## 2 Model and Band Structures  
### Three-Orbital Tight-Binding Model: Previous Studies  
We first introduce a model for the phase without PLD, and this is the tight-binding Hamiltonian used in ref. [38]. This model involves three \(d\)-orbitals (\(3z^{2}-r^{2}\), \(x^{2}-y^{2}\), and \(xy\)) of Ta ions on the two-dimensional triangular lattice, and is defined as  
\[\hat{H}_{\rm TB}^{0}=-\sum_{\langle i,j\rangle}\sum_{m,m^{\prime},\sigma}t_{ im\sigma}^{0,ij}c_{im\sigma}^{\dagger}c_{jm^{\prime}\sigma}+\sum_{i}(\sum_{m}d_{m} \hat{n}_{im}+\hat{\varsigma}\hat{L}_{i}\cdot\hat{S}_{i}), \tag{1}\]  
where the summation \(\sum_{\langle i,j\rangle}\) runs over the nearest neighbor site pairs. Here, \(\hat{c}_{im\sigma}\) denotes the electron annihilation operator at site \(i\), in orbital \(m\) (\(=3z^{2}-r^{2}\), \(xy\), \(xy\)), with spin \(\sigma\) (\(=\uparrow,\downarrow\)), and \(\hat{n}_{im}\equiv\sum_{\sigma}\hat{c}_{im\sigma}^{\dagger}\hat{c}_{im\sigma}\) is the orbital density operator. The operators \(\hat{L}_{i}\) and \(\hat{S}_{i}\) represent the orbital and the spin angular momenta, respectively, and their expressions are given in Section 6.1. The parameters \(d_{m}\) are the crystal field energies, while the hopping integrals \(t_{mm^{\prime}}^{0,ij}\) are obtained as linear combinations of the Slater-Koster parameters [39; 40] shown in Table 1. The spin-orbit interaction energy is set as  
Figure 1: (**a**) PLD in hexagram configuration in real space (exaggerated). The red, blue, and orange dots show ”A”, ”B”, and ”C” Ta atoms, respectively. For reference, Ta atoms without PLD are shown by the gray dots; (**b**) parameters \(r_{B}\) and \(r_{c}\) characterizing distorted hexagram configuration. The configuration is assumed to be symmetric under the \(\pi/3\) rotation and the mirror transformation about the \(x\)-axis. The site indices will be used in Section 4. \(\xi=0.3129\) eV. The number of electrons is one per site, and the Fermi energy or the chemical potential is determined accordingly. The effect of the PLD is taken into account by changes in the transfer integrals [37]. Let \(\mathbf{r}_{i}^{\text{eq}}\) denote the position of the \(i\)-th Ta atom in the absence of a PLD, while \(\mathbf{r}_{i}\) is the position when a PLD is present. Upon the change in atomic positions, we set the transfer integral between sites \(i\) and \(j\) as  
\[t_{mm^{\prime}}^{ij}(\{\mathbf{r}_{i}\})=\left(\frac{|\mathbf{r}_{i}^{\text{eq}}-\mathbf{r }_{j}^{\text{eq}}|}{|\mathbf{r}_{i}-\mathbf{r}_{j}|}\right)^{5}t_{mm^{\prime}}^{0,ij}, \tag{2}\]  
following the empirical law for the \(d\)-electrons. Thus, the Hamiltonian with taking account of a PLD is given by  
\[\hat{H}_{\text{TB}}=-\sum_{\langle i,j\rangle}\sum_{m,m^{\prime},\sigma}t_{ mm^{\prime}}^{ij}(\{\mathbf{r}_{i}\})c_{im^{\prime}\sigma}^{\dagger}c_{jm^{ \prime}\sigma}+\sum_{i}(\sum_{m}d_{m}\hat{n}_{im}+\xi\mathbf{\hat{L}}_{i}\cdot\mathbf{ \hat{S}}_{i}). \tag{3}\]  
Note that we neglect inhomogeneous changes in local potential \(\Delta d_{im}(\{\mathbf{r}_{j}\})\) associated with the PLD. Following the previous studies [37; 38], we consider a specific PLD mode of the hexagram shape shown in Figure 1a. Here, the unit cell has 6-fold rotation symmetry and contains 13 sites categorized into three types as shown in Figure 1b; one \"A\" (red), six \"B\" (blue), and six \"C\" (orange) sites. The distances from A to B and C sites are denoted by \(r_{B}\) and \(r_{C}\) and these are parameterized by one parameter \(x\) quantifying the amplitude of the PLD as  
\[r_{B}=1+0.064x;\quad r_{C}=\sqrt{3}+0.072x. \tag{4}\]  
Here, the length unit is the lattice constant 3.36 A in the absence of a PLD. In [38], the value \(x=-1\) corresponding to the values \((r_{B},r_{C})=(0.936,1.66)\) is used to be consistent with experimental results. We note that \(x<0\) (\(x>0\)) corresponds to a PLD with shrinking (expanding) hexagrams.