# A Lattice non-perturbative definition

1. Oct 15, 2017

### shiraz

Dear All
I would like to understand a paper for Xiao-Gang Wen " A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model" .
Can any one help me by suggesting another reference that is a little bit easier.
Thank you

2. Oct 15, 2017

### DeathbyGreen

This paper is a merger of a lot of complicated issues. For other readers, the paper is posted on arXiv here: https://arxiv.org/pdf/1305.1045.pdf
Basically the problem they are addressing is that the standard model can only be described perturbatively (as a coupling constant expansion), keeping the first one or two perturbative terms. They look to turn this into a "Hamiltonian quantum theory" (a term I have not heard before, but as they define in the paper a theory which is described by operators and has a time evolution described by a quantity such as $U(t,t') = \mathcal{T}exp(-i\int_{t_0}^tdt'V_I(t'))$) in order to get an exact description of a SO(10) gauge theory by placing it on a finite lattice.

He also mentions a finite dimensional Hilbert space, but I assume this means they are introducing boundary conditions somewhere. The most helpful places to look are in the appendices, as this is where they show the math. It goes into the Clifford algebra of the hopping elements and how they place this chiral gauge theory on a lattice.

But to answer your first question, you need to clarify yourself. This paper combines many advanced topics, so just asking for a general reference to understand it isn't answerable. What specifically are having trouble understanding in the paper? Also, if your specific difficulty involves standard model questions, you should move over to the high energy forum. If you have questions regarding the lattice model their using and how gauge theories can be described on it, then this is the right forum.

3. Oct 15, 2017

### shiraz

Dear DeathbyGreen
I am trying to understand how the mechanism of how mirror fermions acquire masses ( gapped) " the first column of page 4" especially when he start talking about solitons.
Thank you

4. Oct 15, 2017

### DeathbyGreen

So the system they are describing is a topological insulator. In an actual example, such as graphene under circularly polarized light (floquet topological insulator) or standard topologically insulating graphene just as a theoretical toy model, the Lagrangian can be described using a QED 2+1 Lagrangian. This is because in graphene you have not only fermion spin degrees of freedom, but also sublattice degrees of freedom (pseudospin). The "psuedospinors" can be described using left and right handed Weyl spinors, each of which would have spin up and down components (think 4 component gamma matrices used in the Dirac equation).

So in their system, they have Weyl fermions which are gapped in the bulk, and gapless on the edges. In order to reflect standard model SO(10) symmetries they introduce 16 Weyl spinors for 16 different introduced field as $\sum_{\alpha}^{16}\phi_{\alpha}$. They then want to break the symmetry of the left handed spinors ("gap them out") but keep the symmetry of the right handed ones. I think this is to reflect the types of symmetry you have in the standard model, but I can't speak much to this as a condensed matter guy. So what they want is to introduce an interaction which can break symmetries in a portion of the system while keeping an overall symmetry intact. This reminds me of how Haldane originally envisioned using spin orbit coupling to induce a topological phase without violating time reversal symmetry of the system (time reversal symmetry in topological insulators is key to having protected edge states).

So they begin considering interactions which satisfy this condition; as in a typical topological insulator, any magnetic impurity will violate time reversal symmetry and thus remove the protected states. They use a soliton as a description of a magnetic impurity, a type of "solitary" spin wave of magnetic moments. So these are types of impurities they worry about which would remove the symmetry they want to keep. They mention seeing these as gravitational anomalies, but I don't know what that means; clarification on that would best be posed in a different forum.

Last edited: Oct 15, 2017
5. Oct 17, 2017

### shiraz

Dear DeathbyGreen
Thank you for your help. I understand the main point which is gapping out the the left handed by intriducing an interaction between them and the scalar fields ( note : we have 10 scalar fields and not 16).why did u write "they introduce 16 Weyl spinors for 16 different introduced field scalar fields although we have only 10). and if you please can u explain for me more about pseudo spinnors?