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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?


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Your word "half-infinite" is quite tricky. A QHS is defined on a closed manifold, a 2-torus. If I open x direction, there is a chiral current in y direction (a circle); if I open both directions, there will be a chiral current around the edge. Yes, for your question, there is a current in y direction, you may just stretch the other sides to infinity.

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Classically it doesn't do any of this--- it just says that you have a current in a direction different from the applied voltage. If you apply a voltage dropping in the x direction, you get an electric field in the x-direction (at first), so the current will have a y-component (at first). If you wait a little, this current will build up charges on the edges which will change the direction of the electric field in the interior, and this will continue until you redirect the current all in the x direction. At this point, you have a y-component of the electric field, and integrating this E field gives the hall voltage.

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.

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