Optical and DC conductivity in the two-dimensional
t−t′−t′′ Hubbard model near the antiferromagnetic quantum critical point
Two-dimensional ferromagnetic electron gases subject to random scalar potentials and Rashba spin-orbit interactions exhibit a striking quantum criticality. As disorder strength
W increases, such a system undergoes two transitions. It first changes from a normal diffusive metal consisting of extended states to a marginal metal consisting of critical states at a critical disorder
Wc,1. Another transition from the marginal metal to an insulator occurs at a stronger disorder
Wc,2. Through highly accurate numerical procedures based on the recursive Green's function and the exact diagonalization methods, we elucidate the nature of the quantum criticality and the properties of the pertinent states. The conductance is described by an unorthodox one-parameter scaling law: Conductance of various system sizes and disorders collapse into two branches of a scaling curve corresponding to diffusive metal and insulating phases with an exponentially diverging correlation length, $\ensuremath{\xi}\ensuremath{\propto}exp[\ensuremath{\alpha}/\sqrt{|W\ensuremath{-}{W}_{c}|}]$, near transition points. Finite-size scaling analysis of the inverse participation ratio reveals that the states of the marginal metal are fractals of a universal fractal dimension $D=1.90\ifmmode\pm\else\textpm\fi{}0.02$, while those of the diffusive metal spread over the whole system with
D=2 and states in the insulating phase are localized with
D=0. The phase diagram for diffusive metals, marginal metals, and the Anderson insulators is plotted in the disorder-magnetic-coupling plane.
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.