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Optical and Hall conductivities of a thermally disordered two-dimensional spin-density wave: two-particle response in the pseudogap regime of electron-doped high-$T_c$ superconductors

We calculate the longitudinal ($σ_{xx}$) and Hall ($σ_{xy}$) optical conductivities for two-dimensional metals with thermally disordered antiferromagnetism using a generalization of an approximation introduced by Lee, Rice and Anderson for the self energy. The conductivities are calculated from the Kubo formula, with current vertex function treated in a conserving approximation satisfying the Ward identity. In order to obtain a finite DC limit, we introduce phenomenologically impurity scattering, with relaxation time $τ$. $σ_{xx}(Ω)$ satisfies the $f$-sum rule. For the infinitely peaked spin correlation function, $χ(\mathbf{q})\propto δ(\mathbf{q}-\mathbf{Q})$, we recover the expressions for the conductivities in the mean-field theory of the ordered state. When the spin correlation length $ξ$ is large but finite, both $σ_{xx}$ and $σ_{xy}$ show behaviors characteristic of the state with long-range order. The calculation runs into difficulty for $Ω\lesssim 1/τ$. The difficulties are traced to an inaccurate treatment of the very low energy density of states within the Lee-Rice-Anderson approximation. The results for $σ_{xx}(Ω)$ and $σ_{xy}(Ω)$ are qualitatively consistent with data on electron-doped cuprates when $Ω>1/τ$.