| This article needs attention from an expert in physics. The specific problem is: Needs summarizing sections, too many equations and not all variables are defined. WikiProject Physics may be able to help recruit an expert. (October 2019) |
The Peierls substitution method, named after the original work by Rudolf Peierls[1] is a widely employed approximation for describing tightly-bound electrons in the presence of a slowly varying magnetic vector potential.[2]
In the presence of an external magnetic vector potential
, the translation operators, which form the kinetic part of the Hamiltonian in the tight-binding framework, are simply
![{\displaystyle \mathbf {T} _{x}=|m+1,n\rangle \langle m,n|e^{i\theta _{m,n}^{x}},\quad \mathbf {T} _{y}=|m,n+1\rangle \langle m,n|e^{i\theta _{m,n}^{y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ea9b2a70c4de29c0097b7c3c4fdb7f17aa4ad6)
and in the second quantization formulation
![{\displaystyle \mathbf {T} _{x}={\boldsymbol {\psi }}_{m+1,n}^{\dagger }{\boldsymbol {\psi }}_{m,n}e^{i\theta _{m,n}^{x}},\quad \mathbf {T} _{y}={\boldsymbol {\psi }}_{m,n+1}^{\dagger }{\boldsymbol {\psi }}_{m,n}e^{i\theta _{m,n}^{y}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8235faacc7d982ad6b1899c9ebadf656d734f3)
The phases are defined as
![{\displaystyle \theta _{m,n}^{x}={\frac {q}{\hbar }}\int _{m}^{m+1}A_{x}(x,n){\text{d}}x,\quad \theta _{m,n}^{y}={\frac {q}{\hbar }}\int _{n}^{n+1}A_{y}(m,y){\text{d}}y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/706106b3c83d618d704f4d4dc53979cc79806f5e)