Domain-wall excitons and optical conductivity in one-dimensional Wigner lattices

Motivated by the recent finding that doped edge-sharing Cu-O chain compounds like Na$_3$Cu$_2$O$_4$ and Na$_8$Cu$_5$O$_{10}$ are realizations of one-dimensional Wigner crystals, we study the optical spectra of such systems. Charge excitations in 1D Wigner crystals are described in terms of domain wall excitations with fractional charge. We investigate analytically and numerically the domain-wall excitations that dominate the optical absorption, and analyse the dispersion and the parameter range of exciton states characteristic for the long-ranged Coulomb attraction between domain walls. Here we focus on the Wigner lattice at quarter-filling relevant for Na$_3$Cu$_2$O$_4$ and analyze in particular the role of second neighbor hopping $t_2$ which is important in edge-sharing chain compounds. Large $t_2$ drives an instability of the Wigner lattice via a soft domain-wall exciton towards a charge-density wave with a modulation period distinct from that of the Wigner lattice. Furthermore we calculate the temperature dependence of the DC-conductivity and show that it can be described by activated behavior combined with a $T^{-\alpha}$ dependent mobility.