# Thermoelectricity Modeling with Cold Dipole Atoms in Aubry Phase of Optical Lattice
## 1 Introduction  
The phenomenon of Aubry transition describes the transport properties of a chain of particles linked by linear springs in a periodic potential. At a small potential amplitude the chain slides freely while above a certain potential amplitude it is pinned by the potential. This system is known as the Frenkel-Kontorova model [1]. The transition takes place at a fixed incommensurate density of particles per period \(\
u\). In fact the equilibrium positions of particles are described by the Chirikov standard map [2; 3; 4] that represents a cornerstone model of systems with dynamical chaos and symplectic maps. Indeed, a variety of physical systems can be locally described by this map [5]. In the frame of map description a density of particles corresponds to a winding number of an invariant Kolmogorov-Arnold-Moser (KAM) curve. Such curves cover the main part of the phase space at small potential amplitudes (small kick amplitudes \(K\) of the map). At large amplitudes the main part of phase space becomes chaotic and the KAM curves are transformed into cantori invariant sets which correspond to the chain ground states at a given density, as it was proved by [PERSON] [6]. In addition to nontrivial mathematical properties, the Frenkel-Kontorova model represents a fundamental interest for incommensurate crystals of interacting particles [7]. An experimental realization of linear spring interactions between particles is not very realistic. Thus in [8] it was proposed to consider a chain of Coulomb charges placed in a periodic potential. It was shown that this system of a one-dimensional (1D) Wigner crystal in a periodic potential can be locally described by the Chirikov standard map and the Frenkel-Kontorova model. Thus the Aubry-liketransition from the sliding KAM phase to the Aubry pinned phase takes place at a certain critical potential amplitude \(K_{c}\). The dependence of \(K_{c}\) on the density \(\
u\) is obtained by a local description in terms of the Chirikov standard map [8; 9]. For an experimental realization of the Aubry-like transition it was proposed to use cold ions placed in both a periodic potential and a global harmonic trap [8]. The experimental studies of such a system had been started in [10; 11]. The first signatures of the Aubry transition have been reported by the [PERSON] group with up to five ions [12]. The chains with a larger number of ions are now under investigations in [13; 14]. A significant interest for a Wigner crystal transport in a periodic potential is related to the recent results showing that the Aubry phase is characterized by remarkable thermoelectric properties with high Seebeck coefficient \(S\) and high figure of merit \(ZT\)[9; 15]. The fundamental grounds of thermoelectricity had been established in far 1957 by [PERSON] [16; 17]. The thermoelectricity is characterized by the Seebeck coefficient \(S=-\Delta V/\Delta T\) (or thermopower). It is expressed through a voltage difference \(\Delta V\) compensated by a temperature difference \(\Delta T\). Below we use units with a charge \(e=1\) and the Boltzmann constant \(k_{B}=1\) so that \(S\) is dimensionless (\(S=1\) corresponds to \(S\approx 88\)\(\upmu\)V/K (microvolt per Kelvin)). The thermoelectric materials are ranked by a figure of merit \(ZT=S^{2}\sigma T/\kappa\)[16; 17] where \(\sigma\) is the electric conductivity, \(T\) the temperature, and \(\kappa\) the material thermal conductivity. At present, the request for efficient energy usage stimulated extensive investigations of various materials with high characteristics of thermoelectricity as reviewed in [18; 19; 20; 21; 22]. The request is to find materials with \(ZT>3\) that would allow an efficient conversion between electrical and thermal forms of energy. The best thermoelectric materials created till now have \(ZT\approx 2.6\). At the same time, the numerical modeling reported for a Wigner crystal in the Aubry phase reached values \(ZT\approx 8\)[9; 15].