Mathematical Physics at Leeds (MaPLe)

Rotwisted calorons are self-dual Yang-Mills connections on R^4 invariant under a glide rotation, and were first introduced in the context of rotating quark-gluon plasmas. Much of the success in the study of self-dual Yang-Mills has been via the existence of a nonlinear transform called the ADHMN (Atiyah-Drinfeld-Hitchin-Manin-Nahm) construction. In this talk, after reviewing the many facets of ADHMN constructions, we shall discuss the formulation of a Nahm transform for rotwisted calorons, identifying them with solutions of an integrable delayed-differential equation. We will also describe some solutions in the simplest non-trivial case. This is based on joint work with Derek Harland.

Joint work with Bernd Schroers (Edinburgh), Tom Winyard (Dundee)

Some thin ferromagnetic films can support topological defects in their magnetization vector called magnetic skyrmions, stabilized by a chiral phenomenon called the Dzyaloshinskii-Moriya (DM) interaction. These have been proposed as bits in next-generation data storage devices. In some circumstances, isolated skyrmions may have (in a meaningful sense) negative total energy. It's then favourable for the system to spawn infinitely many skyrmions which pack together into a periodic lattice. But this might not be the only negative energy configuration available to the system, so may or may not be its ground state.

I will present a careful analysis of a simple but rather general model of thin ferromagnets including a completely general DM interaction and the Zeeman energy of an arbitrary applied magnetic field. The aim is to understand the parameter domain on which the ground state of the system is a skyrmion lattice, and, in this case, what the unit cell of the lattice looks like.

Nonstabilizerness or quantum magic is a fundamental resource required for universal quantum computation, however it’s role in many-body quantum systems, especially near criticality, remain poorly understood. Here, we develop a spectral transfer-matrix framework for the stabilizer Renyi entropy (SRE) in infinite matrix product states, showing that its spectrum contains universal subleading information. In particular, we identify an SRE correlation length – distinct from the standard correlation length –which diverges at continuous phase transitions and governs the spatial response of the SRE to local perturbations. We derive exact SRE expressions for the bond dimension χ = 2 MPS “skeleton” of the cluster–Ising model, and we numerically probe its universal scaling along the Z2 critical lines in the phase diagram. These results demonstrate that nonstabilizerness captures signatures of criticality and local perturbations, providing a new lens on the interplay between computational resources and emergent phenomena in quantum many-body systems.

Understanding how complex systems transition between order and chaos is a central challenge of nonequilibrium physics. While weak perturbations of classical integrable systems give rise to a mixed phase space of coexisting regular and chaotic trajectories, analogous behavior in interacting quantum many-body systems has remained elusive. Here we develop and experimentally implement a hybrid quantum–classical feedback protocol that autonomously discovers and stabilizes long-lived regular trajectories in a superconducting quantum processor. Each iteration combines short-time quantum evolution with classical optimization that projects the dynamics back onto a low-entanglement variational manifold, effectively distilling coherence from chaotic evolution. The stabilized trajectories reveal a quantum many-body mixed phase space emerging from nonlinear variational dynamics, without a direct analogue in classical or few-body quantum systems. Our results establish a versatile framework for algorithmic discovery and control of coherent dynamics previously inaccessible to experiment.

Despite wave-particle duality being the cornerstone of quantum physics, we do not use the same formalism to describe photons and quantum mechanical particles [1]. To elevate wave-particle duality, we recently introduced a local photon model [2] which allows us to do quantum mechanics with photons. To illustrate the usefulness of our approach, we first have a closer look at the modelling of a single-photon emitter [3]. In addition, we consider an example, where quantum physics is used to answer a longstanding controversy in classical physics. More concretely, we show that our approach allows us to define the momentum of light in a unique way, namely as the generator for spatial translation. When analysing the momentum dynamics of photonic wave packets which transition from air into a dielectric medium, we obtain an answer to the Abraham-Minkowski controversy. Although our results align with Minkowski’s theory, there are also some crucial differences [4].

[1] A. Bukhari, D. Hodgson, S. Kanzi, R. Purdy, and A. Beige, Enhancing wave-particle duality, New J. Phys. 27, 084501 (2025).
[2] D. Hodgson, J. Southall, R. Purdy, and A. Beige, Local photons, Front. Photon. 3, 978855 (2022).
[3] T. Hartwell, D. Hodgson, H. Alshemmari, G. Jose, and A. Beige, Photon emission without quantum jumps, New J. Phys. 28, 014510 (2026).
[4] G. Waite, D. Hodgson, B. Lang, V. Alapatt, and A. Beige, Local photon model of the momentum of light, Phys. Rev. A. 111, 023703 (2025).

Motivated by the appearance of the pentagon equation in the algebraic structures underlying integrable systems, and in the spirit of this mathematical physics seminar, my aim in this talk is also to explore what kind of interaction there can be between these very algebraic constructions and questions coming from integrable systems. I will give a complete description of finite bijective set-theoretic solutions to the pentagon equation in terms of matched products of groups, based on joint work with Okninski and Van Antwerpen. The emphasis will be on the underlying group-theoretic structure: I will explain the characterisation via matched products, describe the role of irretractable solutions as building blocks, and show how they can be used to obtain all bijective solutions. Based on ongoing joint work with Janssens, I will also briefly indicate how this description, together with the link between pentagon solutions and Hopf algebras, leads to constructions of Hopf algebras with a positive basis in the sense of Lu–Yan–Zhu.

Free theories are landmarks in the landscape of quantum field theories: their exact solvability serves as a pillar for perturbative constructions of interacting theories.
Fuzzy sphere regularization, which combines quantum Hall physics with state-operator correspondence, has recently been proposed as a promising framework for simulating three-dimensional conformal field theories (CFTs), but so far it has not provided access to free theories. We overcome this limitation by designing a bilayer quantum Hall system that hosts an Ising tricritical point—a nontrivial fixed point where first-order and second-order transitions meet—which flows to the conformally coupled scalar theory in the infrared. The critical energy spectrum and operator structure match those at the Gaussian fixed point, providing nonperturbative evidence for the emergence of a free scalar CFT.
Our results expand the landscape of CFTs realizable on the fuzzy sphere and demonstrate that even free bosonic theories – previously inaccessible – can emerge from interacting electrons in this framework.

NOTES: Special external MaPLe seminar. Unusual day, time, and room.
This talk explores non-integrable solitons in 1+1 dimensions, focusing on kink-antikink interactions and their rich, often chaotic dynamics. Depending on the initial velocity, collisions may lead to annihilation, with or without formation of a long-lived oscillatory bound state (a bion or an oscillon), or inelastic scattering accompanied by radiation. Most intriguingly, in an intermediate velocity regime, the outcome alternates between annihilation and escape, producing resonance windows. These arise when the colliding pair temporarily stores kinetic energy in internal vibrational modes and later releases it, allowing the kinks to escape. I will discuss several mechanisms that enable this energy exchange, including the role of fermionic fields and quasinormal modes of the kink-antikink system.

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In previous work with Norris and Silvestri, we have shown that the global fluctuations present in these models exhibit behaviour that can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks. In this talk I will discuss work in progress with Larissa Richards in which we explore how the correlation structure of local fluctuations near the cluster boundary changes at the point of phase transition.

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.