# Lattice hopping at boundary in graphene lattice with magnetic field

Let's say I have a tight binding model for graphene, where I have a two-atom basis and three nearest neighbor vectors. I've applied a homogenous magnetic field $$B$$ in the z-axis, and can take the vector potential $$\vec{A} = (0, Bx, 0)$$.

I'm familiar with the Peierl's substitution, where I apply $$t\rightarrow t \exp{\int_{{\delta_i}}\vec{A}\cdot d\vec{r}}$$

where $$\delta_i$$ is the NN vector in question. My issue is, if I'm applying periodic boundary conditions, what should I do at the boundary? I've looked into the magnetic translation operators, but I've made limited progress and have been unable to understand how to do this at the boundary in a way that doesn't lead to strange discontinuities. As I understand, I want to set up my model such that moving across a boundary adds a phase of $$2\pi n i$$ ($$n$$ is some integer), but do I do this by adjusting the gauge or the size of the lattice?

• Perhaps you want to specify your problem a bit more: are you thinking of a homogeneous magnetic field perpendicular to the graphene plane? Also, what is $n$ in your $2\pi n i$ expression? – flaudemus Mar 13 '19 at 13:27
• With periodic boundary conditions, the net magnetic flux through the system has to be an integer multiple of $2\pi$. If you impose that, you should be able to find a gauge without discontinuities. – Dominic Else Mar 13 '19 at 15:21
• I've edited the original comment (I was interested in a homogeneous magnetic field). @DominicElse Can you point me to a resource on how one goes from a total magnetic flux that is an integer multiple of $2\pi$ to finding the appropriate gauge? – user147177 Mar 13 '19 at 15:23