# Photocontrollable Mixed-Valent States in Platinum-Halide Ladder Compounds
## 3 Competing Mixed-Valent States and Their Optical Properties  
Various possible electronic states of the \(MX\) ladders were revealed through a group-theoretical analysis [27]. When \(M=\text{Pt}\), electron-lattice couplings cause dimerization of the halogen sublattice, which is accompanied by a CDW on the metal sublattice. The interchain valence arrangement is either IP or OP, as shown in Figure 2a. In the strong-coupling limit, we have a difference in energy between OP CDW and IP CDW within the single-band approximation [29],  
\[\frac{E_{\text{OP}}-E_{\text{IP}}}{N}=2(2V_{M\!M}^{\times}-V_{M\!M}^{\perp})- \frac{2\left(t_{M\!M}^{\perp}\right)^{2}}{4\beta_{M}u-U_{M}+V_{M\!M}^{\perp}+4 (V_{M\!M}^{\parallel}-V_{M\!M}^{\times})}, \tag{5}\]  
where the energy gain by the interchain electron hopping is taken into account within the second-order perturbation scheme, while \(u\equiv|u_{n:lX}-u_{n:lM}|=|u_{n+1:lM}-u_{n:lX}|\) indicates the bond variable in isolated \(MX\) chains. The first term on the right-hand side of Equation (5) corresponds to the naivest Coulomb energy difference between OP CDW and IP CDW. In the absence of Coulomb interactions, the energy of OP CDW is always lower than that of IP CDW by the energy gain \(\propto(t_{M\!M}^{\perp})^{2}\). Figure 2b shows ground-state phase diagrams of the present model [Equation (1)] in the moderate-coupling regime at the sufficiently low temperature \(k_{\text{B}}T/t_{M\!X}^{\parallel}=0.001\), which are numerically obtained. The lattice displacements are adiabatically treated and numerically determined to minimize free energy. Considering that Coulomb force depends on electron-electron distance in general, the condition \(V_{M\!M}^{\perp}>V_{M\!M}^{\times}\) is suitable for the lattice structure. The obtained phase boundaries are in this region and agree well with the above simple calculation Equation (5). Let us observe polarized optical conductivity spectra of the IP-CDW and OP-CDW ground states,  
\[\sigma^{\gamma}(\omega)=\frac{\pi}{N\omega}\sum_{\lambda}|\langle E_{\lambda}| \mathcal{J}^{\gamma}|E_{0}\rangle|^{2}\delta(E_{\lambda}-E_{0}-\hbar\omega). \tag{6}\]  
\(|E_{\lambda}\rangle\) is a one-electron-hole pair excited state from \(|E_{0}\rangle\) corresponding to either OP CDW or IP CDW, and the current-density operators parallel (\(\gamma=\parallel\)) and perpendicular (\(\gamma=\perp\)) to the leg direction are defined by  
\[ \begin{array}{ll}\mathcal{J}^{\parallel}&=\frac{i\alpha c}{\hbar}\sum\limits _{n,l,s}\left\{t_{\text{MX}}^{\parallel}+\alpha(u_{n:lX}-u_{n+1:lM})]c_{n+1:lM s}^{\dagger}c_{n:lXs}\right.\\ &\left.+\left[t_{\text{MX}}^{\parallel}-\alpha(u_{n:lX}-u_{n:lM})]c_{n:lXs}^{ \dagger}c_{n:lMs}-\text{H.c.}\right\},\\ &\mathcal{J}^{\perp}&=\frac{i\alpha c}{\hbar}\sum\limits_{n,s}(t_{\text{MM}}^{ \perp}c_{n:2 Ms}^{\dagger}c_{n:1 Ms}-\text{H.c.}),\end{array}  \tag{7}\]  
where the \(M\)-\(X\) distance along the leg and the \(M\)-\(M\) one along the rung are taken as \(\bar{a}=a/2\) and \(c=a\), respectively. In Figure 2c, \(\sigma^{\parallel}\) shows a single-peaked structure in OP CDW at each \(t_{\text{MM}}^{\perp}\), whereas it shows a shoulder-like one in IP CDW at \(t_{\text{MM}}^{\perp}/t_{\text{MX}}^{\parallel}=0.4\). In OP CDW, an electron transition from the highest-occupied molecular orbital (HOMO) to the lowest-unoccupied molecular orbital (LUMO) contributes to the major optical absorption (see the solid arrow in the upper panel of Figure 2d). In IP CDW, on the other hand, a HOMO-to-LUMO electron transition is optically forbidden and another two excitation modes predominantly contribute to the optical absorption (see solid arrows in the lower panel of Figure 2d). These transition energies are different from each other due to electron-hole symmetry breaking [29], which becomes obvious with increasing \(t_{\text{MM}}^{\perp}\). \(\sigma^{\perp}\) is significantly affected by an interchain valence arrangement. In IP CDW, \(\sigma^{\perp}\) shows no absorption because electrons on metal ions have no chance of hopping along the rung.