In this project, we examine the transport and optical properties of weakly doped Wigner crystals. For concreteness, we restrict ourselves to an effective two-dimensional tight-binding model on a triangular lattice with a screened Coulomb potential and fermions without internal degrees of freedom. Following the example of the material $1T-TaS_2$, we focus on average fillings of one thirteenth of an atomic orbital, as well as weak electron or hole doping. The state of the system is described using an inhomogeneous Hartree–Fock theory, while the optical and transport properties are calculated using linear response theory. In addition, we take into account the effects of thermal fluctuations and non-trivial configurations using the Metropolis–Hastings algorithm. We confirm the presence of a first-order phase transition at finite temperature between phases with and without long-range order. Furthermore, doping and the presence of domain walls have a non-trivial effect on the transport and optical properties of the material.
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