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

Properties

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  1. 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.
  2. The flux quanta per plaquette   is related to the accumulated phase of a single particle state,   surrounding a plaquette:
 

Justification

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Here we give three derivations of the Peierls substitution, each one is based on a different formulation of quantum mechanics theory.

Axiomatic approach

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Here we give a simple derivation of the Peierls substitution, which is based on The Feynman Lectures (Vol. III, Chapter 21).[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   and  .

Semi-classical approach

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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, 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

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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 functions. 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[4] - 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  [4][5][6][7]

References

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  1. ^ Peierls, R (1933). "On the theory of diamagnetism of conduction electrons". Z. Phys. 80 (11–12): 763–791. Bibcode:1933ZPhy...80..763P. doi:10.1007/bf01342591. S2CID 119930820.
  2. ^ a b Hofstadter, Douglas R. (Sep 1976). "Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields". Phys. Rev. B. 14 (6): 2239–2249. Bibcode:1976PhRvB..14.2239H. doi:10.1103/PhysRevB.14.2239.
  3. ^ The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity
  4. ^ a b Luttinger, J. M. (Nov 1951). "The Effect of a Magnetic Field on Electrons in a Periodic Potential". Phys. Rev. 84 (4): 814–817. Bibcode:1951PhRv...84..814L. doi:10.1103/PhysRev.84.814.
  5. ^ Kohn, Walter (Sep 1959). "Theory of Bloch Electrons in a Magnetic Field: The Effective Hamiltonian". Phys. Rev. 115 (6): 1460–1478. Bibcode:1959PhRv..115.1460K. doi:10.1103/PhysRev.115.1460.
  6. ^ Blount, E. I. (Jun 1962). "Bloch Electrons in a Magnetic Field". Phys. Rev. 126 (5): 1636–1653. Bibcode:1962PhRv..126.1636B. doi:10.1103/PhysRev.126.1636.
  7. ^ Wannier, Gregory H. (Oct 1962). "Dynamics of Band Electrons in Electric and Magnetic Fields". Rev. Mod. Phys. 34 (4): 645–655. Bibcode:1962RvMP...34..645W. doi:10.1103/RevModPhys.34.645.