The Peierls substitution method, named after the original work by
Rudolf Peierls
Sir Rudolf Ernst Peierls, (; ; 5 June 1907 – 19 September 1995) was a German-born British physicist who played a major role in Tube Alloys, Britain's nuclear weapon programme, as well as the subsequent Manhattan Project, the combined Allied ...
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
In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic v ...
, the translation operators, which form the kinetic part of the Hamiltonian in the
tight-binding
In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each ...
framework, are simply
:
and in the
second quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
formulation
:
The phases are defined as
:
Properties
#The number of flux quanta per plaquette
is related to the lattice curl of the phase factor,
and the total flux through the lattice is
with
being the magnetic flux quantum in
Gaussian units
Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
.
# The flux quanta per plaquette
is related to the accumulated phase of a single particle state,
surrounding a plaquette:
:
Justification
Here we give three derivations of the Peierls substitution, each one is based on a different formulation of quantum mechanics theory.
Axiomatic approach
Here we give a simple derivation of the Peierls substitution, which is based on The Feynman Lectures (Vol. III, Chapter 21).
[The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity](_blank)
/ref> 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:
:
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
In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model (physics), lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approxi ...
:
:
where the effective mass is and .
Semi-classical approach
Here we show that the Peierls phase factor originates from the propagator of an electron in a magnetic field due to the dynamical term appearing in the Lagrangian. In the path integral formalism, which generalizes the action principle of classical mechanics, the transition amplitude from site at time to site at time is given by
:
where the integration operator, denotes the sum over all possible paths from to and is the classical action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
, which is a functional that takes a trajectory as its argument. We use to denote a trajectory with endpoints at . The Lagrangian of the system can be written as
:
where 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.
Another derivation
The Hamiltonian is given by
:
where 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 are known as Wannier function
The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. Wannier functions are the localized molecular orbitals of crystalline systems.
The Wannier functions fo ...
s. The corresponding eigenvalues , 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 is the charge of the particle. 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 , with the same energy as before. To see this we first use 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. Since we assume is approximately uniform at the lattice scale - the scale at which the Wannier states are localized to the positions - we can approximate , 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. For electrons () it amounts to replacing the hopping term with [
][
]
References
{{Reflist
Electronic structure methods
Electronic band structures