4D beyond-cohomology topological phase protected by C2C_2 symmetry and its boundary theories

We study bosonic symmetry protected topological (SPT) phases with $C_{2}$ rotational symmetry in four spatial dimensions which is not captured by the group cohomology classification. By using the topological crystal approach, we show that the topological crystalline state of this SPT phase is given by placing an $E_{8}$ state on the two dimensional rotational invariant plane, which provides a simple physical picture of this phase. Based on this understanding, we show that a variant of QED4 with charge-1 and charge-3 Dirac fermions is a field theoretical description of the three dimensional boundary. We also discuss the connection to symmetric gapped boundary with topological order and its anomaly signature.Comment: 6 pages, 1 figur

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

Development of Computer Vision-Enhanced Smart Golf Ball Retriever

An automatic vehicle system was developed to assist golfers in collecting golf balls from a practice field. Computer vision methodology was utilized to enhance the detection of golf balls in shallow and/or deep grass regions. The free software OpenCV was used in this project because of its powerful features and supported repository. The homemade golf ball picker was built with a smart recognition function for golf balls and can lock onto targets by itself. A set of field tests was completed in which the rate of golf ball recognition was as high as 95%. We report that this homemade smart golf ball picker can reduce the tremendous amount of labor associated with having to gather golf balls scattered throughout a practice field

Encountering Rigidity In Topological Land: From Topological Crystals To Fractons

Topological phases of matter are quantum phases with an energy gap. Due to the presence of the gap, many properties of a topological phase are stable against continuous and small deformations. This property makes a topological phase behaves more like a liquid since, as a vague image of liquids we all have, liquids are shapeless and freely to be deformed. In this thesis, we study two different kinds of topological phases where, surprisingly, rigidity plays distinct and fundamental roles. The first kind of topological phases is crystalline symmetry protected topological (SPT) phases. We show that all these symmetry protected topological states can be adiabatically deformed into a real-space crystalline pattern of lower-dimensional topological states, which we refer to as a topological crystal. We then develop a classification of these phases of matter in terms of topological crystals. This approach not only gives a classification scheme but also provides a clear physical picture for SPT phases protected by crystalline symmetry. The other kinds of topological phases are a novel class of gapped quantum phases in three spatial dimensions, dubbed fracton topological orders. These exotic phases host point-like excitations which are fundamentally immobile or which are confined to move only along sub-dimensional manifolds. We find several examples of fracton topological orders in which these excitations with restricted mobility are non-Abelian. We further show that these non-Abelian excitations are a fundamentally three-dimensional phenomenon instead of the results of trivial possibilities such as stacking of two-dimensional states.</p