A quasi-isometry between the universal covers of n-dimensional special cube complexes preserves n-dimensional flats up to finite Hausdorff distance. From this fact we can define an intersection complex of the cover, which turns out to be a quasi-isometry invariant. In this talk, I will introduce an isometry between intersection complexes induced from the quasi-isometry and talk about its applications to 2-dimensional right-angled Artin groups and graph 2-braid groups.