Time reversal symmetry breaking

[hbaromega]

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]

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]

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?

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]

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.

I though it should be equivalent. When symmetry is held, the laws would be the same after symmetry operation. Both in Hamiltonian or the force.

Even if I consider the Hamiltonian, doesn't it come like

H=(p-eA/c)^2 ?

Now when we apply T-reversal operator p and A both change sign. Then overall Hamiltonian remain invariant. Then how do we get T-reversal symmetry breaking in quantum Hall effect after applying magnetic field?

 

[hbaromega]

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.

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]

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]

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.

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.

 

[hbaromega]

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

I think, there should be an extra conjugation operator as well (as K in Sakurai's Mod. Q. M. Chapter 4). But still I'm not clear how to operate to show whether TR symm. broken. I can only understand that it can change up-spin ket to a down-spin ket.

Thanks for the reference. I looked at the paper though need to spend time fully understand. But I think, he is one of the proposers of TKNN invariant in QHE. I roughly understand the topology argument which connected to Berry phase that appears due to periodicity of the Bloch wave function.

Actually my question arises due to topological insulator context. People say that TR symmetry is broken in QHE, where spin degeneracy is maintained, but TR is symm. is preserved in QSHE where spin degeneracy is lifted due to strong spin orbit coupling.

Now spin orbit coupling looks like S.L .Since S and L both are odd to TR, together they become invariant.

However, how can I show that this cannot happen to Ferromagnet or QHE where we expect similar interaction term in the Hamiltonian?

Sound pretty confusing to me!

 

[hbaromega]

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.

So you mean in the process of T-reversal I'm not looking at the magnetic field ! So is it like what happens if I put a magnetic field in the same direction when the system is under T-reversal ?