Topological Phases in Quantum Systems with Quantum Group Symmetries

This is a project in mathematical physics. During this project, I will tackle two research problems related to one-dimensional many-body quantum systems with Uq(sl2) quantum group symmetry, particularly in the context of what is known as the q-deformed AKLT model.

1. My supervisor’s recent publication hypothesised that the ground states ground states of the q-deformed AKLT model fall into two classes. I will verify this by constructing examples, using interpolations on matrix product states to help with the classification task. If this goes well, I may be able to demonstrate the result holds in full generality.

2. I will attempt to find a suitable analogue of the concept of Renyi entropy for the q-deformed AKLT model. In the undeformed case, the Renyi entropy encodes all knowledge of the entanglement present in the quantum system. This motivates the question as to whether a similar (and similarly useful) metric exists in the deformed case, and whether it is readily computable.

Approximately two weeks will be devoted to developing a thorough understanding of the q-deformed AKLT model. This will transition into the main tasks described above, which are legitimate research questions yet accessible at an advanced undergraduate level.

Michael Law

The University of Melbourne

Michael grew up in Hong Kong and began his undergraduate studies at the University of Melbourne in 2018. Initially starting out in a business-related field, he pivoted to studying mathematics in pursuit of his lifelong passion. Michael is interested in a range of pure and applied mathematical fields, and he has recently been drawn towards geometry, topology and mathematical physics in particular. Michael is looking forward to pursuing Honours in 2021 followed by postgraduate studies, with the eventual goal of becoming a researcher in the mathematical sciences.

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