Mathematical Physics at Leeds (MaPLe)

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It has long been known that some important spin systems on lattices, such as the spin-O(N) model, have representations in terms of discrete structures (loops, paths etc) on the lattice. This has been a fruitful connection, however many discrete models on lattices, such as percolation, the dimer model,  spatial random permutations, and others, are also of considerable interest. In these cases there was no precise link to spin systems, either because the motivation came from elsewhere, or because these are toy models that have taken on a life of their own. In this talk I will present a spin system where spins take continuous complex values. By choosing the measures appropriately, virtually any discrete system of interest can be obtained as a special case. These spin systems then lend themselves naturally to several of the well developed tools of classical spin systems. This is joint work with Volker Betz.

Topological phases of matter in (2+1)D should naturally form a 3-category, in which k-morphisms represent defects of codimension k. By the cobordism hypothesis, the 3-categories of (2+1)D topological order with a fixed anomaly are each equivalent to the 3-category of fusion categories enriched over a corresponding UMTC. In ongoing work with Fiona Burnell, we introduce algebraic techniques for concrete computations in 3-categories of (2+1)D topological order, including a tunneling approach to the classification of point defects which generalizes the use of braiding to identify anyon types. We then apply these techniques to compute ground state degeneracy and classify low energy excitations in a class of fracton-like (2+1)D topological defect networks.defect networks have been proposed as a mechanism for describing fracton orders. However, determining the excitation spectra and fracton phase from the topological defect network remains a highly nontrivial task. In ongoing work with Fiona Burnell, we study a class of fracton-like defect networks in (2+1)D which give examples of non-Abelian lineons. We describe algebraic techniques for computing the excitation spectrum and ground state degeneracy and explore the interplay between the two in examples.

Understanding the universal properties of continuous phase transitions has been a long-standing area of focus. A powerful tool in this endeavor have been conformal field theories (CFTs) — a class of interacting field theories with a rich symmetry structure that can emerge in statistical mechanics models tuned to a critical point. The recently introduced “fuzzy sphere” method has enabled accurate numerical regularizations of certain three-dimensional (3D) CFTs. The regularization is provided by the non-commutative geometry of the lowest Landau level filled by electrons, such that the charge sector is trivially gapped due to the Pauli exclusion principle at filling factor ν = 1, while the electron spins encode the desired CFT. In this talk, along with key concepts for CFT in 3D, I will provide an overview of the fuzzy sphere method and its application to the paradigmatic 3D Ising CFT. I will also present recent results for encoding the same CFT using strongly correlated fractional quantum Hall states, setting the stage for the fuzzy-sphere exploration of conformal critical points between topologically ordered states.

Partitions of integers play a role in a variety of fields including number theory, representation theory and random matrix theory as well as being of independent interest in enumerative combinatorics. I will present several key concepts and discuss various places where partitions have arisen in my research.

Magnetic buoyancy is the phenomenon for strong magnetic fields to reduce the pressure of electrically-conducting plasma, which can lead to gravitational instabilities. Starting from a toy model primarily of academic concern, magnetic buoyancy was seen as an interesting phenomenon with no known applications. Just one year after its conception, Eugene N. Parker hypothesised that magnetic buoyancy could be a component of the solar dynamo, explaining how the Sun uses this mechanism to redistribute its magnetic field generated deep beneath its surface. The first part of this talk will be a brief literature review summarising how our knowledge of magnetic buoyancy evolved over time, and how its applications were discovered.

The second part of the talk will focus on overstability, i.e., states which exhibit oscillations which grow in time. There are two well-known physical mechanisms within the magnetic buoyancy instability (MBI) which drive overstable modes. I will describe these physical mechanisms and present the findings of my second paper, including the discovery of a third mechanism for overstability. Furthermore, generalising MBI to include variable diffusion restricts overstability, and our newly discovered mechanism is the only one possible in solar interiors. However, our fluid model does not provide sufficient physical insight to describe it, and we instead create a secondary flux tube model to capture the physics of the problem.

Vortices are 2-dimensional topological solitons defined on a Riemann surface in the context of the Abelian Higgs model. Physically, they model magnetic flux tubes in superconductors. At critical coupling, they satisfy a 1st order system of PDEs called the Bogomolny equations. I will first review the derivation of these equations using a Bogomolny argument, and then derive the Taubes equation. Next, I will introduce a generalised Abelian Higgs energy functional, which gives rise to 5 different vortex equations (this has been carefully investigated by Nicholas Manton). For a constant curvature base surface, these equations turn out to be integrable, reducing to a Liouville equation. One can obtain further integrable vortex equations by choosing suitable conformal factors, and in this case, the Taubes equation becomes the sinh-Gordon or Tzitzeica equation. If we assume radial symmetry, these are equivalent to a Painlevé III ODE. I will discuss the construction of these vortices for the rest of the talk, which is a joint work with Maciej Dunajski.

The induced connection is a natural connection on a subbundle of a vector bundle. In physics, it is known as the Berry connection, and its parallel transport operators give rise to the Berry phase. In this talk I will explain exactly what the Berry/induced connection is and present some work I have done on finding numerical approximations to its parallel transport. This will lead to some interesting(?) questions for the algebraists in the audience!