Topological phases protected by point group symmetry

We consider symmetry protected topological (SPT) phases with crystalline point group symmetry, dubbed point group SPT (pgSPT) phases. We show that such phases can be understood in terms of lower-dimensional topological phases with on-site symmetry, and can be constructed as stacks and arrays of these lower-dimensional states. This provides the basis for a general framework to classify and characterize bosonic and fermionic pgSPT phases, that can be applied for arbitrary crystalline point group symmetry and in arbitrary spatial dimension. We develop and illustrate this framework by means of a few examples, focusing on three-dimensional states. We classify bosonic pgSPT phases and fermionic topological crystalline superconductors with $Z_2^P$ (reflection) symmetry, electronic topological crystalline insulators (TCIs) with ${\rm U}(1) \times {Z}_2^P$ symmetry, and bosonic pgSPT phases with $C_{2v}$ symmetry, which is generated by two perpendicular mirror reflections. We also study surface properties, with a focus on gapped, topologically ordered surface states. For electronic TCIs we find a $Z_8 \times Z_2$ classification, where the $Z_8$ corresponds to known states obtained from non-interacting electrons, and the $Z_2$ corresponds to a "strongly correlated" TCI that requires strong interactions in the bulk. Our approach may also point the way toward a general theory of symmetry enriched topological (SET) phases with crystalline point group symmetry.Comment: v2: Minor changes/additions to introduction and discussion sections, references added, published version. 21 pages, 11 figure

Bulge formation from SSCs in a responding cuspy dark matter halo

We simulate the bulge formation in very late-type dwarf galaxies from circumnuclear super star clusters (SSCs) moving in a responding cuspy dark matter halo (DMH). The simulations show that (1) the response of DMH to sinking of SSCs is detectable only in the region interior to about 200 pc. The mean logarithmic slope of the responding DM density profile over that area displays two different phases: the very early descent followed by ascent till approaching to 1.2 at the age of 2 Gyrs. (2) the detectable feedbacks of the DMH response on the bulge formation turned out to be very small, in the sense that the formed bulges and their paired nuclear cusps in the fixed and the responding DMH are basically the same, both are consistent with $HST$ observations. (3) the yielded mass correlation of bulges to their nuclear (stellar) cusps and the time evolution of cusps' mass are accordance with recent findings on relevant relations. In combination with the consistent effective radii of nuclear cusps with observed quantities of nuclear clusters, we believe that the bulge formation scenario that we proposed could be a very promising mechanism to form nuclear clusters.Comment: 27 pages, 11 figures, accepted for publication in Ap