Anisotropic conductivity tensor on a half-filled high Landau level

We study two-dimensional interacting electrons in a weak perpendicular magnetic field with the filling factor $\nu ⪢1$ and in the presence of a quenched disorder. As it is known, the unidirectional charge density wave state can exist near a half-filled high Landau level at low temperatures if disorder is weak enough. We show that the existence of the unidirectional charge density wave state at temperature $T<T_c$ where $T_c$ is the transition temperature leads to the anisotropic conductivity tensor. We find that the anisotropic part of conductivity tensor is proportional to $(T_c-T)∕T_c$ below the transition. The order parameter fluctuations wash out the mean-field cusp at $T=T_c$ and the conductivity tensor becomes anisotropic even above the mean-field transition temperature $T_c$.

Figure 1

Mean-field phase diagram for electrons on half-filled high Landau levels, $N⪢1$.

Figure 2

Diagrams for corrections to the conductivity tensor ${\sigma }_{ab}$. Solid line denotes Green’s function, $N∕p∕p^{\prime }$ Landau level indices, dashed line the induced potential $\lambda \left(\mathit{r}\right)$, and shaded block the impurity ladder.

Figure 3

Impurity ladder. Frequency ${\omega }_n+{\nu }_n$ runs from right to left, whereas ${\omega }_n$ runs from left to right.

Figure 4

Functions $h\left(z\right)$, $h_{xx}\left(z\right)$ and $h_{xy}\left(z\right)$.

Figure 5

Functions $H\left(z\right)$, $H_{xx}\left(z\right)$, and $H_{xy}\left(z\right)$.

Figure 6

The dependence of anisotropic part of conductivity $({\sigma }_{yy}-{\sigma }_{xx})∕2{\sigma }_{xx}^{\left(0\right)}$ on temperature for $1∕4\pi T_c\tau =0.24$ and $\eta =0.01$. Dashed line corresponds to mean-field result (67).