• Wanner waves, tight-binding model, and quantum geometry
  • Wannier waves
  • Quantum metric: lower bound of quadratic spread
  • Second perspective: quantum metric from multi-band models

Wanner waves, tight-binding model, and quantum geometry

Wannier waves

We begin with the single-particle Schroedinger equation in \(d\) spatial dimensions

\[H\vert\psi\rangle=\left[-\frac{(\hbar\nabla)^{2}}{2m}+V(\mathbf{r})\right]\vert\psi\rangle, \label{eq:Hcont}\]

where \(V(\mathbf{r}+\mathbf{a}_{i})=V(\mathbf{r})\) represents a periodic potential, and \(\mathbf{a}_{i}\) (\(i=1,\cdots,d\)) defines a lattice system. According to the Bloch theorem, the solutions, known as Bloch waves, for an energy band \(n\) can be expressed as:

\[\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}),\]

where \(u_{n\mathbf{k}}(\mathbf{r})\) is a periodic Bloch function satisfying \(u_{n\mathbf{k}}(\mathbf{r})=u_{n\mathbf{k}}(\mathbf{r}+\mathbf{a}_{i})\), and \(\mathbf{k}\) is the Bloch wavevector. The normalization condition for \(u_{n\mathbf{k}}(\mathbf{r})\) is given by:

\[\int_{\text{u.c.}}d^{d}\mathbf{r} \vert u_{n\mathbf{k}}(\mathbf{r})\vert^{2}=1,\]

where the integral is taken over one unit cell. Here, u.c. represents the unit cell with volume \(\mathcal{A}_\mathrm{uc}\). The energy \(\epsilon_n(\mathbf{k})\) satisfies periodicity with respect to the reciprocal lattice vectors \(\mathbf{G}_i\), given by the condition \(\mathbf{a}_i \cdot \mathbf{G}_j = 2\pi \delta_{ij}\). In other words, the energy is invariant under translations by the reciprocal lattice vectors.

We consider composite bands labeled by the band index \(n\) in a specific subset \(\mathcal{V}\), which is separated from other bands by sufficiently large band gaps. In this case, we can construct a set of Wannier basis states \(\{\vert\mathbf{r}_i\alpha\rangle\}\) that span the same sub-Hilbert space as the Bloch waves corresponding to the bands with indices \(n\in\mathcal{V}\). The Wannier basis states can be expressed as follows:

\[\begin{align} \vert\mathbf{r}_i\alpha\rangle &= \frac{\mathcal{A}_\mathrm{uc}}{(2\pi)^d}\int_\mathrm{BZ} d^d\mathbf{k}\, e^{i\mathbf{k}\cdot(\mathbf{r}-\mathbf{r}_i)} \sum_{n\in\mathcal{V}} (\mathcal{U}_\mathbf{k})_{n,\alpha} \vert u_{n\mathbf{k}}\rangle, \label{eq:wannier_Bloch1} \\ \vert u_{n\mathbf{k}}\rangle &= \sum_{\mathbf{r}_i}\sum_\alpha e^{-i\mathbf{k}\cdot(\mathbf{r}-\mathbf{r}_i)} (\mathcal{U}_\mathbf{k}^\dagger)_{\alpha,n} \vert\mathbf{r}_i\alpha\rangle. \label{eq:wannier_Bloch2} \end{align}\]

Here, \(\mathcal{A}_\mathrm{uc}\) is the volume of the unit cell, and \(\mathbf{r}_i\) represents a lattice site spanned by the lattice vectors \(\mathbf{a}_i\) \((i=1,\cdots,d)\). The integration over momentum is performed over the first Brillouin zone (BZ). The unitary matrix \(\mathcal{U}_\mathbf{k}\) is chosen to optimize the localization of the Wannier functions. The Wannier function \(\langle\mathbf{r}\vert\mathbf{r}_i\alpha\rangle \equiv w_\alpha(\mathbf{r}-\mathbf{r}_i)\) is localized around the lattice site \(\mathbf{r}_i\). It turns out to be the Fourier transformation of the corresponding Bloch wave and thus inherits the orthonormality properties of the Bloch functions.

Quantum metric: lower bound of quadratic spread

The unitary matrix \(\mathcal{U}_{\mathbf{k}}\) is chosen to maximize the localization of Wannier functions by minimizing a localization functional, as introduced by Marzari and Vanderbilt in their seminal work ¹. The localization functional is given by

\[\begin{align} F & =\sum_{\alpha\in\mathcal{V}}\left[\langle\mathbf{0}\alpha\vert r^{2}\vert\mathbf{0}\alpha\rangle-\vert\langle\mathbf{0}\alpha\vert\mathbf{r}\vert\mathbf{0}\alpha\rangle\vert^{2}\right] =F_{I}+\delta F~. \end{align}\]

Both parts, \(F_I\) and \(\delta F\), are non-negative, where

\[\begin{align} F_{I} & =\sum_{\alpha\in\mathcal{V}}\left[\langle\mathbf{0}\alpha\vert r^{2}\vert\mathbf{0}\alpha\rangle-\sum_{\mathbf{r}_{i}}\sum_{\beta}\vert\langle\mathbf{r}_{i}\beta\vert\mathbf{r}\vert\mathbf{0}\alpha\rangle\vert^{2}\right],\\ \delta F & =\sum_{\mathbf{r}_{i}(\neq\mathbf{0})}\sum_{\beta(\neq\alpha)}\vert\langle\mathbf{r}_{i}\beta\vert\mathbf{r}\vert\mathbf{0}\alpha\rangle\vert^{2}. \end{align}\]

The optimization of the unitary matrix \(\mathcal{U}_{\mathbf{k}}\) aims to minimize the localization functional \(F\), leading to the construction of maximally localized Wannier functions. The term \(F_I\) is independent of the unitary transformation \(\mathcal{U}_{\mathbf{k}}\) and therefore gauge invariant. This allows us to choose \(\mathcal{U}_{\mathbf{k}}\) as an identity matrix with components \((\mathcal{U}_{\mathbf{k}})_{\alpha,n}=\delta_{\alpha,n}\) when calculating \(F_I\). Then from the relation in Eqs.\(~(\ref{eq:wannier_Bloch1})\) and \((\ref{eq:wannier_Bloch2})\), we have

\[\begin{align} \langle u_{n\mathbf{k}}\vert u_{m\mathbf{k+q}}\rangle & =\sum_{\mathbf{r}_{i}}e^{-i\mathbf{k}\cdot\mathbf{r}_{i}}\langle\mathbf{r}_{i}n\vert e^{-i\mathbf{q}\cdot\mathbf{r}}\vert\mathbf{0}m\rangle ,\label{eq:uuq} \end{align}\]

By taking the derivative with respect to \(\mathbf{q}\) on both sides of Eq.~\((\ref{eq:uuq})\), we obtain a series of relations in the limit \(q\rightarrow 0\). For example, taking the first and second derivatives with respect to \(\mathbf{q}\) gives

\[\begin{align} \langle u_{n\mathbf{k}}\vert\nabla_{\mathbf{k}}u_{m\mathbf{k}}\rangle & =-i\sum_{\mathbf{r}_{i}}e^{-i\mathbf{k}\cdot\mathbf{r}_{i}}\langle\mathbf{r}_{i}n\vert\mathbf{r}\vert\mathbf{0}m\rangle ,\\ \langle u_{n\mathbf{k}}\vert\nabla_{\mathbf{k}}^{2}u_{m\mathbf{k}}\rangle & =-\sum_{\mathbf{r}_{i}}e^{-i\mathbf{k}\cdot\mathbf{r}_{i}}\langle\mathbf{r}_{i}n\vert\mathbf{r}^{2}\vert\mathbf{0}m\rangle, \end{align}\]

Similarly, we can establish the converse relations

\[\begin{align} \langle\mathbf{r}_{i}n\vert\mathbf{r}\vert\mathbf{0}m\rangle & =i\frac{\mathcal{A}_{\mathrm{uc}}}{(2\pi)^{d}}\int_\mathrm{BZ} d^{d}\mathbf{k}e^{i\mathbf{k}\cdot\mathbf{r}_{i}}\langle u_{n\mathbf{k}}\vert\nabla_{\mathbf{k}}u_{m\mathbf{k}}\rangle ,\\ \langle\mathbf{r}_{i}n\vert\mathbf{r}^{2}\vert\mathbf{0}m\rangle & =\frac{\mathcal{A}_{\mathrm{uc}}}{(2\pi)^{d}}\int_\mathrm{BZ} d^{d}\mathbf{k}e^{i\mathbf{k}\cdot\mathbf{r}_{i}}\langle\nabla_{\mathbf{k}}u_{n\mathbf{k}}\vert\nabla_{\mathbf{k}}u_{m\mathbf{k}}\rangle. \end{align}\]

Therefore, we can simplify \(F_{I}\) as

\[\begin{align} F_{I} & =\sum_{\alpha\in\mathcal{V}}\left[\langle\mathbf{0}\alpha\vert r^{2}\vert\mathbf{0}\alpha\rangle-\sum_{\mathbf{r}_{i}}\sum_{\beta}\vert\langle\mathbf{r}_{i}\beta\vert\mathbf{r}\vert\mathbf{0}\alpha\rangle\vert^{2}\right]\\ & =\frac{\mathcal{A}_{\mathrm{uc}}}{(2\pi)^{d}}\int_\mathrm{BZ}d^{d}\mathbf{k}\sum_{n\in\mathcal{V}}\mathrm{Re}\langle\nabla_{\mathbf{k}}u_{n\mathbf{k}}\vert(\mathbb{I}_{\mathcal{V}}-\vert u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}\vert)\vert\nabla_{\mathbf{k}}u_{n\mathbf{k}}\rangle~, \end{align}\]

where \(\mathbb{I}_{\mathcal{V}}\) is the identity operator in the sub-Hilbert space spanned by bands carrying indices in \(\mathcal{V}\). This expression clearly shows that \(F_{I}\) is expressed in terms of the quantum metric. Since \(\delta F\geq 0\), we have the inequality relation,

\[F\geq F_{I}~.\]

Hence, we can conclude that the quantum metric characterizes an obstruction to finding a complete set of exponentially localized Wannier functions. When \(F_{I}\) is finite, it indicates that more bands need to be included in the composite bands in order to construct a complete set of exponentially localized Wannier functions.

Second perspective: quantum metric from multi-band models

Another perspective on the quantum metric arises from considering a multiband tight-binding model. Assuming we have already obtained a complete set of exponentially localized Wannier functions constructed from composite bands, we can approximate the continuum Hamiltonian in Eq.\(~(\ref{eq:Hcont})\) with a tight-binding model. In the language of second quantization, the continuum model in Eq.\(~(\ref{eq:Hcont})\) can be expressed as

\[H=\int d^{d}\mathbf{r}\psi^{\dagger}(\mathbf{r})\left[-\frac{(\hbar\nabla)^{2}}{2m}+V(\mathbf{r})\right]\psi(\mathbf{r})~. \label{eq:2ndH}\]

We then expand the field operator \(\psi(\mathbf{r})\) in the basis of Wannier functions

\[\psi(\mathbf{r})=\sum_{\mathbf{r}_{i}}\sum_{\alpha\in\mathcal{V}}w_{\alpha}(\mathbf{r}-\mathbf{r}_{i})a_{i\alpha}+\sum_{\mathbf{r}_{i}}\sum_{\beta\in\mathcal{V}^{\perp}}w_{\beta}^{\perp}(\mathbf{r}-\mathbf{r}_{i})b_{i\beta}~,\]

where \(w_{\beta}^{\perp}(\mathbf{r}-\mathbf{r}_{i})\) denotes Wannier functions associated with the complementary band set \(\mathcal{V}^{\perp}\). By substituting the expansion into the Hamiltonian in Eq.~\((\ref{eq:2ndH})\), we can derive a tight-binding model defined on the lattice \(\{\mathbf{r}_{i}\}\)

\[\begin{align} H & =\sum_{\alpha\beta\in\mathcal{V}}\sum_{\mathbf{r}_{i},\mathbf{r}_{j}}\langle\mathbf{r}_{i}\alpha\vert H\vert\mathbf{r}_{j}\beta\rangle a_{i\alpha}^{\dagger}a_{j\beta}^{}+\sum_{\alpha^{\prime},\beta^{\prime}\in\mathcal{V}^{\perp}}\sum_{\mathbf{r}_{i},\mathbf{r}_{j}}\langle\mathbf{r}_{i}\alpha^{\prime}\vert H\vert\mathbf{r}_{j}\beta^{\prime}\rangle b_{i\alpha^{\prime}}^{\dagger}b_{j\beta^{\prime}}^{}\nonumber \\ & =\sum_{\alpha\beta\in\mathcal{V}}\sum_{\mathbf{r}_{i},\mathbf{r}_{j}}t_{ij,\alpha\beta}a_{i\alpha}^{\dagger}a_{j\beta}^{}+\sum_{\alpha^{\prime},\beta^{\prime}\in\mathcal{V}^{\perp}}\sum_{\mathbf{r}_{i},\mathbf{r}_{j}}t_{ij\alpha^{\prime}\beta^{\prime}}^{\perp}b_{i\alpha^{\prime}}^{\dagger}b_{j\beta^{\prime}}^{}~, \label{eq:tbhfull} \end{align}\]

where no mixing term between indices from \(\mathcal{V}\) and \(\mathcal{V}^{\perp}\). Up to this point, all the derivations have been rigorous, and the expression in Eq.~\((\ref{eq:tbhfull})\) includes all bands. However, since our interest lies solely in the bands belonging to \(\mathcal{V}\), we can utilize a complete set of exponentially localized Wannier functions to approximate the Hamiltonian in Eq.\(~(\ref{eq:Hcont})\) with a multi-band tight-binding model \(H_{\mathrm{tb}}\) by disregarding the \(t^\perp\) terms

\[\begin{align} H_{\mathrm{tb}} & =\sum_{\alpha\beta\in\mathcal{V}}\sum_{\mathbf{r}_{i},\mathbf{r}_{j}}t_{ij,\alpha\beta}a_{i\alpha}^{\dagger}a_{j\beta} =\sum_{\alpha\beta\in\mathcal{V}}\sum_{\mathbf{k}}h_{\alpha\beta}(\mathbf{k})a_{\mathbf{k}\alpha}^{\dagger}a_{\mathbf{k}\beta}^, \label{eq:tb} \end{align}\]

where \(t_{ij,\alpha\beta}\) exponentially decays with the distance \(\vert\mathbf{r}_{i}-\mathbf{r}_{j}\vert\). In Eq.~\((\ref{eq:tb})\), we have further introduced the Fourier transformation \(a_{\mathbf{k}\alpha}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{\mathbf{r}_{i}}a_{i\alpha}^{\dagger}e^{i\mathbf{k}\cdot\mathbf{r}_{i}}\), where \(N\) represents the total number of lattice sites. In our specific setup, where there is a significant gap between the targeted band and the others, we can project onto the targeted band using the following expressions

\[a_{i\alpha}\rightarrow\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}_{i}}u_{\mathbf{k}}^{*}(\alpha)c_{\mathbf{k}}~,\]

or

\[\psi({\mathbf r})\rightarrow\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}_{i}}g_{\mathbf{k}}^{*}(\alpha)c_{\mathbf{k}}~,\]

where \(u_{\mathbf{k}}\) represents an eigenvector of \(h_{\alpha\beta}(\mathbf{k})\), and \(c_{\mathbf{k}}\) annihilates an electron in the targeted band. It is important to note that the index \(\alpha\) appearing in both \(a_{i\alpha}\) and \(g_{\mathbf{k}}(\alpha)\) arises from the realization of a multiband tight-binding model, which accounts for the nontrivial quantum metric or Wannier obstruction. This can be inferred from the quantum metric associated with \(g_{\mathbf{k}}\).