Peierls substitution explained

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]

A , the translation operators, which form the kinetic part of the Hamiltonian in the tight-binding framework, are simply

$$\backslash mathbf\_x\; =\; |m+1,n\backslash rangle\backslash langle\; m,n|e^,\; \backslash quad\; \backslash mathbf\_y\; =\; |m,n+1\backslash rangle\backslash langle\; m,n|e^$$ and in the second quantization formulation

$$\backslash mathbf\_x\; =\; \backslash boldsymbol^\backslash dagger\_\backslash boldsymbol\_e^,\; \backslash quad\; \backslash mathbf\_y\; =\; \backslash boldsymbol^\backslash dagger\_\backslash boldsymbol\_e^.$$The phases are defined as

$$\backslash theta^x\_\; =\; \backslash frac\backslash int\_m^\; A\_x(x,n)\backslash textx,\; \backslash quad\; \backslash theta^y\_\; =\; \backslash frac\backslash int\_n^\; A\_y(m,y)\; \backslash texty.$$

Properties

1. The number of flux quanta per plaquette

\phi _{mn} is related to the lattice curl of the phase factor, and the total flux through the lattice is $\backslash Phi\; =\; \backslash Phi\_0\backslash sum\_\backslash phi\_$ with \Phi _{0} = hc/e being the magnetic flux quantum in Gaussian units.

1. The flux quanta per plaquette

\phi _{mn} is related to the accumulated phase of a single particle state, |\psi\rangle = \boldsymbol{\psi} _{i,j} | 0 \rangle 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).^[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:

$$H\_0\; =\; \backslash sum\_\backslash bigg(-te^\backslash vert\; m\backslash !+\backslash !a,n\; \backslash rangle\; \backslash langle\; m,n\backslash vert\; -te^\backslash vert\; m,n\backslash !+\backslash !a\backslash rangle\backslash langle\; m,n\backslash vert\; -\backslash epsilon\_0\backslash vert\; m,n\backslash rangle\backslash langle\; m,n\backslash vert\backslash bigg)+\; \backslash text.$$The translation operator

\vert m +1 \rangle\langle m\vert 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,

$$\backslash vert\; m\backslash !+\backslash !a\backslash rangle\backslash langle\; m\backslash vert\; =\; \backslash exp\backslash vert\; m\backslash rangle\backslash langle\; m\backslash vert\; =\; \backslash left(1-\backslash fraca\; -\backslash fraca^2+\backslash mathcal(a^3)\; \backslash right)\backslash vert\; m\backslash rangle\backslash langle\; m\backslash vert$$and in a 2D lattice

\vert m + a\rangle\langle m\vert \longrightarrow\vert m + a,n\rangle\langle m,n\vert . 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 - q A ) ^{2+ \tilde{\epsilon} _{0 }} where the effective mass is m = \hbar ^{2/2ta} ^{2} and \tilde{\epsilon} _{0} = \epsilon _{0-4t} .

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

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

$$\backslash langle\backslash mathbf\_i,t\_i|\backslash mathbf\_j,t\_j\backslash rangle\; =\; \backslash int\_^\; \backslash mathcal[\backslash mathbf\{r\}(t)]e^,$$where the integration operator,

\int ^{r (t _{j)}} _{r (t _{i)}} l{D}[ r (t)] denotes the sum over all possible paths from r (t _{i)} to r (t _{j)} and l{S}[ r _{ij} ] = \int ^{t _{j}} _{t _{i}} 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 _{ij} to denote a trajectory with endpoints at r(t _{i),} r(t _{j)} . The Lagrangian of the system can be written as

$$L\; =\; L^+q\backslash mathbf\backslash cdot\backslash mathbf,$$where

L ^{(0)} is the Lagrangian in the absence of a magnetic field. The corresponding action reads

$$S[\backslash mathbf\{r\}\_\{ij\}]\; =\; S^[\backslash mathbf\{r\}\_\{ij\}]+q\backslash int\_^dt\backslash left(\backslash frac\backslash right)\backslash cdot\backslash mathbf\; =\; S^[\backslash mathbf\{r\}\_\{ij\}]+q\backslash int\_\backslash mathbf\backslash cdot\backslash text\backslash mathbf$$Now, assuming that only one path contributes strongly, we have

$$\backslash langle\backslash mathbf\_i,t\_i|\backslash mathbf\_j,t\_j\backslash rangle\; =\; e^\backslash int\_^\; \backslash mathcal[\backslash mathbf\{r\}(t)]e^$$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

$$H\; =\; \backslash frac+U\backslash left(\backslash mathbf\backslash right),$$ where

U\left( r \right) is the potential landscape due to the crystal lattice. The Bloch theorem asserts that the solution to the problem:H\Psi _{k} ( r ) = E\left( k \right)\Psi _{k} ( r ) , is to be sought in the Bloch sum form

$$\backslash Psi\_(\backslash mathbf)\; =\; \backslash frac\backslash sum\_e^\backslash phi\_\backslash mathbf\backslash left(\backslash mathbf\backslash right),$$

where

N is the number of unit cells, and the \phi _{R} are known as Wannier functions. The corresponding eigenvalues E\left( k \right) , which form bands depending on the crystal momentum k , are obtained by calculating the matrix element

$$E\backslash left(\backslash mathbf\backslash right)\; =\; \backslash int\; d\backslash mathbf\backslash \; \backslash Psi\_\backslash mathbf^*(\backslash mathbf)H\backslash Psi\_\backslash mathbf(\backslash mathbf)\; =\; \backslash frac\backslash sum\_e^\backslash int\; d\backslash mathbf\backslash \; \backslash phi^*\_\backslash mathbf\backslash left(\backslash mathbf\backslash right)H\backslash phi\_\backslash left(\backslash mathbf\backslash right)$$

and ultimately depend on material-dependent hopping integrals

$$t\_\; =\; -\backslash intd\backslash mathbf\backslash \; \backslash phi^*\_\backslash left(\backslash mathbf\backslash right)H\backslash phi\_\backslash left(\backslash mathbf\backslash right).$$

In the presence of the magnetic field the Hamiltonian changes to

$$\backslash tilde(t)\; =\; \backslash frac+U\backslash left(\backslash mathbf\backslash right),$$where

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

$$\backslash begin\backslash tilde\_\backslash mathbf(\backslash mathbf)\; =\; e^\; \backslash phi\_\backslash mathbf(\backslash mathbf),\backslash end$$

where

\phi _{R} \equiv \tilde{\phi} _{R(A\to} 0 ) . This makes the new Bloch wave functions

$$\backslash tilde\_\backslash mathbf(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_\; e^\; \backslash tilde\_\backslash mathbf(\backslash mathbf),$$

into eigenstates of the full Hamiltonian at time

t , with the same energy as before. To see this we first use p = - i \hbar \nabla to write

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

where we have defined

\Phi _{R',r,R} = \oint _{R'\to} 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 \Phi _{R,r,R'} ≈ 0 , yielding the desired result,$$\backslash tilde\_(t)\; \backslash approx\; t\_\; e^.$$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 (q = - e ) it amounts to replacing the hopping term t _{ij} with t _{ij} e ^{- i (e)/(\int ^{j A ⋅} _{i} d l \hbar)} ^{[4] [5] [6] [7]}

Notes and References

1. Peierls, R . 1933 . On the theory of diamagnetism of conduction electrons . Z. Phys. . 80 . 11–12 . 763–791 . 10.1007/bf01342591 . 1933ZPhy...80..763P. 119930820 .
2. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields . Hofstadter, Douglas R. . Sep 1976 . Phys. Rev. B . 14 . 6 . 2239–2249 . 10.1103/PhysRevB.14.2239. 1976PhRvB..14.2239H .
3. https://feynmanlectures.caltech.edu/III_21.html The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity
4. Luttinger, J. M. . Nov 1951 . The Effect of a Magnetic Field on Electrons in a Periodic Potential . Phys. Rev. . 84 . 4 . 814–817 . 10.1103/PhysRev.84.814. 1951PhRv...84..814L .
5. Kohn, Walter . Sep 1959 . Theory of Bloch Electrons in a Magnetic Field: The Effective Hamiltonian . Phys. Rev. . 115 . 6 . 1460–1478 . 10.1103/PhysRev.115.1460. 1959PhRv..115.1460K .
6. Blount, E. I. . Jun 1962 . Bloch Electrons in a Magnetic Field . Phys. Rev. . 126 . 5 . 1636–1653 . 10.1103/PhysRev.126.1636. 1962PhRv..126.1636B .
7. Wannier, Gregory H. . Oct 1962 . Dynamics of Band Electrons in Electric and Magnetic Fields . Rev. Mod. Phys. . 34 . 4 . 645–655 . 10.1103/RevModPhys.34.645. 1962RvMP...34..645W .