Simple systems with only 3 or 4 interacting particles can behave in a bizarre, counter-intuitive manner. Examples to be discussed in this Colloquium include ultra-long-range Rydberg molecules with enormous electric dipole moments, as well as states of a few neutral particles that resonantly form a cluster. Moreover, insights from a few-body viewpoint can help to understand some of the rich many-body systems being actively explored, from the unitary Bose gas to the fermionic or bosonic flavors of the fractional quantum Hall effect.
Materials around us posses a space group symmetry once the presence of disorders and the finiteness of the system size are neglected. When this spatial symmetry is taken into account in addition to usual internal symmetries, it is natural to expect a richer structure of topological phases and, at the same time, stronger constraints on possible phases. In this talk, we will discuss our recent studies in this direction for noninteracting systems [1-3] and interacting systems [4,5].  H. C. Po, HW, M. P. Zaletel, and A. Vishwanath, Sci. Adv. 2, e1501782 (2016).  HW, H. C. Po, M. P. Zaletel, and A. Vishwanath, Phys. Rev. Lett. 117, 096404 (2016).  H. C. Po, A. Vishwanath, and HW, arxiv:1703.00911  HW, H. C. Po, A. Vishwanath, and M. P. Zaletel, PNAS 112, 14551 (2015).  H. C. Po, HW, C.-M. Jian, M. P. Zaletel, arxiv:1703.06882
In plain words chaos refers to extreme dynamical instability and unpredictability. Yet in spite of such inherent instability, quantum systems with classically chaotic dynamics exhibit remarkable universality. In particular, their energy levels often display the universal statistical properties which can be effectively described by Random Matrix Theory. From the semiclassical point of view this remarkable phenomenon can be attributed to the existence of pairs of classical periodic orbits with small action differences. So far, however, the scope of this theory has, by and large, been restricted to single-particle systems. I will discuss recent efforts to extend this program to a class of systems with a very large number of particles. The crucial ingredient of our approach is a two-dimensional symbolic dynamics which allows an effective representation of periodic orbits and their pairings. I will illustrate the theory with a specific model of coupled cat maps, where such a symbolic dynamics can be constructed explicitly.
RNA nanotechnology has the great potential to allow us to produce well-defined nanostructures and devices inside cells and thus open up a wide range of design opportunities in synthetic biology. To achieve this goal we need to understand the design principles of geometry, folding kinetics and topology that will allow us to genetically encode well-defined RNA nanostructures that self-assemble during the transcription process. We have recently introduced the single-stranded RNA origami method and validated the architecture by transcribing RNA tiles that assemble into lattices of different geometries. I will introduce new software tools that allow interactive design of RNA origami structures using a library of functional modules and new sequence design approaches that allow large structures to be designed. Also I will show our latest progress in developing larger three-dimensional RNA origami structures and functional RNA nanodevices with applications in biosensing and diagnostics.
Deconfined criticality is a concept that has emerged in recent years to describe quantum phase transitions beyond the Landau-Ginzburg paradigm. Its basic idea is that a continuous quantum phase transition between two different symmetry broken phases is generically possible, if it is driven by the proliferation of topological defects which carry quantum numbers related to the order parameter of the other phase. The prime example for a deconfined quantum critical point is the SU(2) Neel - valence bond solid (VBS) transition on the square lattice. In this talk I will describe a generalization to a deconfined quantum critical point in SU(3) antiferromagnets on the triangular lattice. Studying the critical theory from both sides by RG methods and analysis of the topological defects, one can provide strong evidence for a continuous phase transition, opposed to a naive Landau-Ginzburg expectation.