# Topological Insulators/Superconductors

In contrast to conventional ordered phases of matter, topological phases demonstrate numerous striking features that are independent of material details. The quantum Hall state (QHS) is the prototype of such novel topological phases where the bulk is gapped while the Hall conductance is precisely quantized to an integer multiple of conductance quantum $\frac{e^2}{h}$, manifestly independent of material parameters. Furthermore, the integer n ∈ ℤ that fully characterizes a QHS, can be interpreted in a twofold manner, (i) the number of filled Landau levels of the bulk, (ii) the number of chiral edge modes that are responsible for the quantized conductance. Such correspondence between boundary and bulk is the central feature of all topological phases where a topological invariant of the bulk is holographically related to the number of gapless fermionic modes on the boundary. In close relation to the QHS (time-reversal symmetry broken, TRB), there exist a quantum spin Hall state (QSHS) in 2d, protected by time-reversal symmetry (TRS) whose edge supports a 1d helical Dirac fermion liquid whereas the bulk is gapped. Yet, time-reversal invariant (TRI) insulators in 2d are classified by a ℤ2 invariant, indicating that there are only two types in general, and we designate the ones supporting helical edge modes as topological insulators while the other ones, the trivial. The massless Dirac fermions actually result from TRS-protected Kramer degeneracy but such degeneracy can be lifted if there are an even number of Dirac cones, which is the underlying reason why TRS protected insulators exhibit ℤ2 feature (even or odd). In analogy to QHS (ℤ) and topological insulators (ℤ2), in the weak pairing limit, topological superconductors were proposed to support chiral/helical Majorana fermion liquids on the boundary. The TRB p + ip chiral superconductors are analogues of QHS and labeled by ℤ while the TRI px ± ipy helical superconductors are counterparts of ℤ2 topological insulators.

Classification of topological matters are reflected by their holographic boundary states, however, the non-trivial topology of the bulk can also be exposed by introducing topological defects, π-flux, dislocations and vortices, to name a few. Moreover, the interplay of defect topology and topology of the original states result in even richer physics. Previous works have been done on the effect of π-flux and dislocations on the Haldane model which indeed trap zero energy bound states. Moreover, according to the index theorem, a vortex in topological superconductors is predicted to harness a Majorana fermion inside the vortex core. In turn, such vortices were found to be non-Abelian anyons which are the holy grail of topological quantum computation.

# Cold Atoms

The other play ground for theoretical physics is probably the ultracold atomic gases. On the one hand, such systems serve as quantum simulators of condensed matter physics and even high energy physics. Recent progresses have been made in artificial gauges. It makes it possible to realize strong "magnetic field" in an atomic gas which is analogous to Laughlin state in solids. Since the interactions between atoms is tunable, atoms in the trap may provide hints for the crossover of weak interaction to strong correlation and hence help qualitatively understand system of strong correlations. The more exotic non-Abelian artificial gauges are also the butterfly in a high energy physicist's stomach. On the other, dilute degenerate gases are exclusive for some exotic phases never observed in condensed matter. It is because the Fermi energy in a solid is way higher than that of a dilute quantum gas. Therefore, weak interactions like magnetic dipole interaction will come to play an important role in the latter.

# Field Theoretic Biology

Apart from the above, I am also interested in interpretation of life phenomena on the basis of emergence and tend to give a mathematical definition of "life" via many-body machinery. Such as path integral, linear response and renormalization group.