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Adam, what were the topic and objectives of your Vacation Scholarship research? The aim of the project was to investigate Kosterlitz-Thouless (KT) transitions in quantum spin chains –1d lattices of magnetically interacting particles. KT transitions are distinguished from conventional phase transitions because they are topological, rather than symmetry-changing, in nature, and possess rather subtle properties. To analyse our model, we used an ansatz developed from matrix product states to describe the transition in terms of an entanglement field. Entanglement is a quantum phenomenon where measurements made on one particle affect the outcome of measurements on another particle, and this allows the possibility of a more meaningful description of the transition than the proliferation of vortices in imaginary time that conventional field theories require. We aimed to demonstrate that this technique is capable of correctly capturing the transition, and so not only improve our understanding of the transition but also establish our techniques within the field.
Why is this subject important? The classification of the phases of matter is a fundamental problem in physics, and its answer requires an understanding of the behaviour of quantum systems at phase transitions, or critical points. The development of new mathematical techniques capable of capturing critical behaviour in a tractable manner and simultaneously yielding new physical insight is a useful result.
What were your findings? We observed that there is a phase transition in the entanglement structure of our model at the point where the KT transition is known to exist. At the most basic level of our analysis we noticed a spurious second transition, but this is an artefact of our inexact ansatz and is removed when we allow for fluctuations of the entanglement field. We have not yet proved that this has all the necessary features of a KT transition, but we aim to do so in the near future.
What have you gained from the scholarship? I have gained significant experience of scientific collaboration, and learned a number of useful mathematical techniques that are also directly relevant to my final year courses. Most helpful, perhaps, is the experience of persevering through calculations much longer than any I have previously dealt with; a skill that will be necessary for my intended PhD and career in theoretical physics.
What advice would you give to future applicants? The experience of research is extremely rewarding, and especially useful for anyone considering a career in academia; it is definitely one of the best ways to spend one’s summer.
Awarded: Undergraduate Vacation Scholarship
University: University of St Andrews