Tintinnabuli

Research

This page presents my research interests, ideas, and path to you who are passionate about physics and math but are not necessarily canonically trained. For a more technical summary of my works see here (placeholder). You can find my publications on my Google Scholar page.

• Quantum magnetism.
  • Theory:

    While magnetism is an intrisinc quantum phenomenon on the microscopic scale, some magnets—what we call quantum magnets—behave far more quantum than others, to the extend that even the macroscopic behavior needs be understood using quantum mechanics.

    An extreme example is quantum spin liquid (QSL)—quantum paramagnetic ground states featuring excitations best described by an emergent gauge field.

    Out of many exotic properties of QSLs, symmetry fractionalization stands out, and serves as a powerful way to distinguish different QSLs (at least theoretically). This motivates our study of symmetric QSLs on prototypical lattices in 2D (PRB) and 3D (PRB1, PRB2, PRB3).

    Sometimes the ground state of a quantum magnet with full crystalline symmetry must be a QSL. A somewhat complete condition for this to happen (the "Lieb–Schultz–Mattis" theorem) has been enumerated by us, applicable to all 3D magnets. Check out our preprint for more details (and useful tables for crystallography!).

    QSLs (and quantum magnetism in general) are intimately related to quantum information, topological physics, non-equilibrium quantum dynamics, integrable systems, disordered systems, and even AI. Which of these branches would you like to embark on?

  • Experiment:

    The ongoing experimental search for QSLs identified many candidate materials. Here is neutron evidence for a disordered ground state in NaYbO2, possibly a U(1) QSL, and match with microscopic modeling (Nat. Phys.).

    Admittedly, experimental situation is far from the idealized theory land: the evidences for QSLs remain inconclusive, and alternative interpretations usually exist. Our experimental colleagues are working hard on making improvements, by synthesizing new magnetic compounds being and developing novel experimental techniques.

    Besides QSLs, magnetic orders can be interesting! See here (PRB) for the observation and theoretical undstanding of a field-induced commensurate-to-incommensurate transition in LiYbO2.

    You can find some of my works on triangular lattice magnets here (BYBO, NaYbO2, BEBO, NaYbSe2, CMAO). Note the intimate relation between the (layered 2D) triagular/kagome lattice and the (intrinsically 3D) pyrochlore lattice. Recent compounds bring excellent opportunity to unify the theory for these three notable frustrated lattices.

• Interacting electrons.

  • Electron is the protagonist of condensed matter physics. An electron is a wild horse—galloping at its own speed, with no will to be tamed. Yet, amazingly, at mesoscopic scale the herd of electrons sometimes march in unison, making the properties of a metal, a semiconductor, or an insulator understandable, measurable, and engineerable.

    Superconductivity has always been a major topic in the study of interaction electrons. Electrons pair up and condense, resulting in zero resistence and complete repulsion of magnetic field lines. We propose an interlayer pairing to understand the superconductivity of 4Hb-TaS2: see our paper (PRB).

    Many exotic electron phases trace their origin back to the instability of metals (or a "Fermi-liquid" in physic parlance). The instability of metal remains one of hardest questions in condensed matter. Recent advances (Son, Goldman...) are worth exploring. Some projects await!

    Outside the realm of (conventional) Fermi-liquid lie semimetals (Weyl, Dirac...) and flat band systems. All feature a versatile mathematical description and exotic transport properties. In presence of interactions, a flat band exhibits a reach phase diagram—as we observed in the Hubbard kagome system (PRB).

    Interacting electrons in an external magnetic field is another steady supply of exotic physics (quantum Hall ferromagnet, fractional quantum Hall/Chern insulators, even the fuzzy sphere construction for 3D CFT...). How can we combine the formal theoretical ideas and cutting edge numerics to make concrete predictions in materials and their universal classes?

• Topological matter and computational homotopy.
  • The physics:

    Topological phases of matter have properties invariant under finite deformations of the system, making topology the appropriate tool in describing them.

    Free fermions in crystals can have nontrivial band topology. A prototypical (but somewhat unconventional) example is a Hopf insulator (3D, two bands). A multiband Hopf insulator requires crystalline symmetry something we showed in our 2016 work ( PRB).

    Berry phase and Chern number are standard tools in characterizing 2D electron topology. For a (less trivial) application in twisted bilayer graphene with possible experimental relevance, see our 2019 papers ( PRB1, PRB2).

    The topological theory extends well beyond free fermions, and depending on what we focus on—topological aspects of symmetry, and/or topological aspects of entanglement—one arrives at different corners of the phase diagram for interacting topological matter: the so-called SPT phases, topological orders, and SET phases. Now we have a somewhat established notion for each of these phases, thanks to algebraic topology.

    Our attempt in connecting different corners of the topological phase diagram can be found here (PRB). We focused on examples in physical systems. For a more general theory, see papers by Thorngren et al.

  • The math:

    Coming to the math side, the theoretical framework behind topological phases is deep, abstract, and still being developed.

    Because calculations for general systems are quite involved, with both conceptual and technical challenges, we are far from obtaining the complete data of topological phases, and for this matter we need to develop new tools in computational topology.

    Recently, we have achieved in calculating the mod-2 cohomology ring of (all but No. 226 of) the 230 3D crystallographic space groups. You can find the paper, the code written in GAP, and the .nb file for the explicit cochain form of the cocycles.

    Our next step goal would be to work out the explicit data for fermionic crystalline band topology and SPTs, and compare with existing data (see e.g. Gu, Qi, Shiozaki). Tools are cohomology, K-theory, corbordism, and their equivariant versions. These tools have also found applications in quantum circuits—another interesting setup to explore.

• Statistical physics and combinatorics.

  • Random matrices, disordered systems, random geometry, representation theory...

    SYK...

Notes

• Ising model and beyond
• Lattice gauge theory
• Topology of electron bands
• Lie algebra (Humphreys, in Chinese, pdf)
• Solutions to problems in point set topology (尤承业, in Chinese, pdf)
• Differential forms in algebraic topology (Bott & Tu, pdf)
• Equivariant cohomology (Loring Tu, pdf)

Better notes

• Arun Debray's notes for math and physics
• Leon's lecture notes on spins and electrons
• Allen Hatcher's chapter 5 on spectral sequences
• Eckhard Meinrenken's short notes on equivariant cohomology
• Stephen Mitchell's notes on Serre fibration and notes on principal bundles