# Thermoelectricity Modeling with Cold Dipole Atoms in Aubry Phase of Optical Lattice
## 3 Results  
### Ground State Properties  
The equilibrium static atom positions are determined by the conditions \(\partial H/\partial x_{i}=0\), \(P_{i}=0\)[6; 8; 9]. In the approximation of nearest neighbor interacting atom, this leads to the symplectic map for recurrent atom positions \(x_{i}\)  
\[p_{i+1}=p_{i}+(K/3)g(x_{i})\,\ \ x_{i+1}=x_{i}+1/p_{i+1}^{1/4}\, \tag{2}\]  
where the effective momentum \(p_{i}=1/(x_{i}-x_{i-1})^{4}\) is conjugated to \(x_{i}\) and the kick function \(g(x)\) is such as \(Kg(x_{i})/3=-dV/dx|_{x=x_{i}}/3=-(K/3)\sin x_{i}\). This map description assumes only nearest neighbor interactions. Below, we show that it well describes the real situation with interactions between all the atoms. This is rather natural since the nearest neighbor approximation worked already well for ions with Coulomb interactions [8; 9], moreover for dipole atoms, the interactions drop even faster with the distance between atoms. As in [9; 15], the validity of the map (2) is checked numerically by finding the ground state configuration using numerical methods of energy minimization described in [6; 8] and taking into account the long range nature of the interactions between all the atoms. The obtained ground state positions, \(\{x_{1},\dots,x_{N}\}\), of the \(N\) atoms allows to recursively determine the \(N\) effective momenta, \(\{p_{1},\dots,p_{N}\}\), by inverting the second equation of the map (2), \(p_{i}=(x_{i+1}-x_{i})^{-4}\). Once obtained, the effective momenta can be used to compute successively \(N\) values \(\{g(x_{1}),\dots,g(x_{N})\}\) of the kick function \(g(x)\), by using the first equation of the map (2), \((K/3)\,g(x_{i})=p_{i}-p_{i+1}\). Figure 1 (top panels)shows that the map (2), obtained in the nearest neighbor approximation, is indeed valid since the \(\{g(x_{1}),\ldots,g(x_{N})\}\) values fall on the top of the function \(g(x)=\sin x\) analytically obtained from the equations of motion derived from the Hamiltonian (1). This means that the description of the positions of the \(N\) atoms in the ground state can be mapped on a dynamical problem of a fictitious kicked particle taking successively the positions, \(\{x_{1},\ldots,x_{N}\}\), and the momenta, \(\{p_{1},\ldots,p_{N}\}\). The dynamics of the fictitious kicked particle is governed by the map (2). Phase space portraits of the map (2) is given in Figure 1 (middle panels). A phase space portrait (green points and curves) is obtained by taking many different initial conditions \((x_{0},p_{0})\) and computing many of the corresponding successive phase space points \((x_{i},p_{i})\). In Figure 1 (middle panels), we show the phase space portrait for KAM phase (\(K=0.02<K_{c}\), left panel) and for the Aubry phase (\(K=0.1>K_{c}\), right panel). We observe that the ground state positions of the atoms (red circles) are located, for \(K=0.02\) (left panel), on the top of an invariant KAM curve which is a situation corresponding to a regular motion of the fictitious kicked particle, and, for \(K=0.1\) (right panel), in the chaotic component of the phase space portrait which is a situation corresponding to a chaotic motion of the fictitious kicked particle. All the chaotic dynamics concepts used in the article are well documented in the literature (see, for example, [3, 4]). Figure 1 (bottom panels) shows the hull function \(h(x)=(x_{i}+\pi)[\text{mod }2\pi]-\pi\) with \(x=(2\pi(i-1)/\
u+\pi)[\text{mod }2\pi]-\pi\). Indeed, for \(K\to 0\), i.e., for a vanishing optical potential, we have \(h(x)=x\). For \(K<K_{c}\), the hull function \(h(x)\) still follows this linear law with smooth deviations, while, for \(K>K_{c}\), we obtain the devil's staircase corresponding to the fractal cantori structure of the chaotic phase space. The transition from the smooth hull function, which is typical for a sliding chain in the KAM phase, to the devil's staircase, which is typical for a pinned chain in the Aubry phase, is visible in Figure 1 (bottom panels).