# Hall Conductivity and Rotational Invariance

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• thatboi
thatboi
I am reading up on QHE from: https://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf
and am confused about the comment: "The action (5.5) has no Hall conductivity because this is ruled out in d = 3+1 dimensions on rotational grounds." Can someone explain why this is the case? My only guess is that the action in (5.5) contains no linear terms, whereas if we look at the general case in IQHE where have an electric and magnetic field, the Hamiltonian would contain terms linear in the magnetic field and electric field. But I cannot understand why this would necessarily rule out the IQHE?

Ok, I think the easiest way to see this is to rewrite (5.5) using the Maxwell tensor ##F_{\mu\nu}^{2} = 2(\partial_{\mu}A_{\nu})^2-2(\partial_{\mu}A_{\mu})^{2}##. Then we note that taking the functional derivative of the Maxwell action with respect to any specific ##A_{\mu}## necessarily evaluates to ##0## so there is no Hall Conductivity.

## What is Hall conductivity?

Hall conductivity is a measure of the induced transverse current in a material when it is subjected to an external magnetic field. It quantifies the Hall effect, where a voltage difference is generated perpendicular to both the current flow and the magnetic field.

## How is rotational invariance related to Hall conductivity?

Rotational invariance refers to the property of a system where its physical laws remain unchanged under rotations. In the context of Hall conductivity, this implies that the Hall response should be the same regardless of the orientation of the material, provided the magnetic field and current directions are appropriately adjusted.

## Why is rotational invariance important in studying Hall conductivity?

Rotational invariance is important because it simplifies the theoretical analysis and ensures that the derived properties of Hall conductivity are intrinsic to the material and not dependent on its orientation. This invariance helps in making universal predictions about the Hall effect in different materials.

## Can Hall conductivity be anisotropic?

Yes, Hall conductivity can be anisotropic in materials that lack rotational symmetry, such as those with a crystalline structure that has different properties along different axes. In such cases, the Hall conductivity may vary depending on the direction of the applied magnetic field and current.

## How do researchers test for rotational invariance in Hall conductivity experiments?

Researchers test for rotational invariance by conducting experiments where they measure Hall conductivity while systematically rotating the sample or changing the orientation of the applied magnetic field and current. Consistent results across different orientations indicate rotational invariance.

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