Quantum lattice models from fusion categories

Thesis

Fusion categories offer a systematic framework for constructing quantum lattice models with non-invertible symmetries and dualities. This thesis examines the quantum phases and transitions realized in a specific 1+1d lattice model and develops these ideas further to study 2+1d models with topological order.
After a brief review of fusion categories, anyon chains, and string-nets in Chapter 2, Chapter 3 investigates a model of Rydberg-blockade atoms on the square ladder with at most one excited atom per square. Along an integrable line, it is equivalent to the 1+1d anyon chain built from the so(3)₂ fusion category and preserves a non-invertible self-duality symmetry that can be spontaneously broken. A duality mapping to the spin-1/2 XXZ chain reveals a critical line governed by the free boson orbifold CFT. A non-invertible remnant of the XXZ chain's U(1) symmetry applies to the full three-parameter space of couplings and distinguishes two different Z₃ ordered phases. Away from integrability, we use perturbation theory and DMRG to explore a rich phase diagram with several density-wave phases separated by critical and first-order transitions.
The second part of this thesis focuses on 2+1d fusion surface models, higher-dimensional extensions of anyon chains built from braided fusion categories. In Chapter 4, we develop systematic generalizations of Kitaev’s honeycomb model, which itself can be formulated as a fusion surface model built from the Ising category. These models exhibit categorical 1-form symmetries and have phase diagrams qualitatively similar to Kitaev’s, including a weakly coupled chains phase capable of hosting chiral topological order, as demonstrated in a Z₃ symmetric example. Chapter 5 shows that dualities between such models arise by refining the input data with module tensor categories, paralleling the construction of dual 1+1d anyon chains. The resulting 2+1d models support both 0-form and 1-form symmetries and are candidates for realizing symmetry-enriched topological phases. Taken together, these results point towards a broader framework for 2+1d quantum lattice models beyond commuting projectors, with Chapter 6 outlining connections to 3d criticality, quantum scars, and quantum error correction.

Authors: Eck, L

• DOI: 10.5287/ora-zozm87o5w
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: quantum criticality; categorical symmetries; topological order
• Subjects: Condensed matter