Charge-stripe phases versus a weak anisotropy of nearest-neighbor   hopping
  Recently, we demonstrated rigorously the stability of charge-stripe phases in quantum-particle systems that are described by extended Falicov-Kimball Hamiltonians, with the quantum hopping particles being either spinless fermions or hardcore bosons. In this paper, by means of the same methods, we show that any anisotropy of nearest-neighbor hopping eliminates the $\pi/2$-rotation degeneracy of the so called dimeric and axial-stripe phases and orients them in the direction of a weaker hopping. Moreover, due to the same anisotropy the obtained phase diagrams of fermions show a tendency to become similar to those of hardcore bosons. 
V. Derzhko, J. Jedrzejewski