Peierls substitution

In condensed matter, we need to solve the Shroedinger equation in the presence of a periodic potential. One may start with a continuous model

\[H=-\frac{\nabla^{2}}{2m}+V(\mathbf{r}) \label{eq:Seq}\]

with a periodic potential \(V(\mathbf{r}+\mathbf{a}_{i})=V(\mathbf{r})\). The periodic potential defines a lattice \(\mathbf{R}\in\{\sum_{i}n_{i}\mathbf{a}_{i}\}\). The famous Bloch theorem tells us that a general solution

\[H\psi_{\mathbf{k}}(\mathbf{r})=E(\mathbf{k})\psi_{\mathbf{k}}(\mathbf{r})\]

to the Schroedinger equation in Eq. \((\ref{eq:Seq})\) can be written in the form as

\[\begin{align} \psi_{\mathbf{k}}(\mathbf{r}) & =e^{i\mathbf{k}\cdot\mathbf{r}}u_{\mathbf{k}}(\mathbf{r}), \end{align}\]

where \(N\) is the number of unit cells and \(\mathbf{k}\) is the crystal momentum related to the lattice. To consider the minimal coupling to the gauge field, it is better to separate the crystal momentum dependence from \(u_{\mathbf{k}}(\mathbf{r})\).It can be achieved by by the Wannier function \(W_{\mathbf{R}}(\mathbf{r})\) that appears as the Fourier transformation of the periodic part of the Bloch wave,

\[\psi_{\mathbf{k}}(\mathbf{r})=\frac{1}{\sqrt{N}}\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{R}}W_{\mathbf{R}}(\mathbf{r}).\label{eq:Wann}\]

Then we can calculate the eigenvalue \(E(\mathbf{k})\) , by calculating the matrix element

\[\begin{align} E(\mathbf{k}) & =\int d\mathbf{r}\psi_{\mathbf{k}}^{*}(\mathbf{r})H\psi_{\mathbf{k}}(\mathbf{r})=\int d\mathbf{r}\frac{1}{N}\sum_{\mathbf{R}^{\prime}\mathbf{R}}e^{i\mathbf{k}\cdot(\mathbf{R}^{\prime}-\mathbf{R})}W_{\mathbf{R}}^{*}(\mathbf{r})HW_{\mathbf{R}^{\prime}}(\mathbf{r})\\ & =\frac{1}{N}\sum_{\mathbf{R}^{\prime}\mathbf{R}}e^{i\mathbf{k}\cdot(\mathbf{R}^{\prime}-\mathbf{R})}t(\mathbf{R}^{\prime}-\mathbf{R}) \end{align}\]

with \(t(\mathbf{R}^{\prime}-\mathbf{R})\) is the hopping integral

\[\begin{align} t(\mathbf{R}^{\prime}-\mathbf{R}) & =\int d\mathbf{r}W_{\mathbf{R}}^{*}(\mathbf{r})HW_{\mathbf{R}^{\prime}}(\mathbf{r}). \end{align}\]

Therefore, we can have a tight-binding model corresponding to the continuous model in Eq.~\((\ref{eq:Seq})\)

\[\begin{align} H_{\mathrm{tb}} & =\frac{1}{N}\sum_{\mathbf{R}^{\prime}\mathbf{R}}e^{i\mathbf{k}\cdot(\mathbf{R}^{\prime}-\mathbf{R})}t(\mathbf{R}^{\prime}-\mathbf{R})c_{\mathbf{R}}^{\dagger}c_{\mathbf{R}^{\prime}}\\ & =\sum_{\mathbf{k}}E(\mathbf{k})c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}}. \end{align}\]

One of the most striking features of Eq.\(~(\ref{eq:Wann})\) is the independence of Wannier function on the crystal momentum. With the preparation above, we can proceed to consider a minimal coupling to a gauge field, with the Hamiltonian

\[H(\mathbf{A})=\frac{(-i\mathbf{\nabla-\mathbf{A}}(\mathbf{r},t))^{2}}{2m}+V(\mathbf{r}).\]

Our aim is to obtain the guiding principles directing minimal coupling in the tight-binding model to the gauge field. That is, we want to express physical quantities in terms of the crystal momentum. We can consider the modified Wannier functions by the gauging principle for a continuous model

\[W_{\mathbf{R}}(\mathbf{r},\mathbf{A})=\exp\left(i\int_{\mathbf{R}}^{\mathbf{r}}\mathbf{A}(\mathbf{r}^{\prime},t)\right)W_{\mathbf{R}}(\mathbf{r}).\]

Obviously, \(W_{\mathbf{R}}(\mathbf{r})=W_{\mathbf{R}}(\mathbf{r},\mathbf{A}\rightarrow0)\). We then have a new solution to \(H(\mathbf{A})\)

\[\psi_{\mathbf{k}}(\mathbf{r},\mathbf{A})=\frac{1}{\sqrt{N}}\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{R}}W_{\mathbf{R}}(\mathbf{r},\mathbf{A})\]

We can check this by

\[\begin{align} & H(\mathbf{A})\psi_{\mathbf{k}}(\mathbf{r},\mathbf{A}) \\ =& \left[\frac{(-i\mathbf{\nabla-\mathbf{A}}(\mathbf{r},t))^{2}}{2m}+V(\mathbf{r})\right]\frac{1}{\sqrt{N}}\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{R}}W_{\mathbf{R}}(\mathbf{r},\mathbf{A})\\ =& \frac{1}{\sqrt{N}}\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{R}}\left[\frac{(-i\mathbf{\nabla-\mathbf{A}}(\mathbf{r},t))^{2}}{2m}+V(\mathbf{r})\right]\left[\exp\left(i\int_{\mathbf{R}}^{\mathbf{r}}\mathbf{A}(\mathbf{r}^{\prime},t)\right)W_{\mathbf{R}}(\mathbf{r})\right]\\ =& \frac{1}{\sqrt{N}}\sum_{\mathbf{R}}\exp\left(i\int_{\mathbf{R}}^{\mathbf{r}}\mathbf{A}(\mathbf{r}^{\prime},t)\right)e^{i\mathbf{k}\cdot\mathbf{R}}\left[\frac{(-i\mathbf{\nabla-\mathbf{A}}(\mathbf{r},t)+\mathbf{A}(\mathbf{r},t))^{2}}{2m}+V(\mathbf{r})\right]W_{\mathbf{R}}(\mathbf{r})\\ =& \frac{1}{\sqrt{N}}\sum_{\mathbf{R}}\exp\left(i\int_{\mathbf{R}}^{\mathbf{r}}\mathbf{A}(\mathbf{r}^{\prime},t)\right)e^{i\mathbf{k}\cdot\mathbf{R}}\left[\frac{(-i\nabla)^{2}}{2m}+V(\mathbf{r})\right]W_{\mathbf{R}}(\mathbf{r}) \end{align}\]

When calculating the hopping integral, we have

\[\begin{align} t_{\mathbf R\mathbf R^{\prime}}(\mathbf{A}) & =\int d\mathbf{r}W_{\mathbf{R}}^{*}(\mathbf{r},\mathbf{A})HW_{\mathbf{R}^{\prime}}(\mathbf{r},\mathbf{A})\\ & =\exp\left[i\int_{\mathbf{R}^{\prime}}^{\mathbf{R}}\mathbf{A}\cdot d\mathbf{r}^{\prime}\right]\int d\mathbf{r}e^{i\Phi_{R^{\prime}rR}}HW_{\mathbf{R}^{\prime}}(\mathbf{r}) \end{align}\]

where \(\Phi_{R^{\prime}rR}=\oint_{R^{\prime}\rightarrow r\rightarrow R\rightarrow R^{\prime}}\mathbf{A}(\mathbf{r}^{\prime})\cdot d\mathbf{r}^{\prime}\), is the flux through the triangular made by three position arguments.Since the \(\mathbf{A}\) is approximately uniform at the lattice scale,— the scale at which the Wannier states are localized to the positions \(\mathbf{R}\), we can than approximate \(\Phi_{R^{\prime}rR}=0\), yielding the desired result.

\[t_{\mathbf R\mathbf R^{\prime}}(\mathbf{A})=\exp\left[i\int_{\mathbf{R}^{\prime}}^{\mathbf{R}}\mathbf{A}\cdot d\mathbf{r}^{\prime}\right]t(\mathbf{R}^{\prime}-\mathbf{R})\]

Therefore, the matrix elements are the same as in the case without magnetic field, apart from the extra phase factor picked up, which is denoted by the Peierls phase factor.

One can generalize the Peierls substitution in a multi-band system.