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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
and in the second quantization formulation
The phases are defined as