terms in the optical conductivity cancel out, leaving a subleading contribution \ensuremathσ(\ensuremathω)\ensuremath∝\ensuremathω2\ensuremathξ4L\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}{\ensuremath{\xi}}^{4}\mathcal{L}\ensuremathσ(\ensuremathω)\ensuremath∝\ensuremathω2\ensuremathξ4L, where L=const\mathcal{L}=\mathrm{const}L=const for a parabolic dispersion and L\ensuremath∝ln\ensuremathω\ensuremathξ3\mathcal{L}\ensuremath{\propto}ln\ensuremath{\omega}{\ensuremath{\xi}}^{3}L\ensuremath∝ln\ensuremathω\ensuremathξ3 in a generic case. For a concave Fermi surface, the leading terms do not cancel, and \ensuremathσ(\ensuremathω)\ensuremath∝\ensuremathξ2\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\xi}}^{2}\ensuremathσ(\ensuremathω)\ensuremath∝\ensuremathξ2. We extrapolate these results to a QCP and obtain \ensuremathσ(\ensuremathω)\ensuremath∝\ensuremathω2/3\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2/3}\ensuremathσ(\ensuremathω)\ensuremath∝\ensuremathω2/3 for a convex Fermi surface and \ensuremathσ(\ensuremathω)\ensuremath∝1/\ensuremathω2/3\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}1/{\ensuremath{\omega}}^{2/3}\ensuremathσ(\ensuremathω)\ensuremath∝1/\ensuremathω2/3 for a concave Fermi surface.
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 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.
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 ZZZ, because the conductivity probes quasiparticles rather than bare electrons. However, in local theories of QCPs, ${m}^{\ensuremath{\star}}/m$, 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 ${m}^{\ensuremath{\star}}/m$ and ZZZ 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 ZZZ factor is canceled out by series of vertex corrections, and ${m}^{\ensuremath{\star}}/m$ does not appear separately from ZZZ. 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.
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.
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.
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 $1/{\ensuremath{\omega}}^{2/3}$ 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.