Lattice Models of Higher Topological Quantum Field Theories

Lead Research Organisation: University of Oxford
Department Name: Oxford Physics

Abstract

Recent work from a number of Groups (Farins, Martins, Bullaivant n Leeds; Wang and Williamson at Microsoft, Cui at Stanford) has proposed a new type of topological quantum field theory based on what is known as a higher category, or a higher gauge theory. These field theories are based on crossed modules (or G-crossed categories). At the same time, the crossed modules are closely related to symetry protected topological (or enriched) phases, a subject of considerable modern interest. We hope to make the connection to symmetry breaking explicit in these models and how they are related to other topological quantum models, including walker wang models.

This direciton of research is of interest to a wide range of researchers, from mathematicians interested in TQFTs to condensed matter physicsists interested in what sort of matter can exist in real physical systems. From a quantum information perspective, this direction of research can be thought of as potentially new models for quantum memories.

Much of what is known about these model is know from a very abstract mathematical perspective which makes the physical properties of such systems hard to unravel. Obtaining a simple geometric picture (similar to what I did a few years ago with Burnell and von Keyserlingk on earlier Walker-Wang models) will be one of our first tasks.

This project is intended to be somewhat forward looking and adventurous. It will be important for the graduate student to also learn more well-grounded condensed matter physics in order to keep contact with the broader field.

This project falls under the EPSRC Quantum Technologies research area.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/R513295/1 01/10/2018 30/09/2023
2114104 Studentship EP/R513295/1 01/10/2018 31/03/2022 Joe Huxford
 
Description So far I, with the guidance of my supervisor, have considered the lattice models based on the higher gauge theory (developed by Bullivant et al.) mentioned in the summary in great detail, in both 2D and 3D. We have explicitly constructed operators that produce and move the simple excitations in this model and have found various properties of these excitations. The most significant such properties are the braiding statistics and topological charge. In 3D there exist both loop-like and point-like excitations, and we have considered the property of both. In particular we have developed techniques for finding the point-like and loop-like topological charges present in the model. Some of the excitations are confined, meaning that it costs energy to separate them which increases with distance, while others are condensed, meaning that they carry trivial topological charge. While many of these properties are necessarily described mathematically, we also have geometric interpretations for many of our results in terms of paths and surfaces.

In the 2D case, we have found that a subset of these models realise symmetry-enriched topological phases that also exhibit spontaneous symmetry breaking, which is not the case for the 3D models we considered. We then studied this phenomena in the 2D case by considering a mapping of some of these 2D models to an existing model for symmetry-enriched topological phases. We related these 2D models to the symmetry-enriched string-net model (developed by Heinrich et al.) for symmetry-enriched topological phases and used this mapping to obtain useful information about both models (such as the presence of confined excitations in the symmetry-enriched string-net model). At the moment, we are working on publishing all of these results, at which point I believe that most of the original objectives of the award will be met.
Exploitation Route We have tried to frame our results in ways that would be accessible to both physicists and mathematicians. There are two main ways in which our results may be used by others. Firstly, the literal results for our particular model should be useful for examining the topological phases represented by the models. For example, it would be interesting to see whether these phases support universal quantum computation or robust (error resistant) quantum memories. Our results could be used to determine which phases support this. Because we have explicitly constructed the operators that produce and move excitations, the operators could be used to set up a protocol for topological quantum computation using these excitations (though we emphasise that finding a physical system that realises the phases we have considered naturally is likely to be difficult, while artificially creating these systems is a possibility for the future). The other way in which the outcomes of this research may be taken forward is that our methods for analysing topological phases may be applied to other models. We believe that our explicit approach gives a wealth of information about these phases that other indirect methods may not provide.
Sectors Digital/Communication/Information Technologies (including Software)