# Thermoelectricity Modeling with Cold Dipole Atoms in Aubry Phase of Optical Lattice
## 2 Methods  
The chain of atoms with magneto-dipole interactions in 1D periodic potential is described by the Hamiltonian:  
\[H = \sum_{i=1}^{N}\left(\frac{p_{i}^{\,2}}{2}+V(x_{i})\right)+U_{I}\,\] \[V(x_{i}) = -K\cos x_{i}\,\qquad U_{I}=\sum_{i>j}\frac{U_{dd}}{\mid x_{i}-x_{j} \mid^{3}} \tag{1}\]  
Here \(x_{i},P_{i}\) are conjugated coordinate and momentum of atom \(i\), and \(V(x)\) is an external periodic optical potential of amplitude \(K\). The magneto-dipole-dipole interactions are given by the \(U_{I}\propto U_{dd}=\mu^{2}/(\ell/2\pi)^{3}\) term with \(\mu\approx 10\mu_{B}\) (for \({}^{164}\)Dy atoms) and \(\mu_{B}\) is the Bohr magneton (we assume all magnetic momenta to be polarized) [25]. The Hamiltonian is written in dimensionless units where the lattice period is \(\ell=2\pi\) and atom mass is \(m=1\). In the following we also take \(U_{dd}=1\) so that \(K\) in (1) is the dimensionless amplitude of the periodic potential and thus the physical interaction is \(U_{A}=U_{dd}K\). In these atomic-type units, the physical system parameters are measured in units of \(r_{a}=\ell/2\pi\) for length, and of \(e_{a}=\mu^{2}/r_{a}{}^{3}=U_{dd}\) for energy. For \(\ell=1\)\(\upmu\)m the dimensionless temperature \(T=1\) (or \(k_{B}T\)) corresponds to the physical temperature \(T=\varepsilon_{a}/k_{B}=100\mu_{B}{}^{2}/(k_{B}(\ell/2\pi)^{3})\approx 25\) nK (nano-Kelvin) at \(\mu=10\mu_{B}\). The thermoelectric properties of model (1) are determined in the framework of the Langevin approach [9; 15] with the equations of motion: \(\dot{P}_{i}=\dot{v}_{i}=-\partial H/\partial x_{i}+f_{dc}-\eta P_{i}+g\xi_{i}( t)\), \(\dot{x}_{i}=P_{i}=v_{i}\). Here \(\eta\) describes phenomenologically the dissipative relaxation processes, and the amplitude of Langevin force \(g\) is given by the fluctuation-dissipation theorem \(g=\sqrt{2\eta T}\); the random variables \(\xi_{i}\) have normal distribution being \(\delta-\)correlated, \(v_{i}\) is atom velocity, \(f_{dc}\) is a static force applied to atoms. As in [9; 15], we use \(\eta=0.02\), the results being not sensitive to this quantity. The computations of \(S\) are done as it is described in [9; 15]. At fixed temperature \(T\), we apply a static force \(f_{dc}\) which creates an energy (voltage) drop \(\Delta V=f_{dc}2\pi L\) and a gradient of atom density \(\
u(x)\) along the chain with \(L\) potential periods and \(N\) atoms. Then, for \(f_{dc}=0\) within the Langevin equations, we impose a linear gradient of temperature \(\Delta T\) along the chain, and in the stabilized steady-state regime, we determine the charge density gradient of \(\
u(x)\) along the chain (see e.g., Figure 2 in [15]). The data are obtained in the linear regime of relatively small \(f_{dc}\) and \(\Delta T\) values. Then, the Seebeck coefficient, \(S=\Delta V/\Delta T\), is computed using values of \(\Delta V\) and \(\Delta T\) for which the density gradient obtained from the application of a voltage \(\Delta V\) compensates the one obtained from the application of a gradient of temperature \(\Delta T\). We used the computation times up to \(t=10^{8}\) to achieve the relaxation of the chain and to reach the required statistical accuracy. We assume that the cold atoms are in contact with an external environment which is able to play the role of thermostat (e.g., residual gas). In order to compute Seebeck coefficient, we need a temperature gradient along the chain. In the Langevin equation, we can impose that the temperature \(T\) is a function of the atom position \(x\) along the chain, \(T=T(x)=T_{0}+gx\), where \(T_{0}\) is the average temperature and \(g=dT/dx\) is a small temperature gradient. In cold atom experiments, such a temperature gradient can be setup by multiple laser beams generating a zero average fluctuating force \(f(x)\). The average of the square of the force, \(f^{2}(x)\), should change linearly along the chain.