Non-Abelian anyon proliferation and self-dual quantum doubles

This work brings together quantum information, high-energy theory, and mathematics to develop a unified framework for non-Abelian anyon proliferation in the non-Abelian Fradkin-Shenker model. We construct the first non-Abelian generalization of the Wen plaquette model: a self-dual lattice realization of the S₃ quantum double, 𝒟(S₃), in which lattice translation exchanges a non-Abelian electric charge and magnetic flux. The model supports sign-problem-free, symmetry-preserving anyon-proliferation deformations, enabling large-scale quantum Monte Carlo studies. Its zigzag boundary is pinned by duality—without fine-tuning—to the tetracritical Ising conformal field theory.

By gauging the electric–magnetic permutation symmetry, we map the lattice transitions to concrete non-Abelian Chern–Simons–Higgs theories. We also formulate a minimal-condensation principle connecting anyon-proliferation dynamics to the categorical structure of condensable algebras, and use Frobenius–Schur indicators to determine the discrete-torsion data of the associated metaplectic modular tensor categories. Extending these results to an infinite family of self-dual dihedral quantum doubles establishes a concrete bridge among lattice models, quantum field theory, and category theory, and shows how generalized symmetry and duality can identify and constrain non-Abelian self-dual multicriticality.

Read arXiv:2608.05294.

Non-invertible symmetry, self-duality, and quantum criticality

A central theme of my research is to turn non-invertible symmetry—whose symmetry operators obey fusion rules rather than group multiplication—into a practical framework for understanding quantum phases and critical phenomena. In Self-duality under gauging a non-invertible symmetry, we constructed the first explicit CFT self-duality obtained by gauging an entire non-invertible symmetry. Along the c = 1 orbifold critical line, gauging Rep(H₈) maps the radius as R ↔ 2/R, leaving the Ising² CFT at R = √2 invariant. Gauging on only half of space produces a new defect that acts as an exact non-invertible symmetry at this fixed point. We determined the fusion rules and associativity data of the enlarged fusion category and showed that it forbids all relevant and marginal symmetry-preserving perturbations, placing unusually strong constraints on the critical theory.

To explain the bulk origin of this self-duality, in SymSETs and self-dualities under gauging non-invertible symmetries we reformulated the problem as a bulk–boundary correspondence. The symmetry of the (1+1)-dimensional system is encoded by a (2+1)-dimensional topological order, while the gauging duality appears as an additional ℤ₂ symmetry of the bulk. The resulting symmetry-enriched topological orders, which we call SymSETs, organize the possible dualities through familiar data such as anyon permutations, symmetry fractionalization, and discrete torsion. Unexpectedly, we found that changing the fractionalization pattern can modify the fusion rules of the boundary duality defects themselves. Thus, bulk symmetry enrichment directly controls both the algebra of critical defects and the possible SPT phases at the boundary.

We then developed a microscopic lattice realization of these ideas in Generalized Kramers-Wannier Self-Duality in Hopf-Ising Models. Using a Hopf-algebraic extension of ZX calculus—a diagrammatic language for tensor networks—we constructed quantum spin chains on ordinary tensor-product Hilbert spaces with explicit matrix-product symmetry operators and gauging maps. When the underlying Hopf algebra is self-dual, gauging becomes a generalized Kramers–Wannier transformation that exchanges ordered and disordered phases. For the Kac–Paljutkin algebra H₈, exact diagonalization, DMRG, and VUMPS reveal four gapped phases, Ising critical lines, and a candidate multicritical point. We also constructed fixed-point Hamiltonians for all six Rep(H₈)-symmetric gapped phases, connecting the abstract classification directly to microscopic models and numerical calculations.

We further developed practical ways to recognize non-invertible SPT phases in lattice systems. In our work on strange correlators and string order parameters, we introduced three complementary diagnostics analogous to those used for conventional SPT phases: strange correlators, symmetry-decorated string order, and symmetry-enforced entanglement-spectrum degeneracies for on-site realizations. We then asked whether every non-invertible SPT is simply a conventional symmetry phase written in a different duality frame. In subsequent work, we showed that the answer is no by constructing an intrinsic non-invertible SPT that cannot be transformed by gauging into any gapped phase protected by ordinary symmetry. Complementarily, our study of non-invertible symmetry breaking identifies the broad class for which such a dual description does exist: under precise conditions, certain non-invertible order–disorder transitions map to deconfined quantum critical points between ordinary symmetry-broken phases with incompatible unbroken subgroups. Together, these results clarify when non-invertible symmetry reveals genuinely new phases and when it provides a new description of familiar beyond-Landau physics.

Read arXiv:2310.19867, arXiv:2501.07787, arXiv:2602.10183, arXiv:2505.00673, arXiv:2511.01965, and arXiv:2605.27672.

Duality and multicritical point in deconfined quantum criticality

Duality can do more than relate equivalent descriptions of a system: it can constrain phase diagrams and predict new critical behavior. We showed that self-duality can protect a multicritical point at which a continuous easy-plane antiferromagnet–VBS deconfined quantum phase transition gives way to a first-order transition. We identified its distinct field theory and traced the first-order behavior to microscopic staggered dimer–dimer interactions. The broader nearby-critical and multicritical picture later received support from high-pressure NMR experiments on SrCu₂(BO₃)₂.

We then generalized this perspective to systems with two conserved U(1) charges by developing an Sp(4, ℤ) duality web that systematically organizes symmetry-breaking, invertible, and Abelian topological phases. We identified an order-five transformation that suggests the possibility of a multicritical point cycling phases in a five-element orbit. Applied to bilayer fractional quantum Hall systems, this framework generates new hierarchy constructions and candidate even-denominator states, showing how abstract duality principles can organize experimentally relevant quantum phases.

Read arXiv:2104.05147 and arXiv:2607.20622.

Fermi-surface anomalies and symmetric mass generation with application to nicklate superconductivity

Fermi surfaces are remarkably robust despite being gapless and lacking a conventional order parameter. We understood this stability through quantum anomalies and developed a universal classification of Fermi-surface anomalies using interacting fermionic symmetry-protected topological phases in phase space. A nonzero anomaly imposes a nonperturbative constraint on the allowed low-energy phases: it obstructs symmetrically gapping the Fermi surface into a featureless state.

When anomalies from multiple Fermi surfaces cancel, this anomaly obstruction is removed. We constructed (1+1)D and (2+1)D models in which nonperturbative interactions gap the entire Fermi surface without symmetry breaking, fermion-bilinear condensation, or topological order—a mechanism known as symmetric mass generation. The (1+1)D construction also provides a one-dimensional lattice regularization of the anomaly-free 3–4–5–0 chiral-fermion model.

We subsequently applied this principle to pressurized La₃Ni₂O₇. In a minimal bilayer model, anomaly cancellation permits a featureless symmetric-mass-generation Mott insulator; doping it yields s-wave interlayer spin-singlet superconductivity, while tuning J/t connects the BCS and BEC regimes. This provides a concrete theoretical link between anomaly constraints, correlated insulators, and high-temperature superconductivity.

Read arXiv:2302.12731, arXiv:2210.16304, and arXiv:2308.11195.