Topological phases of matter are particularly intriguing because they are robust to perturbations, such as thermal fluctuations and impurities, and exhibit emergent properties not affected by specific details such as the purity or the geometry of the system. This makes them especially promising for applications in quantum computing. Current quantum computers are limited by decoherence, where even minor perturbations can erase the quantum properties of the system. By contrast, topological phases are inherently resistant to such perturbations, making them a promising avenue for the development of stable, large-scale quantum computers.
There are many other theoretical reasons to study topological phases, one being that the study of topological phases has led to many new understandings about phase transitions. Traditionally, phase transitions were understood through the Landau-Ginzburg-Wilson paradigm, which heavily relied on symmetry analysis. However, our exploration of topological phases has revealed that there is more to phase transitions than meets the eye, prompting us to redefine the very concept of symmetry in this context
There are two principal approaches to understanding topological phases. Since topological properties are invariant under “zooming in/out”, it is natural to start with a lattice system at infinite volume. The mathematical properties of infinite volume systems are quite rich and are described by operator algebras. An important object in this study is the algebra of quasi-local observables. This approach lends itself quite nicely to study the properties of definite microscopic systems and describe the many features of topological phases such as understanding how certain features remain stable when transitioning from a microscopic to a macroscopic perspective.
One may also study topological phases using first principles, and forgetting any system details altogether and simply relying on the macroscopic properties of topological phases. This is the approach taken by category theorists, and the resulting structures describing topological phases are given by (higher) braided monoidal categories. This approach has been quite successful in understanding the general properties of and giving a classification of topological phases.
A natural question then is to ask if these two approaches are equivalent. The answer is an affirmative, at least in the 2+1D gapped topological order. This programme has primarily been headed by the various works of Pieter Naaijkens and Yoshiko Ogata, adapting the Axiomatic-QFT approach to the lattice. Though very successful so far, the programme is not yet finished, and there are many avenues left to explore.
So far my research has been in building the dictionary between the two approaches given above. My principle contributions to this programme are two-fold:
First is the incorporation of symmetry enrichment into the study of topological phases. It's heuristically believed that in the presence of a symmetry, topological phases exhibit an even richer structure than that of braided tensor categories, since the action of the symmetry group can non-trivially permute anyons. The resulting structure is called a G-crossed braided tensor category. We were able to show that when one starts from a microscopic system, it is possible to recover this structure. As a very nice bonus, we were able to give a prescription for calculating symmetry defects in the bulk (arXiv:2410.23380).
Second is the proof of a long-standing physics conjecture for Kitaev's quantum double model. That the global anyon types, called anyon sectors, indeed posses the structure of a braided tensor category, and moreover this category should be equivalent to the representation category of the quantum double of G. Showing this conjecture was a two parter, in the first part we showed that each irreducible representation of the quantum double of G has a corresponding anyon sector associated to it, and moreover that this label exhausts all possible anyon sectors (arXiv:2310.19661). The second part was about establishing the braided tensor category structure to these anyon sectors (arXiv: 2503.15611). The novelty in these works was in solving the non-abelian group case. For non-abelian groups, the string operators are no longer unitary. This is because of the existence of a local quantum dimension of non-abelian anyons. In order to solve this issue, we used something called amplimorphisms, which are matrix amplifications of the quasi-local algebra of operators. We were then able to write a coherent string operator that is unitary. This has implications toward the unitary transport of non-abelian anyons.
It should be no surprise that someone working on topological phases is also interested in the applications of topological phases in performing fault-tolerant quantum computation. There are various fascinating ideas born from this field that have insightful interplay with my study. The key difference is that instead of assuming the existence of a Hamiltonian, and studying the error-resistant ground-state thereof, the state becomes the key object of study, and the resulting excitations, called syndromes, are then dynamically corrected using quantum algorithms. There are many curious conjectures and theorems born from this perspective: the property F conjecture, and the fact that one can implement a fault-tolerant universal quantum computer using braid operations on fibonacci anyons.