Topological phase and spontaneous symmetry breaking coexist?

[SoumiGhosh]

As we know topological phases cannot be explained using spontaneous symmetry breaking and order parameter. But can they coexist? Suppose there is a system which is undergoing quantum phase transition to a anti-ferromagnetic phase from a disordered phase. So in the anti-ferromagnetic phase continuous rotational symmetry is spontaneously broken. Can this anti-ferromagnetic phase be a topological insulator as well? Or one need to tune some parameter within the anti-ferromagnetic phase to make it a anti-ferromagnetic topological insulator.

 

[radium]

I have heard of antiferromagnetic TIs. Look at http://arxiv.org/abs/1004.1403.

I am not quite clear about how you are thinking about a topological phase. It's a very broad term. Not all topologically nontrivial systems have edge states. For example, not all Z2 spin liquids have edge states (certain ones do if also considering symmetry enriched phases). Furthermore, you must make a distinction between topologically ordered phases and symmetry protected topological phases (SPTs). Topologically ordered systems need not have any symmetry. Think of the FQHE or symmetric spin liquids. Some need a broken symmetry. Chiral SLs need broken time reversal (the FQHE does too, that's why the edge states are chiral). Others are enriched by the presence of symmetries, called symmetry enriched topological phases (SETs) which is actually a component of my current work. It means that if you consider symmetries in SLs, you can have different phases which cannot be tuned into one another even if the different states have the same symmetries.

By contrast, SPTs need some certain symmetry for the system to be nontrivial. For example, in a garden variety QSHE (K-M model, Mercury telluride) or 3D TI (take bismuth selenide as an example), you need to have time reversal symmetry. If you have that, when the spin orbit term is strong enough to invert the bands (an odd number specifically) they can't uninvert because you can't scatter between Kramers pairs for an odd number of excitation. Kramers degeneracy means that systems with TR symmetry and an odd number of electrons are at least two fold degenerate. This means there is a two fold degeneracy at the TR invariant points in the BZ of two orthogonal states. You can't break it with a local perturbation without breaking TR. The gap can't close so the edge states are protected. You would go from trivial to nontrivial by tuning the spin orbit term in this case. The gap closes at the transition.

An antiferromagnetic breaks both time reversal (we know for a fact magnetic impurities in a typical Z2 topological insulator will gap the edges) and translation since it enlarges the unit cell (this is a discrete symmetry). However, the combination of these operators is still a symmetry so you can have protected edge states which are protected by this symmetry. From the paper above it seems like there are some cases where the edges are gapped though. I think this is because our symmetry is the product of a crystal symmetry and time reversal it is sensitive to how the edges are terminated.

For states with bands inverted by spin orbit, the value of the spin orbit coupling is what is responsible for the topological classification. As I said above, if you start from a trivial insulator and turn on spin orbit, you will reach a point where the gap closes (the dirac point in graphene without considering SO) and then reopens with the bands inverted. The case of graphene is simple to see because if you look at the Haldane or KM (technically a copy of the Haldane model for each spin), you will find the masses are opposite around the k points where the Dirac points were. Look up the Haldane model online and plot it with different values of the mass M and you'll see what I mean about the transition.

So long story short, all of the SPT systems I have heard of with some symmetry breaking are cases where if you break two symmetries but still have the product of them, you can still have a nontrivial state. As you see, topological phases of matter are very subtle and highly nontrivial in many cases.