Peierls substitution

Properties of the Peierls substitution

1. The number of flux quanta per plaquette ϕ m n is related to the lattice curl of the phase factor,

and the total flux through the lattice is Φ = Φ 0 ∑ m , n ϕ m , n with Φ 0 = h c / e being the magnetic flux quantum in Gaussian units.

2. flux quanta per plaquette ϕ m n is related to the accumulated phase of a single particle state, | ψ ⟩ = ψ i , j | 0 ⟩ surrounding a plaquette:

Justification of Peierls substitution

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

The axiomatic approach

Here we give a simple derivation of the Peierls substitution, which is based on the Feynman's Lectures (Vol. III, Chapter 21) . This derivation postulate 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 | m + 1 ⟩ ⟨ m | 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 | m + a ⟩ ⟨ m | ⟶ | m + a , n ⟩ ⟨ m , n | . 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:

H 0 = 1 2 m ( p − e A ) 2 + ϵ 0 ~

where the effective mass is m = ℏ 2 / 2 t a 2 and ϵ ~ 0 = ϵ 0 − 4 .

The semi-classical approach

Here we show that the Peierls phase originates from the propagator of an electron in a magnetic field due to the dynamical term q v ⋅ A appearing in the Lagrangian. In the path integral formalism, which generalizes the action principle of classical mechanics, the transition amplitude from site j at time t j to site i at time t i is given by

where the integration operator, ∫ r ( t i ) r ( t j ) D [ r ( t ) ] denotes the sum over all possible paths from r ( t i ) to r ( t j ) and S [ r i j ] = ∫ t i t j L [ r ( t ) , r ˙ ( t ) , t ] d t is the classical action, which is a functional that takes a trajectory as its argument. We use r i j to denote a trajectory with endpoints at r ( t i ) , r ( t j ) . The Lagrangian of the system can be written as

where L ( 0 ) 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.

A rigorous derivation

The Hamiltonian is given by

where U ( r ) is the potential landscape due to the crystal lattice. The Bloch theorem asserts that the solution to the problem: H Ψ k = E ( k ) Ψ k , is to be sought in the Bloch sum form

where N is the number of unit cells, and ϕ are known as Wannier states. The corresponding eigenvalues E ( k ) , which form bands depending on the crystal momentum k , are obtained by calculating the matrix element

and ultimately depend on material-related hopping integrals

In the presence of the magnetic field the Hamiltonian changes to

where q is the charge of the particle. To amend this, consider changing the Wannier states to

where | ϕ R ⟩ ≡ | ϕ ~ R ( A → 0 ) ⟩ . This makes the new Bloch wave functions

into eigenstates of the full Hamiltonian at time t , with the same energy as before. To see this we first use p = − i ℏ ∇ to write

Then when we compute the hopping integral in quasi-equilibrium (assuming that the vector potential changes slowly)

where we have defined Φ R , r , R ′ = ∫ R → r → R ′ → R A ( r ′ , t ) ⋅ d r ′ , the flux through the triangle made by the three position arguments. Since we assume A ( r , t ) is approximately uniform at the lattice scale - the scale at which the Wannier states are localized to the positions R - we can approximate Φ R , r , R ′ ≈ 0 , yielding the desired result,

t i j t i j e i e ℏ ∫ i j A ⋅ d l