# Hall conductivity and edge response

The hall conductivity $\sigma_{xy}$ seems to reflect to some extent the response of a system in direction $\hat{y}$ to certain perturbation (electric field for example) restricted in $\hat{x}$ direction.

My question is, does a nonzero $\sigma_{xy}$ imply anything about the the physics of edge response, i.e. if given a half infinite system with an edge at $x=0$, what would be the effect on y direction? Would there be a current in y direction along the edge?

Thanks!

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There are a lot of interesting quantum edge effects in quantum hall effect materials, but these are not accessible just from the classical $\sigma_{xy}$. This quantity is the tensor component that describes how the E-field direction is tilted away from the current direction. Mathematically speaking, the tensor that you multiply $E_x,E_y$ by to get $J_x,J_y$ is symmetric, and is the conductivity $\sigma$ on the diagonal, and $\sigma_{xy}$ off diagonal.