Peierls substitution

The Peierls substitution method, named after the original work by Rudolf Peierls is a widely employed approximation for describing tightly-bound electrons in the presence of a slowly varying magnetic vector potential.

In the presence of an external magnetic vector potential ${\displaystyle \mathbf {A} }$, the translation operators, which form the kinetic part of the Hamiltonian in the tight-binding framework, are simply

$${\displaystyle \mathbf {T} _{x}=|m+1,n\rangle \langle m,n|e^{i\theta _{m,n}^{x}},\quad \mathbf {T} _{y}=|m,n+1\rangle \langle m,n|e^{i\theta _{m,n}^{y}}}$$

and in the second quantization formulation

$${\displaystyle \mathbf {T} _{x}={\boldsymbol {\psi }}_{m+1,n}^{\dagger }{\boldsymbol {\psi }}_{m,n}e^{i\theta _{m,n}^{x}},\quad \mathbf {T} _{y}={\boldsymbol {\psi }}_{m,n+1}^{\dagger }{\boldsymbol {\psi }}_{m,n}e^{i\theta _{m,n}^{y}}.}$$

The phases are defined as

$${\displaystyle \theta _{m,n}^{x}={\frac {q}{\hbar }}\int _{m}^{m+1}A_{x}(x,n){\text{d}}x,\quad \theta _{m,n}^{y}={\frac {q}{\hbar }}\int _{n}^{n+1}A_{y}(m,y){\text{d}}y.}$$