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hbaromega
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We know velocity/momentum and magnetic field both are odd to time-reversal operation. Then how is the time-reversal symmetry broken in quantum Hall effect since magnetic field is always coupled with velocity/momentum?
hbaromega said:We know velocity/momentum and magnetic field both are odd to time-reversal operation. Then how is the time-reversal symmetry broken in quantum Hall effect since magnetic field is always coupled with velocity/momentum?
stone said:In quantum mechanics, the effect of magnetic field is included by using the vector potential (A). This is done by replacing the momentum operator p by (p-eA/c). Now both p and A are odd under time reversal.
It's not the force which determines whether the system is time reversal invariant or not.
weejee said:You are right indeed, but here, the B field is usually considered 'external'.
With the direction of the 'external' B field fixed, the system certainly breaks time-reversal.
hbaromega said:Sorry didn't get what you meant by 'external'. Doesn't it become a part of the Hamiltonian? In Landau level we indeed solve the Hamiltonian
H=(p-eA/c)^2/2m
* Sorry ! I missed the 2m term in earlier post.
stone said:Okay..sorry I was a bit sloppy in the last post!
To determine if a system has time reversal symmetry, what should we do?
We must construct the time reversal operator (T). For fermions (as in this case) this would be exp(i*pi*Sy).
Where Sy is the pauli matrix. Now if the commutator of Hamiltonian H and T vanishes, then we can conclude that the system has time reversal symmetry. Not otherwise!
Now for this (QHE) case let us choose the Landau gauge A=(By,0,0) and thus the magnetic field is constant and in the z direction.
If you calculate the commutator you will realize that it is non zero. And hence time reversal symmetry is broken in this system.
Another way to argue this is to use what are called the Chern numbers, which in this case are the quantized values of conductivity. I suggest that you go through this paper:
Mahito Kohmoto, Topological invariant and the quantization of the Hall conductance
Annals of Physics
Volume 160, Issue 2, 1 April 1985, Pages 343-354
weejee said:That means, when you perform the time reversal, you don't change the direction of the magnetic field (or equivalently, A).
Suppose you are measuring the Hall conductivity of a sample. If you just look at the sample, its behavior breaks the time-reversal symmetry. (B field is a fixed quantity here.) However, if you take the sample plus the magnet as your system, it preserves the time-reversal symmetry as a whole.
Time reversal symmetry breaking is a phenomenon in physics where the laws of physics appear to behave differently when time is reversed. This means that certain physical processes, such as the direction of heat flow or the rotation of a spinning object, can only occur in one direction in time, rather than being able to occur in both the forward and backward directions.
Time reversal symmetry breaking occurs when a system is not in equilibrium or when there is an external influence, such as an external force or a temperature gradient, acting on the system. These factors cause the system to evolve in a specific direction in time, breaking the symmetry that would otherwise allow for the same processes to occur in both directions.
Time reversal symmetry breaking has significant implications for our understanding of fundamental physics and the behavior of matter and energy. It helps explain why certain physical laws appear to only work in one direction in time, and it also plays a role in understanding the behavior of systems at the micro and macro scales.
Yes, time reversal symmetry breaking has been observed in various experiments and natural phenomena. One example is the weak nuclear force, which only operates in one direction in time, breaking the time reversal symmetry. Other examples include the directional flow of heat and the chirality of molecules.
Time reversal symmetry breaking is closely related to the concept of entropy, which is a measure of the disorder or randomness in a system. As time reversal symmetry breaking causes systems to evolve in a specific direction in time, it also leads to an increase in entropy. This is because the system becomes more disordered as it moves away from its original state, in which time reversal symmetry was not broken.