Vector potential with an angle to the periodic direction

One of the most difficult aspects in Tight Binding models is the incorporation of the magnetic field. And that is because a lot of the things that exist in simple analytical expressions in quantum mechanics, change when we are talking about a Tight Binding model, and especially one derived from First Principles using Density Functional Theory as that discussed here.

One of the problems that could emerge is that there exists an angle \theta between the periodic direction and the direction the magnetic field is applied. This differs from the case of the Peierls phase, defined as,

\phi_{ij}^{lg} = \left(e/\hbar\right)\int_{\mathbf{r_j}}^{\mathbf{r_i}}\mathbf{A} \cdot d\mathbf{r}

where r_i and r_j are sites in the TB model.

In that case, there is an elegant solution, which I have found in

Graphene-Based Heterojunction between Two Topological Insulators Oleksii Shevtsov, Pierre Carmier, Cyril Petitjean, Christoph Groth, David Carpentier, and Xavier Waintal Phys. Rev. X 2, 031004

where the phase is changed so that it supports an angle to the periodic direction. The phase becomes,

\phi = \phi_{ij}^{lg} + \Phi_i - \Phi_j

where \Phi_i = B\left(1-\cos{2\theta}\right)\frac{x_iy_i}{2} + B\sin{2\theta}\frac{x_i^2-y_i^2}{4}

The relevant vector potential is


To check its correctness,I plotted this vector potential in Python, with a 20 degrees angle with the x axis:

Vector potential translationally invariant at an angle of 20 degrees from the x axis.

When plotting the vector potential, notice that the arrows point in the direction of translational invariance of the TB model.

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