QHE: rotational invariance, no terms linear in E or B

In order for the equations to be rotationally invariant, the terms must also be rotationally invariant, meaning they should only depend on the magnitudes of the vectors. This is why terms linear in E or B cannot be written down, since they depend on the direction of the vectors. Instead, the first terms that can be written down are quadratic in E and B, such as E.E and B.B, which only depend on the magnitudes and are therefore rotationally invariant. This is important in the context of d = 3 + 1 dimensions, as explained in David Tong's notes on the Quantum Hall Effect.
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binbagsss
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'Let’s first see what all of this means in the context of d = 3 + 1 dimensions. If we have rotational invariance then we can’t write down any terms linear in E or B. The first terms that we can write down are instead ...'

Why is this? I don't understnad . My thoughts would be pictruing the set up needing to be rotationally invariant, and since E and B are perpendicular a linear term alone wouldn't do this, but this wouldn't explaiin why E.E and B.B are invariant, I don't really know what I'm talking about, as you can tell

many thanks in advancehttp://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf , page 145 David Tong notes QHE , chapter 5 (eq. 5.5)
 
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binbagsss said:
Why is this?

Because vectors are not invariant under rotations: the rotation makes them point in a different direction. But the magnitudes of vectors are invariant under rotations, since rotation doesn't change the length of a vector, only its direction.
 
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1. What is QHE (Quantum Hall Effect)?

The Quantum Hall Effect (QHE) is a phenomenon in condensed matter physics where a strong magnetic field applied perpendicular to a two-dimensional electron gas leads to the quantization of the Hall conductivity, resulting in a series of plateaus in the Hall resistance at certain values of the magnetic field.

2. What is rotational invariance in QHE?

Rotational invariance in QHE refers to the fact that the quantized Hall conductivity is independent of the orientation of the magnetic field with respect to the sample. This means that the QHE is a robust and universal phenomenon, not affected by the direction of the magnetic field.

3. Why are there no terms linear in E or B in QHE?

In QHE, the Hall conductivity is quantized and does not depend on the strength of the magnetic field or the electric field. This is due to the fact that the electrons in the two-dimensional gas are confined to Landau levels, which are discrete energy levels. As a result, the Hall conductivity remains constant and does not vary with changes in the electric or magnetic field.

4. What is the significance of QHE in condensed matter physics?

QHE is a significant discovery in condensed matter physics as it provides a clear example of a topological phase transition, where the properties of a material change without any change in its symmetry. It has also led to the discovery of new states of matter, such as the fractional quantum Hall effect, and has potential applications in quantum computing and metrology.

5. How is QHE experimentally observed?

QHE is experimentally observed by measuring the Hall resistance of a two-dimensional electron gas at low temperatures and high magnetic fields. The quantized plateaus in the Hall resistance can be detected using sensitive electronic equipment, confirming the presence of the QHE. This experimental technique has been crucial in studying the properties of QHE and its various applications.

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