What breaks time reversal symmetry in ferromagnets

[taishizhiqiu]

I am told that in ferromagnets, time reversal symmetry is broken. However, I don't know any hamiltonian terms in solid that can break time reversal symmetry. So is there a hamiltonian term I don't know or is there any subtlety in ferromagnets?

 

[DrDu]

In spontaneous symmetry breaking, there is never a term which explicitly breaks the symmetry in the Hamiltonian

 

[taishizhiqiu]

OK, I finally learned spontaneous symmetry breaking. However, I am stilled confused. Supposing two local minimum in ferromagnets, time reversal symmetry will transform one into another, both of which are ground states of the Hamiltonian. Thus, it seems that $T$ is actually commutable with $H$. Then what does time reversal symmetry breaking mean here?

 

[DrDu]

T commutes with H, but the ground state is not symmetric under T. That's the general situation in spontaneous symmetry breaking. Likewise, rotational symmetry is also spontaneously broken in a ferromagnet. While the hamiltonian is symmetric under rotations, each of the ground states which are intertransformed under rotations shows a preferred orientation of the spin directions.

 

[DrDu]

In fact, the spontaneously broken symmetries cannot be represented as operators in a Hilbert space, as all matrix elements involving two different ground states vanish. That's the formal definition of a spontaneously broken symmetry.

 

[taishizhiqiu]

In fact, the spontaneously broken symmetries cannot be represented as operators in a Hilbert space, as all matrix elements involving two different ground states vanish. That's the formal definition of a spontaneously broken symmetry.

I don't understand this. Can you show me an example?

 

[radium]

The different grounds states belong to what are called different superselection sectors. It means that you basically can't get from one to the other in the thermodynamic limit. If you think of the Ising model, the ordered spin up state and ordered spin down states are degenerate without a magnetic field. However, you cannot get between the two ordered states through spin flips in a finite time in an infinite system.

By the way, the reason that AFs break time reversal is that time reversal flips the spins on each site. AFs also break translation (double the unit cell) but the combination of translation and TR is preserved.