## 3 Lattice Degree of Freedom  
To discuss cooperative phenomena, we now generalize our model and treat the lattice distortion \(x\) as a dynamical variable. We discuss its statics in this section and shall do its dynamics in Section 4.  
We parametrize the elastic energy per two-hexagram unit cell accompanied by a PLD as  
\[V_{\text{lat}}(x)=\frac{K_{2}}{2}x^{2}+\frac{K_{3}}{3}x^{3}+\frac{K_{4}}{4}x^ {4}, \tag{7}\]  
where \(K_{2},K_{3}\), and \(K_{4}\) are positive parameters so that \(V_{\text{lat}}(x)\) is convex. Equation (7) is a Taylor series in \(x\), which starts from \(x^{2}\) since \(x=0\) represents the equilibrium position at high temperatures. We take account of the \(x^{3}\) term, since the asymmetry between \(x<0\) and \(x>0\), or shrinking and expanding hexagrams, become nonnegligible for larger \(x\).  
We set the parameters \(K_{2},K_{3}\), and \(K_{4}\) so as to satisfy the following three conditions: (i) the CDW phase appears at \(T\sim 500\,\)K, (ii) \(x=-1.5\) in equilibrium at \(T=0\,\)K, and (iii) the free energy difference between the CCDW (\(x=-1.5\)) and the uniform (\(x=0\)) states is not very large, or typically less than \(1\,\)eV at \(T=0\,\)K per two-hexagram unit cell. Here, the free energy means the sum of \(V_{\text{lat}}(x)\) and the electronic free energy \(F_{\text{el}}(x)\) calculated for \(\hat{H}_{\text{TB}}(x)+\hat{H}_{\text{int}}\) within the mean-field approximation (6). The condition (i) determines \(K_{2}\), and then the conditions (ii) and (iii) do \(K_{3}\) and \(K_{4}\). We have chosen \(K_{2}=12.1\,\)eV, \(K_{3}=5.44\,\)eV, and \(K_{4}=2.54\,\)eV.  
Figure 4a shows the electronic energy \(F_{\text{el}}(x)\) at \(T=0\,\)K and \(V_{\text{lat}}(x)\) as functions of \(x\). We note that the free energy coincides with the energy at \(T=0\,\)K. The PLD decreases \(F_{\text{el}}(x)\) and increases  
Figure 3: Charge (**a**) and spin (**b**) densities in two-hexagram unit cell in presence of both PLD (\(x=-1.5\)) and Coulomb interaction (\(U=3J=0.816\,\)eV) calculated within mean-field approximation at \(T=0\,\)K. The charge–density distributions of the two hexagrams are identical, whereas their spin–density distributions have opposite signs.  
\(V_{\rm lat}(x)\) in both directions \(x<0\) and \(x>0\). The competition between these gain and cost results in the minimum of the total free energy \(F_{\rm el}(x)+V_{\rm lat}(x)\) at \(x=-1.5\) as shown in Figure 4b. There also exists a very shallow local minimum at \(x\sim 0.2\) corresponding to an expanding hexagram. At higher temperatures above \(\sim\)800 K, the total free energy \(F_{\rm el}(x)+V_{\rm lat}(x)\) has only one minimum at \(x=0\) and the CDW is not present. We note that this transition temperature is probably overestimated since we have performed the mean-field approximation (6) and neglected phonon fluctuations.  
We make a remark on the simplification that we have made on the PLD. We have treated one specific mode of the hexagram shape in lattice distortion, but the distortion has many more modes. In fact, 1T-TaS\({}_{2}\) shows an incommensurate CDW below 540 K and a nearly commensurate CDW below 350 K, which cannot be described within the present model. Thus, we do not further discuss the free energy at intermediate temperatures, and shall focus on the CCDW state thus obtained at \(T=0\) K.