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We investigate the ground state of a one-dimensional lattice system thathosts two different kinds of excitations (species) which interact with apower-law potential. Interactions are only present between excitations of thesame kind and the interaction strength can be species-dependent. For the casein which only one excitation is permitted per site we derive a prescription fordetermining the ground state configuration as a function of the fillingfractions of the two species. We show that depending on the filling fractionscompatible or incompatible phases emerge. Furthermore, we discuss in detail thecase in which one species is strongly and the other one weakly interacting. Inthis case the configuration of the strongly interacting (strong) species can beconsidered frozen and forms an effective inhomogeneous lattice for the other(weak) species. In this limit we work out in detail the microscopic groundstate configuration and show that by varying the density of the weak species aseries of compatible--incompatible transitions occurs. Finally we determine thestability regions of the weak species in the compatible phase and compare itwith numerical simulations.

Crystalline structures in a one-dimensional two-component lattice gas with $1/r^{\alpha}$ interactions