Appendix H. Wannier functions of minimal width
https://doi.org/10.48550/arXiv.2403.13981
H.1 Introduction
Wannier function of minimal width is simply another term for maximally localized Wannier function [Marzari and Vanderbilt, 1997]. The title of this section acknowledges the work of Ferreira and Parada, on which it is based [Ferreira and Parada, 1970]. My presentation differs from theirs in several ways, the most deliberate of which is my avoidance of quantum mechanical perturbation theory. This is to demonstrate that there is nothing specific to quantum mechanics in the theory of Wannier functions and their relationships with Bloch functions.All of this section would apply to the eigenfunctions,
\begin{align*}
b_{\alpha k}(x)\equiv \braket{x}{b_{\alpha k}} \equiv b_{\alpha}(k,x),
\end{align*}
of any bounded and self-adjoint $\volume$-periodic operator, \begin{align*}
\hamsmallx:\lebesgue(\onetorus)&\to\lebesgue(\onetorus),
\\
\hamsmallx(x)\equiv \expvaltwo{\hamsmall}{x}&=\hamsmallx(x+m\volume), \;\forall m\in\integer,
\end{align*}
where ${\ket{b_{\alpha k}}=\ket{b_\alpha(k)}}$ is an eigenstate of \begin{align*}
\hamsmall = \int_\onetorus \dd{x} \hamsmallx(x) \dyad{x},
\end{align*}
and the circumference of $\onetorus$ is ${\bulksize=\Nunitcell\volume}$, where ${\Nunitcell\in\integerpos}$.
For example, consider a classically-modelled process in a crystal whose bulk is represented in $\onetorus$. If ${\pdf(x)}$ is a a one-particle position probability density function that has the crystal’s $\volume$-periodicity, it could be specified by a smooth function ${\psi(x)=\sqrt{\pdf(x)}e^{i\theta(x)}}$; and $\psi$ could be expanded in either a Bloch basis or a Wannier basis. Either basis could be used to build a basis of many-particle states to represent a function $\Psi$ whose square modulus is a classical many-particle position probability density function.
H.2 Theoretical setup
In this appendix it will be assumed that ${\{b_{\alpha k}\}}$ is an orthonormal set of single-particle Bloch functions,\begin{align*}
b_{\alpha k}(x)\equiv b_\alpha(k,x) \equiv e^{ikx}u_{\alpha k}(x)\equiv e^{ikx}u_\alpha(k,x),
\end{align*}
which is a complete basis of ${\lebesgue(\onetorus)}$. The subscripts $\alpha$ and $k$ of the set ${\{b_{\alpha k}\}}$ indicate that its elements include those with all band indices $\alpha$, and all wavevectors $k$ that are elements of ${\BZ}$ (see Appendix B). Orthonormality of the set of Bloch functions can be expressed as \begin{align*}
\braketT{b_{\alpha k}}{b_{\beta k'}}\equiv\int_\onetorus\dd{x} b_{\alpha k}^*(x) b_{\beta k'}\dd{x}=\delta_{\alpha\beta}\delta_{kk'}.
\end{align*}
As discussed in Appendix G, at each point ${x\in\onetorus}$, the set ${\{b_{\alpha k}(x)\}_k}$ of all Bloch functions whose band indices are $\alpha$ can be regarded as a function ${b_\alpha(k,x)}$ of $k$, which is ${\hreciplatt}$-periodic, meaning that
\begin{align*}
b_\alpha(g+G,x)=b_\alpha(g,x),
\end{align*}
for any ${g\in\reciplattg}$ and any reciprocal lattice vector ${G\in\reciplatt}$; and it will be assumed that the periodic Bloch functions ${u_{\alpha k}(x)=u_\alpha(k,x)}$ are real valued for all ${k\in\BZ}$. These assumptions are discussed and justified in Appendix G, and the ${\hreciplatt}$-periodicity of ${b_\alpha(g)}$ is shown to imply that \begin{align*}
u_\alpha(g+G,x)=e^{-iGx}u_{\alpha}(g,x),
\end{align*}
for any ${g\in\reciplattg}$ and any reciprocal lattice vector ${G\in\reciplatt}$.
This appendix considers a general Wannier transformation of the set ${\{b_{\alpha k}\}_k\subset\{b_{\alpha k}\}_{\alpha k}}$ to a Wannier function ${w_\alpha(x)}$. The general Wannier transformation will be used to deduce the specific Wannier transformation that minimizes the spread of ${w_\alpha(x)}$.
We have assumed that the set of Bloch functions is orthonormal and complete. Therefore there exist an infinite number of Hermitian operators ${\expvaltwo{\hamsmall}{x}}$, where
\begin{align*}
\hamsmall\equiv\sum_{\alpha k}\epsilon_{\alpha k}\dyad{b_{\alpha k}},
\end{align*}
of which they are eigenfunctions. These operators differ only by their sets of eigenvalues, ${\{\epsilon_{\alpha k}\}_{\alpha k}}$.
One of them will be denoted by $\hamsmallx$, and it will be assumed that ${\hamsmallx}$ varies smoothly with $x$. This facilitates the assumption that all required partial derivatives of each Bloch function with respect to $k$ and/or $x$ exist, where, as in Sec. 7.5.1,
\begin{align*}
\partial_k b_\alpha(k,x) \equiv \lim_{\bulksize\to\infty} \hbulksize^{-1}
\left[b_\alpha\left(k+\hbulksize,x\right)-b_\alpha\left(k,x\right)\right].
\end{align*}
H.3 The most localizing Wannier transformation
Consider the general Wannier transformation,\begin{align*}
w_\alpha(x) = \intbz f_\alpha(k) b_\alpha(k,x)\dd{k},
\tag{148}
\end{align*}
where, as in Sec. 7.5.1, ${\intbz\dd{k}}$ denotes the sum ${\hbulksize\sum_{k\in\BZ}}$; and ${f_\alpha:\reciplattg\to\complex}$ is ${\hreciplatt}$-periodic and normalized on $\BZ$. That is, for any ${g\in\reciplattg}$ and any ${G\in\reciplatt}$, ${f_\alpha(g+G) = f_\alpha(g)}$ and \begin{align*}
\intbz f^*_\alpha(k) f_\alpha(k)\dd{k} &=
\intbz f^*_\alpha(g+k) f_\alpha(g+k)\dd{k} = 1.
\end{align*}
As we are interested in finding the most localized Wannier function, let us assume that $f_\alpha(g)$ is a smooth function of $g$, and that ${b_\alpha(g,x)}$ is a smooth function of both $g$ and $x$.
We will be assuming that we are in the limit ${\bulksize\to\infty}$, which means that the set of points in $\BZ$ is quasicontinuous. It also means that we can assume that the width of $w_\alpha$ is much smaller than $\bulksize$. Therefore when we calculate the spread,
\begin{align*}
W_\alpha(X) = \int_\onetorus \abs{w_\alpha(x)}^2(x-X)^2\dd{x},
\end{align*}
of $w_\alpha$ about an arbitrary point $X\in\onetorus$, or when we calculate any integral of a localized function of position in $\onetorus$, $\int_\onetorus$ should always be taken to mean ${\int_{x_0}^{x_0+\bulksize}}$, where $x_0$ is chosen such that the integrand is negligible at the point ${x_0=x_0+\bulksize\in\onetorus}$.
Let us define a generating function ${G_\alpha(s,X)}$ from which $W_\alpha(X)$ can be calculated, as follows:
\begin{align*}
G_\alpha(s,X) &\equiv \int_\onetorus \abs{w_\alpha(x)}^2 e^{-is(x-X)} \dd{x}
\tag{149}
\\
\implies
W_\alpha(X)
&= \lim_{s\to 0}\int_\onetorus \abs{w_\alpha(x)}^2 \left(x-X\right)^2 e^{-is\left(x-X\right)}\dd{x}
\\
&= -\lim_{s\to 0} \partial_s^2 G_\alpha(s,X)
\tag{150}
\end{align*}
${G_\alpha(s,X)}$ is a Fourier transform of ${\abs{w_\alpha(x)}^2}$ after it has been displaced by ${-X}$. Therefore, if $X$ was the center of $w_\alpha$, ${G_\alpha(s,X)}$ would be the Fourier transform of $w_\alpha$ after its center had been moved to the origin.
Inserting Eq. (148) into Eq. (149) gives
\begin{align*}
G_\alpha(s,X)
=
&
\intbz\dd{k'}
\intbz\dd{k}
\fca(k')
f_\alpha(k+s)
e^{isX}
\\
&\times
\int_\onetorus\dd{x}
\bca(k',x)
b_\alpha(k+s,x)
e^{-isx},
\tag{151}
\end{align*}
where the ${\hreciplatt}$-periodicities of $b_\alpha$ and $f_\alpha$ have been used to shift the domain of the integration over $k$ from $\BZ$ to $\BZ+s$.
The set ${\left\{u_{\alpha k}\right\}_{\alpha}}$ of all periodic Bloch functions at $k$ is orthonormal because we have chosen the set of all Bloch functions to be orthonormal, i.e.,
\begin{align*}
\braket{b_{\alpha k}}{b_{\beta k'}}
&=
\int_\onetorus\dd{x} b_{\alpha k}^*(x) b_{\beta k'}(x)=
\delta_{\alpha\beta}\delta_{kk'}
\\
\implies
\braket{b_{\alpha k}}{b_{\beta k}}&=
\braket{u_{\alpha k}}{u_{\beta k}}=\delta_{\alpha\beta}.
\end{align*}
Furthermore, the set ${\{u_{\alpha k}\}_\alpha}$ of periodic Bloch functions at $k$ is a complete basis of ${\lebesgue(\onetorus)}$ because it is the set of eigenfunctions of an operator ${e^{-ikx}\hamsmallx e^{ikx}}$, which is Hermitian because $\hamsmallx$ is Hermitian.
The completeness of ${\{u_{\alpha k}\}_\alpha}$ allows us to express ${u_\alpha(k+s,x)}$ as
\begin{align*}
u_\alpha(k+s,x) & = \sum_{\beta} C_{\alpha\beta}(k,s)u_\beta(k,x)
\tag{152}
\\
\implies
b_\alpha(k+s,x)&=e^{isx}\sum_\beta C_{\alpha\beta}(k,s)b_\beta(k,x),
\tag{153}
\end{align*}
where \begin{align*}
C_{\alpha\beta}(k,s) &\equiv \braket{u_\beta(k)}{u_\alpha(k+s)} \\
& \equiv \int_{\onetorus} u^*_\beta(k,x)u_\alpha(k+s,x)\dd{x}.
\tag{154}
\end{align*}
Eq. (154) implies that the orthonormality of the set of periodic Bloch functions at wavevector $k$ can be expressed as ${C_{\alpha\beta}(k,0)=\delta_{\alpha\beta}}$.
Inserting Eq. (153) into Eq. (151) gives
\begin{align*}
G_\alpha(s,X) =
e^{isX}
\sum_{\beta}
&
\intbz\dd{k'}
\intbz\dd{k}
\fca(k')
f_\alpha(k+s) \\
&\times C_{\alpha\beta}(k,s)
\braket{b_{\alpha k'}}{b_{\beta k}},
\tag{155}
\end{align*}
and the orthogonality of the set of Bloch functions allows this to be simplified to \begin{align*}
G_\alpha(s,X)& =
e^{isX}
\intbz\dd{k} \fca(k) f_\alpha(k+s) C_{\alpha\alpha}(k,s).
\tag{156}
\end{align*}
It follows that
\begin{align*}
\partial^2_s G_\alpha(s,X)
=
e^{isX}
\intbz\dd{k} \fca(k)
\bigg[
&C_{\alpha\alpha}(k,s)
\partial^2_s f_\alpha(k+s)
+
2\partial_s f_\alpha(k+s) \partial_s C_{\alpha\alpha}(k,s)
+
2iX f_\alpha(k+s) \partial_s C_{\alpha\alpha}(k,s)
\\
+
&2iX C_{\alpha\alpha}(k,s) \partial_s f_\alpha(k+s)
-X^2 f_\alpha(k+s) C_{\alpha\alpha}(k,s)
+ f_\alpha(k+s) \partial_s^2 C_{\alpha\alpha}(k,s)
\bigg]
\tag{157}
\end{align*}
Before calculating ${W_\alpha(X)=-\lim_{s\to 0}\partial_s^2 G_\alpha(s,X)}$ some useful results will be deduced and derived.
The first thing to note is that, because ${u_{\alpha k}}$ inherits smoothness from ${b_{\alpha k}}$,
\begin{align*}
\lim_{s\to 0}\partial_s^n u_\alpha(k+s,x)=
\partial_k^n u_\alpha(k,x), \;\forall n\in\integerpos.
\end{align*}
Therefore the Taylor expansion of ${u_\alpha(k+s,x)}$ about ${s=0}$ is \begin{align*}
u_\alpha(k+s,x)
=
u_\alpha(k&,x)+s\partial_k u_\alpha(k,x)
\\
&+\frac{1}{2} s^2 \partial^2_k u_\alpha(k,x) + \order{s^3}.
\tag{158}
\end{align*}
Inserting this into Eq. (154) gives \begin{align*}
C_{\alpha\beta}(k,s) = \delta_{\alpha\beta}&+s\braket{u_{\beta k}}{\partial_k u_{\alpha k}}
\\
&+\frac{1}{2}s^2\braket{u_{\beta k}}{\partial_k^2 u_{\alpha k}}
+\order{s^3}
\\
\implies C_{\alpha\alpha}(k,s) & = 1 +
\frac{1}{2}s^2\braket{u_{\alpha k}}{\partial_k^2 u_{\alpha k}}+\order{s^3},
\end{align*}
where the term ${s\braket{u_{\alpha k}}{\partial_k u_{\alpha k}}}$ vanishes because the only way for the normalization ${\braket{u_{\alpha k}}{u_{\alpha k}}=1}$ to be preserved as $k$ varies is if ${\partial_k u_{\alpha k}}$ is orthogonal to ${u_{\alpha k}}$. It follows that the first derivative of ${C_{\alpha\alpha}(k,s)}$ with respect to $s$ is \begin{align*}
\partial_s C_{\alpha\alpha}(k,s)
& = s\braket{u_{\alpha k}}{\partial_k^2 u_{\alpha k}}
+\order{s^2},
\end{align*}
which vanishes in limit ${s\to 0}$ because all terms on the right hand side are proportional to powers of $s$. The second derivative of ${C_{\alpha\alpha}(k,s)}$ with respect to $s$ is \begin{align*}
\partial^2_s C_{\alpha\alpha}(k,s) & = \braket{u_{\alpha k}}{\partial_k^2 u_{\alpha k}}
+\order{s},
\end{align*}
and all terms with powers of $s$ as prefactors vanish in the limit ${s\to 0}$. By integrating by parts it is found that \begin{align*}
\braket{u_{\alpha k}}{\partial_k^2 u_{\alpha k}}
&=-\braket{\partial_k u_{\alpha k}}{\partial_k u_{\alpha k}},
\end{align*}
where the boundary term vanishes because the limits of integration, $x_0$ and ${x_0+\bulksize}$, are the same point in $\onetorus$. It follows that \begin{align*}
\lim_{s\to 0} \partial^2_s C_{\alpha\alpha}(k,s) & = -\braket{\partial_k u_{\alpha k}}{\partial_k u_{\alpha k}}.
\end{align*}
The following results will help us to take the ${s\to 0}$ limit of Eq. (157): \begin{align*}
&\lim_{s\to 0} C_{\alpha\alpha}(k,s)=1,
\\
&\lim_{s\to 0} \partial_s C_{\alpha\alpha}(k,s)=0,
\\
&\lim_{s\to 0}\partial_s^2 C_{\alpha\alpha}(k,s)
=-\braket{\partial_k u_{\alpha k}}{\partial_k u_{\alpha k}},
\\
&\lim_{s\to 0} \partial^n_s f_\alpha(k+s)=\partial^n_k f_\alpha(k).
\end{align*}
We find that \begin{align*}
W_\alpha&(X) =-\lim_{s\to 0} G_\alpha(s,X)
\\
=&
\intbz\dd{k} \fca(k)
\bigg[
-
\partial^2_k
-
2iX \partial_k
+X^2
+ t_\alpha(k)
\bigg]
f_\alpha(k),
\end{align*}
where ${t_\alpha(k)\equiv
\braket{\partial_k u_{\alpha k}}{\partial_k u_{\alpha k}}}$.
By defining the function ${g_\alpha(k)\equiv e^{ikX} f_\alpha(k)}$, we can express ${W_{\alpha}(X)}$ as
\begin{align*}
W_\alpha(X) = \intbz\gca(k)
\left[
-\partial_k^2
+
t_\alpha(k)
\right]g_\alpha(k)\dd{k}.
\tag{159}
\end{align*}
This is stationary with respect to norm-preserving variations of $g_\alpha$ when $g_\alpha$ is any eigenfunction of \begin{align*}
\recham_\alpha(k)\equiv\expvaltwo{\recham_\alpha}{k}\equiv-\partial_k^2+t_\alpha(k),
\end{align*}
and the stationary values of $W_\alpha(X)$ are the eigenvalues of $\recham_\alpha(k)$. Note that the relationship in reciprocal space between ${u_\alpha(k,x)}$ and $t_\alpha(k)$ is the same, up to a multiplicative constant, as the relationship in real space between the Bloch function ${b_{\alpha k}}$ and what would be referred to within quantum mechanics as its kinetic energy. Also note that the operator $\recham_\alpha(k)$ is like a Hamiltonian in reciprocal space; and that ${t_\alpha(k)}$ plays the role of a positive potential in reciprocal space for ${g_{\alpha}}$.
If we now set ${g_\alpha(k)\equiv \hreciplatt^{-1} e^{i\theta_\alpha(k)}}$, ${f_\alpha}$ has the required normalization, Eq. (148) becomes
\begin{align*}
w_\alpha(x)
&= \hreciplatt^{-\frac{1}{2}}\intbz e^{-ikX} e^{i\theta_\alpha(k)} b_\alpha(k,x)\dd{k}
\end{align*}
and Eq. (159) becomes \begin{align*}
W_\alpha(X) = \frac{1}{\hreciplatt}\intbz
\left[
\abs{\partial_k\theta_\alpha}^2
+
t_\alpha(k)
\right]\dd{k}.
\tag{160}
\end{align*}
This has its minimum value when $\theta_\alpha$ is a constant, and this minimum value is \begin{align*}
W_\alpha^{\textrm{min}}(X) = \frac{1}{\hreciplatt} \intbz \braket{\partial_ku_{\alpha k}}{\partial_ku_{\alpha k}} \dd{k}.
\end{align*}
Finally, let us return to equations Eq. (149) and Eq. (156) to find the center ${\bar{x}_{\alpha X}}$ of the Wannier function whose spread about point $X$ is minimal: \begin{align*}
\bar{x}_{\alpha X}&= X + i\lim_{s\to 0}\partial_s G_\alpha(k,s)
\\
& = X + i\intbz f^*_\alpha(k)\left[\partial_k + i X\right] f_\alpha(k)\dd{k}
\\
& = \intbz f^*_\alpha(k)\left(i\partial_k f_\alpha(k)\right)\dd{k} = X
\end{align*}
I emphasize that this section does not have any quantum mechanical content. I have presented a derivation that is as applicable within classical statistical mechanics as it is within quantum mechanics.
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