Berry curvature and quantum metric in copper-substituted lead phosphate apatite

Our starting point is the $2\times 2$ spinless tight-binding model that takes the form of a Dirac Hamiltonian [8]

$$\begin{align} H_{0}\left(\textbf{k}\right) = \left(d_{0}-\mu\right)\textbf{I}+\textbf{d}\cdot{\boldsymbol\sigma}, \end{align} \tag{ 1 }$$

where $\left\{\sigma_{1},\sigma_{2},\sigma_{3}\right\}$ are Pauli matrices. The d-vector is parametrized by

$$\begin{align} d_{0}\left(\textbf{k}\right) &= -\frac{1}{4}\sum_{i = 1}^{3}\text{cos}\left(\mathbf{k\cdot b}_{i}\right)-\frac{3}{2}-2t_{z}\text{cos} k_{z}, \nonumber \\ d_{1}\left(\textbf{k}\right) &= -\frac{\sqrt{3}}{4}\left[-\text{cos}\left(\mathbf{k\cdot b}_{1}\right)+\text{cos}\left(\mathbf{k\cdot b}_{2}\right)\right], \nonumber \\ d_{2}\left(\textbf{k}\right) &= -\frac{\sqrt{3}}{4}\sum_{i = 1}^{3}\text{sin}\left(\mathbf{k\cdot b}_{i}\right), \nonumber \\ d_{3}\left(\textbf{k}\right) &= -\frac{1}{2}\text{cos}\left(\mathbf{k\cdot b}_{3}\right)+\frac{1}{4}\left[\text{cos}\left(\mathbf{k\cdot b}_{1}\right)+\text{cos}\left(\mathbf{k\cdot b}_{2}\right)\right]. \end{align} \tag{ 2 }$$

In units of bond length a of the honeycomb lattice, the 2D vectors in real space are parametrized by

$$\begin{align} \textbf{b}_{1} = \left(-\frac{\sqrt 3}{2},-\frac{3}{2}\right),\;\; \textbf{b}_{2} = \left(-\frac{\sqrt 3}{2},\frac{3}{2}\right),\;\; \textbf{b}_{3} = \left(\sqrt{3},0\right). \nonumber \\ \end{align} \tag{ 3 }$$

The hexagonal BZ and the resulting band structure at $k_{z} = 0$ are shown in figure 1. Note that the Hamiltonian is time-reversal (TR) symmetric $TH_{0}(\textbf{k})T^{-1} = H_{0}(-\textbf{k})$, with the TR operator implemented by complex conjugation. This seems to already imply that the band structure alone cannot produce a magnetization. Secondly, we also note that the flat bands are entirely contributed from the hopping between Cu substitutions. This is very different from another well-known mechanism for the flat bands where the substituted atoms or periodic vacancies produce unequal amount of sublattices on a bipartite lattice [23, 24], which has also been propose to stabilize superconductivity [25]. Finally, we emphasize that this two-band tight-binding model is constructed purely for the flat bands near the Fermi surface that are mainly composed of Cu d_zx and d_zy orbitals, while the valence and conduction bands far from the Fermi surface cannot be captured by this model. Nevertheless, magnetism and superconductivity are most likely contributed from the bands near the Fermi surface, which motivates us to investigate this model.

Zoom In Zoom Out Reset image size

The Dirac form of the Hamiltonian and the fact that $(d_{1},d_{2},d_{3})$ only depend on the planar momentum $(k_{x},k_{y})$ motivate us to investigate the Berry curvature and quantum metric of this model, as it bears a striking similarity with the 2D Chern insulator [12, 13]. Note that the TR symmetry explained after equation (3) implies that the Berry curvature integrated over momentum $(k_{x},k_{y})$ at a given k_z must vanish, in contrast to 2D Chern insulators that can have a finite Chern number. Nevertheless, locally at a specific momentum k, the Berry curvature can be nonzero. In addition, an obvious difference is that the two-band model is metallic [8], in contrast to the Chern insulator that has a band gap throughout the BZ. As a result, the Berry curvature and quantum metric of this two-band model is only defined in the regions of BZ where the lower band $E_{-}(\textbf{k})$ is below the Fermi surface and hence serves as a valence band, and the higher band $E_ {+}(\textbf{k})$ is above the Fermi surface and hence plays the role of a conduction band (in the regions where the two bands are both above or both below the Fermi surface, the Berry curvature and quantum metric are zero within this two-band model). In these gapped regions, we denote the valence and conduction band states to be $|n\rangle$ and $|m\rangle$, respectively. Denoting the derivative over momentum as $\partial_{\mu} = \partial/\partial k_{\mu}$ and $d\equiv\sqrt{d_{1}^{2}+d_{2}^{2}+d_{3}^{2}}$, our interest is the Berry curvature of the filled valence band state $|n\rangle$, which is calculated by

$$\begin{align} \Omega_{\mu\nu} &= i\langle\partial_{\mu}n|\partial_{\nu}n\rangle -i\langle\partial_{\nu}n|\partial_{\mu}n\rangle \nonumber \\ & = \frac{1}{2d^{3}}\varepsilon^{ijk}d_{i}\partial_{\mu}d_{j}\partial_{\nu}d_{k}, \end{align} \tag{ 4 }$$

where ε^ijk is the fully antisymmetric Levi-Civita symbol. As we shall see below, the Berry curvature has only the Ω_xy component, so it contributes to an orbital magnetization along ${\hat{\mathbf{z}}}$ direction at momentum k described by [14, 15]

$$\begin{align} m^{z}\left(\mathbf{k}\right) &= -i\frac{e}{2\hbar}\left[\langle\partial_{x}n|\left(H-E_{-}\right)|\partial_{y}n\rangle\right. \nonumber \\ &\quad \left.-\langle\partial_{y}n|\left(H-E_{-}\right)|\partial_{x}n\rangle\right] \nonumber \\ & = \frac{e}{2\hbar}\left(E_{+}-E_{-}\right)\Omega_{xy} \nonumber \\ & = -\left(\frac{e}{\hbar}\right)\frac{1}{2d^{2}}\varepsilon^{ijk}d_{i}\partial_{\mu}d_{j}\partial_{\nu}d_{k}. \end{align} \tag{ 5 }$$

The net orbital magnetization of the material is then given by the momentum integration of $m^{z}(\textbf{k})$.

On the other hand, the quantum metric is defined from the overlap of the valence band state at slightly different momenta [26] $|\langle n(\mathbf{k})|n(\mathbf{k+\delta k})\rangle| = 1-g_{\mu\nu}\delta k_{\mu}\delta k_{\nu}$, which in the $2\times 2$ Dirac Hamiltonian has the expression [27–29]

$$\begin{align} g_{\mu\nu} &= \frac{1}{2}\langle\partial_{\mu}n|\partial_{\nu}n\rangle +\frac{1}{2}\langle\partial_{\nu}n|\partial_{\mu}n\rangle -\langle\partial_{\mu}n|m\rangle\langle m|\partial_{\nu}n\rangle \nonumber \\ & = \frac{1}{4d^{2}}\left\{\sum_{i = 1}^{3}\partial_{\mu}d_{i}\partial_{\nu}d_{i} -\partial_{\mu}d\,\partial_{\nu}d\right\}. \end{align} \tag{ 6 }$$

Note that the d₀ component does not influence the eigenstates, so the Berry curvature and quantum metric are entirely determined by $\left\{d_{1},d_{2},d_{3}\right\}$. However, the d₀ component does influence which states are above and below the Fermi surface and hence whether Berry curvature and quantum metric are nonzero, as we shall see below.

Analytical results near high-symmetry points (HSPs) of the BZ can be given to draw some theoretical understanding. In particular, the K and Kʹ points defined by (in units of $\hbar/a$)

$$\begin{align} \textbf{K} = \left(\frac{2\pi}{3\sqrt{3}},\frac{2\pi}{3}\right),\;\;\;\textbf{K}^{^{\prime}} = \left(-\frac{2\pi}{3\sqrt{3}},\frac{2\pi}{3}\right), \end{align} \tag{ 7 }$$

on the hexagonal plane of $k_{z} = 0$ belong to the gapped regions, and are of particular interest since they are analogous to those in graphene (even though graphene is not gapped), so we seek for analytical result near these points. A straightforward expansion of the momentum near these points $\textbf{k} = \textbf{K}^{(^{\prime})}+\delta\textbf{k}$ yields

$$\begin{align} K\;{\textrm{point}}:\;\;\;d_{1} &= -\frac{9}{8}\delta k_{y},\;\;\; d_{2} = -\frac{9}{8},\;\;\; d_{3} = \frac{9}{8}\delta k_{x}, \nonumber \\ K^{^{\prime}}\;{\textrm{point}}:\;\;\;d_{1} &= \frac{9}{8}\delta k_{y},\;\;\; d_{2} = \frac{9}{8},\;\;\; d_{3} = -\frac{9}{8}\delta k_{x}. \end{align} \tag{ 8 }$$

Firstly, converting $\delta k_{x} = \delta k\text{sin}\theta\text{cos}\phi$ and $\delta k_{y} = \delta k\text{sin}\theta\text{sin}\phi$ into spherical coordinate, the vanishing integration of the Berry curvature over a spherical surface of radius δk

$$\begin{align} \frac{1}{2\pi}\int_{0}^{2\pi}\textrm{d}\phi\int_{0}^{\pi}\textrm{d}\theta\,\Omega_{\theta\phi} = 0, \end{align} \tag{ 9 }$$

indicating that there is no topological charge at K and Kʹ points, defined analogous to that of the 3D Weyl semimetals [30–32]. On the other hand, in the Cartesian coordinate $\partial_{\mu} = \partial/\partial\delta k_{\mu}$, the Berry curvature and quantum metric are

$$\begin{align} \Omega_{xy} &= \pm\frac{1}{2\left(1+\delta k^{2}\right)^{3/2}}, \nonumber \\ g_{\mu\nu} &= \left(\begin{array}{cc} g_{xx} & g_{xy} \\ g_{yx} & g_{yy} \end{array}\right) \nonumber \\ & = \frac{1}{4\left(1+\delta k^{2}\right)^{2}}\left(\begin{array}{cc} 1+\delta k_{y}^{2} & -\delta k_{x}\delta k_{y} \\ -\delta k_{x}\delta k_{y} & 1+\delta k_{x}^{2} \end{array}\right), \end{align} \tag{ 10 }$$

where the upper and lower signs in Ω_xy are for the K and Kʹ points, respectively. The results in equation (10) satisfies a metric-curvature correspondence

$$\begin{align} \sqrt{\text{det} g_{\mu\nu}} = \frac{1}{2}|\Omega_{xy}| = \frac{1}{4\left(1+\delta k^{2}\right)^{3/2}}, \end{align} \tag{ 11 }$$

a relation that is satisfied for TIs and TSCs described by Dirac models in any dimension and symmetry class [27], hinting that it may be valid even beyond topological materials. It should be noted, though, that the material at hand is 3D but the relation is satisfied in a 2D manner, in the sense that g_µν has no g_zz component and the determinant is completely calculated from the planar components.

Numerical results for Ω_xy , m^z , g_xx and g_yy are shown in figure 2, where we plot them as functions of planar momentum $(k_{x},k_{y})$ at two different values of out-of-plane momentum $k_{z} = 0$ and π. Notice that the momentum profile of these quantities should not evolve with k_z since the eigenstates $|n\rangle$ and $|m\rangle$ are determined by $(d_{1},d_{2},d_{3})$ that do not depend on k_z . Nevertheless, as mentioned above, these quantities are nonzero only when $E_{-}\lt0$ and $E_{+}\gt0$, so they are zero in the region surrounding the $\Gamma-A$ line where $\left\{E_{+},E_{-}\right\}\lt0$. This region shrinks as k_z moves from 0 to π, which also changes the overall scale of the plots between the upper panels and the lower panels in figure 2, since these quantities have a larger magnitude around the $\Gamma-A$ line that is gradually revealed as k_z moves from 0 to π (moving from Γ to A). At the boundary of this $\left\{E_{+},E_{-}\right\}\lt0$ region, these quantities has a discrete jump from zero to a finite value, which many have some interesting consequences that shall be further explored.

Zoom In Zoom Out Reset image size

Concerning the momentum dependence of the Berry curvature Ω_xy and the orbital magnetization m^z shown in figures 2(a) and (b), we find that their values around K − H and $K^{^{\prime}}-H^{^{\prime}}$ lines are always of equal magnitude and opposite signs, and hence cancel out after a momentum integration, indicating no net orbital moment. This behavior is very similar to that in transition metal dichalcogenides (TMDs), which have a two-dimensional (2D) hexagonal BZ and also show this type of cancellation of orbital moment and Berry curvature [33, 34]. In addition, the Berry curvature around the $\Gamma-A$ line is actually much higher than at K − H and $K^{^{\prime}}-H^{^{\prime}}$ lines, and maintains the pattern of alternating signs and lobes of the six-fold symmetry of the BZ, although they are only revealed as k_z approaches π such that the $E_{-}\lt0$ and $E_{+}\gt0$ condition is satisfied (as moving from Γ to A), as discussed above.

On the other hand, the quantum metric g_xx and g_yy shown in figures 2(c) and (d) are both positively defined, and their values are the same at K − H and $K^{^{\prime}}-H^{^{\prime}}$ lines. Their patterns are two-fold symmetric in the BZ, consistent with the analytical formula given in equation (10). The quantities are also found to be much more enhanced around the $\Gamma-A$ line, with a pattern of g_xx that has lobes along $\pm k_{y}$ directions and those of g_yy are along $\pm k_{x}$ directions, and the lobes of g_xx and g_yy are roughly of the same magnitude. To further quantify the difference between g_xx and g_yy , we perform a momentum-integration of these quantities over the 3D BZ to obtain

$$\begin{align} {\cal G}_{\mu\mu} = \int\frac{\textrm{d}^{3}\textbf{k}}{\tilde{V}_{BZ}}\,g_{\mu\mu}, \end{align} \tag{ 12 }$$

which is a measure of the average distance between valence band states $|n\rangle$ along the k_µ direction in the $E_{-}\lt0$ and $E_{+}\gt0$ regions of the 3D BZ, a quantity that has been called the fidelity number [29]. Here $\tilde{V} = 16\pi^{3}/3\sqrt{3}$ is the dimensionless volume of the 3D BZ. The numerical result yields ${\cal G}_{xx}\approx 0.3742$ and ${\cal G}_{yy}\approx 0.3734$ in units of $\hbar/c$, where c is the lattice constant along ${\hat{\mathbf{z}}}$ direction, indicating that the two planar directions have roughly the same quantum geometrical properties. In contrast, quantum metric along the ${\hat{\mathbf{z}}}$ direction vanishes everywhere $g_{zz} = g_{xz} = g_{yz} = 0$, and hence if the quantum metric really plays a role in some pairing mechanisms, such as the flat band superconductivity [18–22], then the superconductivity likely only occurs in the ${\hat{\mathbf{x}}}$ and ${\hat{\mathbf{y}}}$ directions despite the band structure is flat in all three directions.

Finally, we remark that in semiconductors and insulators, one can define a Berry curvature spectral function [28] $\Omega_{xy}(\textbf{k},\omega)$ that frequency-integrates to the Berry curvature $\Omega_{xy}(\textbf{k}) = \int \textrm{d}\omega\,\Omega_{xy}(\textbf{k},\omega)$ and momentum-integrates to the Chern number spectral function ${\cal C}(\omega) = \int\frac{\textrm{d}^{2}\textbf{k}}{(2\pi)^{2}}\Omega_{xy}(\textbf{k},\omega)$, with the later being related to the circular dichroism of 2D materials [35]. Likewisely, one can also introduce a quantum metric spectral function [28] $g_{\mu\nu}(\textbf{k},\omega)$ that frequency-integrates to the quantum metric $g_{\mu\mu}(\textbf{k}) = \int \textrm{d}\omega\,g_{\mu\mu}(\textbf{k},\omega)$ and is proportional the optical conductivity $g_{\mu\mu}(\textbf{k},\omega)\propto\sigma_{\mu\mu}(\textbf{k},\omega)/\omega$ at momentum k, which can possibly be detected in pump-probe type of experiments [27]. Furthermore, one can integrate the quantum metric spectral function over momentum to obtain what we call the fidelity number spectral function ${\cal G}_{\mu\mu}(\omega) = \int\frac{\textrm{d}^{D}\textbf{k}}{(2\pi)^{D}}g_{\mu\mu}(\textbf{k},\omega)\propto \sigma_{\mu\mu}(\omega)/\omega$ that corresponds to optical conductivity (and hence the optical absorption power) $\sigma_{\mu\mu}(\omega)$ measured in real space divided by frequency, which serves as a practical way to measure the quantum geometrical properties of insulators and semiconductors [29]. These spectral functions, however, do not apply to the present work simply because the material under question is metallic, and hence does not have optical absorption since infrared light cannot penetrate through. It remains to be elucidated whether there is some simple methods to detect the quantum metric and Berry curvature for this material.