Topological insulators with curved interfaces, [1], [2], [3], [4];
Localization for random/quasiperiodic Schrodinger operators, [5], [6], [7], [8];
Anomalous spectral properties of Moire materials, [9], [10].
Topological insulators (TIs) refer to insulators with non-trivial topology, which can be mathematically characterized by a non-zero integer called the bulk index. When two insulators with distinct bulk indices are glued together, the material becomes a conductor -- edge currents emerge and propagate along the interface, and the edge conductance equals the difference between the distinct bulk indices when the edge is straight. This is the key property of TIs, called the Bulk-edge correspondence.
Topological edge spectrum along curved interfaces (with Alexis Drouot). Int. Math. Res. Not. IMRN (2023). Preprint
The bulk-edge correspondence for curved interfaces (with Alexis Drouot). Submitted. (2024).
Absolutely continuous spectrum for truncated topological insulators (with Alexis Drouot and Jacob Shapiro). Submitted. (2024).
Edge spectrum for truncated Z2-insulators (with Alexis Drouot and Jacob Shapiro) Preprint. (2025).
In [1], we prove the emergence of edge currents for TIs with curved interfaces, see Figure 1 below. In [2], we discover, for the first time, that the curved edge conductance for TI contains both geometric and electronic information that can be fully separated: it is a product of two integers, each representing its own aspect of the system. In [3], we prove that edge currents do transport to infinity under very mild geometric conditions. In [4], we proved emergence of edge spectrum for Z2-topological insulators with curved interfaces.
Figure 1: The first three cases admit the edge currents. The last one does not.
One key effect of randomness in condensed matter physics is that it often prohibits propagation of wave functions - this is known as localization for random Schrodinger operators, proposed by Anderson in his well-known paper ``more is Different''.
5. Large Deviations of the Lyapunov Exponent and Localization for the 1D Anderson Model (with S. Jitomirskaya). Comm. Math. Phys. 370, 311–324 (2019).
6. Localization for random CMV matrices. J. Approx. Theory Preprint (2021).
7. Anderson Localization for Schrödinger Operators with Monotone Potentials over Circle Homeomorphisms (with J. Kerdboon). J. Spectr. Theory Preprint(2023).
8. Dynamical localization for the singular Anderson model in Z^d (with N. Rangamani). To appear. J. Math. Phys. Preprint(2023).
In [5] we developed a non-perturbative method in proving localization for 1d random Schr\"odinger operators with arbitrary single-site distribution. The approach replaces the induction steps of multi-scale analysis by leveraging the positivity of the Lyapunov exponent and large deviation estimates. In [7], we prove Anderson localization for Schrodinger operators over a large class of circle maps with weakly Liouville rotation numbers, which was previously only known for irrational rotations. In [8], we build a general framework to prove strong dynamical localization for those energies where a multi-scale analysis result holds.
Figure 2: Hofstadter Butterfly.
Figure 3: Moire pattern.
When two materials are slightly mismatched, a larger-scale periodicity, known as the moir\'e pattern, emerges, see Figure 3. Moire materials have led to significant discoveries including unconventional super-conductance, anomalous fractional quantum Hall effect, and fractional quantum spin Hall effect, due to their rich, tunable phase transitions. Twisted bilayer graphene (TBG) is a special kind of Moire material that exhibits perfectly flat spectral bands at certain magic angles.
9. Magnetic response of twisted bilayer graphene (with S. Becker and J. Kim). Ann. Henri Poincaré Preprint (2022).
10. Spectral theory of twisted bilayer graphene in a magnetic field (with Simon Becker). Submitted. (2024).
In [9], we show that chiral potential plays a key role in preserving the perfectly flat bands when adding the magnetic field by studying the asymptotic expansion of the density of states in the semiclassical limit. In [10], we show that one can discriminate between flat bands of different multiplicities by turning on the magnetic field.