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 Here we give three derivations of the Peierls substitution, each one is based on a different formulation of quantum mechanics theory.
Here we give a simple derivation of the Peierls substitution, which is based on The Feynman Lectures (Vol.
[3] This derivation postulates that magnetic fields are incorporated in the tight-binding model by adding a phase to the hopping terms and show that it is consistent with the continuum Hamiltonian.
Thus, our starting point is the Hofstadter Hamiltonian:[2] The translation operator
can be written explicitly using its generator, that is the momentum operator.
Under this representation its easy to expand it up to the second order, and in a 2D lattice
Next, we expand up to the second order the phase factors, assuming that the vector potential does not vary significantly over one lattice spacing (which is taken to be small) Substituting these expansions to relevant part of the Hamiltonian yields Generalizing the last result to the 2D case, the we arrive to Hofstadter Hamiltonian at the continuum limit: where the effective mass is
Here we show that the Peierls phase factor originates from the propagator of an electron in a magnetic field due to the dynamical term
In the path integral formalism, which generalizes the action principle of classical mechanics, the transition amplitude from site
is the classical action, which is a functional that takes a trajectory as its argument.
to denote a trajectory with endpoints at
is the Lagrangian in the absence of a magnetic field.
The corresponding action reads Now, assuming that only one path contributes strongly, we have Hence, the transition amplitude of an electron subject to a magnetic field is the one in the absence of a magnetic field times a phase.
is the potential landscape due to the crystal lattice.
The Bloch theorem asserts that the solution to the problem:
, is to be sought in the Bloch sum form where
is the number of unit cells, and the
, which form bands depending on the crystal momentum
, are obtained by calculating the matrix element and ultimately depend on material-dependent hopping integrals In the presence of the magnetic field the Hamiltonian changes to where
To amend this, consider changing the Wannier functions to where
This makes the new Bloch wave functions into eigenstates of the full Hamiltonian at time
to write Then when we compute the hopping integral in quasi-equilibrium (assuming that the vector potential changes slowly) where we have defined
, the flux through the triangle made by the three position arguments.
is approximately uniform at the lattice scale[4] - the scale at which the Wannier states are localized to the positions
, yielding the desired result,
Therefore, the matrix elements are the same as in the case without magnetic field, apart from the phase factor picked up, which is denoted the Peierls phase factor.
This is tremendously convenient, since then we get to use the same material parameters regardless of the magnetic field value, and the corresponding phase is computationally trivial to take into account.
) it amounts to replacing the hopping term