Current Research Directions

Our group uncovered that band degeneracies in crystals can exhibit non-Abelian topological characteristics beyond the usual Abelian (commutative) invariants like quantized Berry phases and Chern numbers. In particular, we showed that pairs of Dirac points or Weyl points can braid around each other in momentum space and that nodal lines can form robust multiband nodal links. Correspondingly, we reported concrete crystalline solids (such as ZrTe, Li₂NaN, and elemental scandium) whose band structures illustrate these non-Abelian phenomena. More recently, we further investigated crystalline constructions dubbed Cayley-Schreier lattices that enable experimental emulation of crystal structures with non-Abelian gauge fields in real space associated with projectively represented space group symmetry.

Non-Abelian band topology in noninteracting metals, Science 365, 1273–1277 (2019) 

Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nat. Phys. 16, 1137–1143 (2020)

From triple-point materials to multiband nodal links Phys. Rev. B 103, L121101 (2021)

Topological non-Abelian Gauge Structures in Cayley-Schreier Lattices, arXiv:2509.25316 (2025)

We explore topological and correlated phases in hyperbolic lattices: regular arrangements of connected sites with emergent negative curvature. Our work significantly expanded the current understanding of band theory in hyperbolic lattices, finding that a two-dimensional crystal in hyperbolic space effectively behaves as if it had higher-dimensional momentum space. We utilized these findings to formulate concrete models of hyperbolic topological insulators, semimetals, and even spin liquids. To make the tools of hyperbolic band theory available to a broader community, we automatized and implemented the group-theoretical reasoning behind the technique in a pair of software packages named HyperCells and HyperBloch (detailed information here). We have also simulated hyperbolic matter experimentally using electric circuits, creating a "hyperbolic drum" with Dirichlet boundary condition as well as a "hyperbolic graphene" without boundaries. As a proxy for a negatively curved space that allows analytic treatment, we also investigated related phenomena on Cayley trees. 

Simulating hyperbolic space on a circuit board, Nat. Commun. 13, 4373 (2022)

Hyperbolic Topological Band Insulators, Phys. Rev. Lett. 129, 246402 (2022)

Hyperbolic matter in electrical circuits with tunable complex phases, Nat. Commun. 14, 622 (2023)

Hyperbolic Spin Liquids, Phys. Rev. Lett. 135, 076604 (2025)

Beyond conventional “stable” topological insulators, we investigate fragile and delicate topological phases that exist only in crystalline systems with a finite number of occupied bands or with a finite-dimensional Hilbert space. In particular, delicate topological insulators are Wannierizable (admitting a band representation through symmetric localized orbitals) but their Wannier orbitals necessarily span multiple unit cells. In addition, they are typically associated with robust boundary states only if the boundaries constitute a sharp termination of the bulk tight-binding Hamiltonian. Simple examples of delicate topology are embodied by the Hopf insulator and the returning Thouless pump. Recently, we expanded the notion of delicate topology from Bloch energy bands to Wannier polarization bands, which promotes the topological in-gap states to higher-order boundaries.

Geometric approach to fragile topology beyond symmetry indicators, Phys. Rev. B 102, 115135 (2020) 

Multicellularity of delicate topological insulators, Phys. Rev. Lett. 126, 216404 (2021) 

Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond, Phys. Rev. B 106, 075124 (2022)

Delicate Wannier insulators, arXiv:2506.05179 (2025)

We extend topological physics to non-Hermitian systems, i.e., those effectively described by Hamiltonians with loss or gain. In this realm, we introduced the concepts of Alice strings (one-dimensional defects that flip the sign of chiral charges upon braiding) and exceptional topological insulator (three-dimensional topological phase with a single exceptional point in the surface Brillouin zone). We also used homotopic techniques to investigate eigenvalue topology captured by braid groups, and to predict a new type of band singularities termed Hopf exceptional points. Another focus is the non-Hermitian skin effect, where bulk states under open boundary condition localize at the system boundary; for example, we found that skin states on a Cayley tree are characterized by multifractal scaling behavior. 

Homotopy characterization of non-Hermitian Hamiltonians, Phys. Rev. B 101, 205417 (2020)

Exceptional Topological Insulators, Nat. Commun. 12, 5681 (2021)

Symmetry breaking and spectral structure of the interacting Hatano-Nelson model, Phys. Rev. B 106, L121102 (2022)

Hopf Exceptional Points, arXiv:2504.13012 (2025)