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We calculate the longitudinal ($\sigma_{xx}$) and Hall ($\sigma_{xy}$)optical conductivities for two-dimensional metals with thermally disorderedantiferromagnetism using a generalization of an approximation introduced byLee, Rice and Anderson for the self energy. The conductivities are calculatedfrom the Kubo formula, with current vertex function treated in a conservingapproximation satisfying the Ward identity. In order to obtain a finite DClimit, we introduce phenomenologically impurity scattering, with relaxationtime $\tau$. $\sigma_{xx}(\Omega)$ satisfies the $f$-sum rule. For theinfinitely peaked spin correlation function, $\chi(\mathbf{q})\propto\delta(\mathbf{q}-\mathbf{Q})$, we recover the expressions for theconductivities in the mean-field theory of the ordered state. When the spincorrelation length $\xi$ is large but finite, both $\sigma_{xx}$ and$\sigma_{xy}$ show behaviors characteristic of the state with long-range order.The calculation runs into difficulty for $\Omega\lesssim 1/\tau$. Thedifficulties are traced to an inaccurate treatment of the very low energydensity of states within the Lee-Rice-Anderson approximation. The results for$\sigma_{xx}(\Omega)$ and $\sigma_{xy}(\Omega)$ are qualitatively consistentwith data on electron-doped cuprates when $\Omega>1/\tau$.

physical review. b, condensed matter and materials physics

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