My research interests relate to Topological Solitons and their links to geometry, Differential Geometry and Gauge Theory. In particular my PhD work focused on studying a geometric interpretation of integrable vortex equations. I have continued working on this as well as studying the relationship between vortices and BPS monopoles. More recently I have worked on solvable models of magnetic skyrmions, as well as domain walls in one dimensional chiral magnets. Three of the main areas of my research are highlighted below.
Topological solitons are particle like lumps of energy that occur in many nonlinear classical field theories. The most famous examples include magnetic vortices, nuclear and magnetic skyrmions, instantons, and kinks. The topology in the name arises because the topology of the space of finite energy, non-singular, field configurations. Roughly speaking, we call a soliton topological if it cannot be continuously deformed to the vacuum. Usually this topology takes the form of the configuration having an associated degree or winding number counting, with sign, the number of zeros that the fields have. My work on topological solitons has primarily focussed on vortices, instantons and monopoles. Typically studying solvable models and exploring the relationship between the solitons and geometric structures.
Chiral magnets are examples of materials where the parity symmetry is broken. Magnetic systems are well known to possess examples of solitons, most famously domain walls, and chiral magnets are also host to a plethora of magnetic solitons. In particular magnetic skyrmions, which are "knotted" configurations in the magnetisation vector field of the magnetic materials. I have studied solvable models of a variety of magnetic solitons, particularly magnetic skyrmions and kinks on domain walls known as Domain wall skyrmions.
As well as the mathematical properties of solitons I am also interested in solitons that can be realised in physical systems. This includes optical skyrmions, which are skyrmion configurations occurring in the polarisation field of lasers, and linked and knotted synthetic gauge fields realised in ultracold atomic gasses.
An up to date list of my publications can be found on arXiv. Below I have split them up into rough subject areas, though there is a lot of overlap between different sections.
Magnetic Zero-Modes, Vortices and Cartan Geometry (with Bernd Schroers). Letters in Mathematical Physics, 108: 949, 2018 online version. arXiv:1705.09632.
Hyperbolic vortices and Dirac fields in 2+1 dimensions (with Bernd Schroers). J Phys A 51: 295202, 2018 online version. arXiv:1803.11120.
Magnetic Impurities, Integrable Vortices and the Toda Equation (with Sven Bjarke Gudnason). Letters in Mathematical Physics, 111: 100, 2021 online version. arXiv:2105.01332.
Cartan Connections and Integrable Vortex Equations. Journal of Geometry and Physics, 2022 online version. arXiv:2112.08328
Linked and knotted synthetic magnetic fields (with Callum Duncan, Niclas Westerberg, Manuel Valiente, Bernd Schroers and Patrik Ohberg). Phys. Rev. A 99, 063613, 2019 online version. arXiv:1808.03655.
Building Paraxial Optical Skyrmions Using Rational Maps (with Claire Cisowski and Sonja Franke-Arnold). Adv. Photonics Res., 4: 2200350, 2023 online version . arXiv:2207.12741
Magnetic Skyrmions at Critical Coupling (with Bruno Barton-Singer and Bernd Schroers). Communications in Mathematical Physics 375, pages 2259–2280(2020) available online here. arXiv:1812.07268.
Skyrmion Interactions and Lattices in Chiral Magnets: Analytical Results (with Norisuke Sakai and Muneto Nitta). JHEP 02 (2021) 095 online version. arXiv:2003.07147.
Exact Ground States and Domain Walls in One Dimensional Chiral Magnets (with Norisuke Sakai and Muneto Nitta). JHEP 12 (2021) 163 online version. arXiv:2012.08800.
Domain Wall Skyrmions in Chiral Magnets (with Muneto Nitta). Phys. Rev. B 107, 024422, 2023 online version. arXiv:2205.11417
Domain-wall skyrmion chain and domain wall bimerons in chiral magnets (with Yuki Amari and Muneto Nitta). Phys. Rev. B 109, 104426, 2024 online version. arXiv:2311.05174
Calorons and constituent monopoles (with Lorenzo Foscolo). Communications in Mathematical Physics (2023) available online here. arXiv:2207.08705
The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space (with Guido Franchetti). SIGMA 19 (2023), 043 online version . arXiv:2302.13792
July 2024, Domain Wall Skyrmions at the Topological Solitons at Edge Hill event held at Edge Hill University
May 2024, Vortices and Cartan Geometry at the York Geometry, Analysis and Mathematical Physics seminar.
June 2023, Instantons and constituents at SIG XI
October 2022, A Gluing Construction for Calorons in the Leeds Geometry Seminars at the University of Leeds.
July 2022 Calorons and Constituent Monopoles at the Geometric Models of Nuclear Matter conference.
March 2022 Calorons and Constituent Monopoles as part of the Solitons at Work seminar series. A recording of the talk is available here.
February 2022 Cartan Connections and Integrable Vortex Equations in the DAMTP Mathematical Physics Seminars in Cambridge
November 2021 Periodic Instantons with Small Period in the Algebra, Geometry and Number Theory seminar at the University of Bath.
November 2021 Integrable Vortices and Flat Connections in the Leeds Geometry Seminars at the University of Leeds.
May 2021 Magnetic Impurities, Integrable Vortices and the Toda Equation as part of the Solitons at Work seminar series. A recording of the talk is available here.
November 2020 Mathematical models of topological solitons at an internal UCC physics seminar.
September 2020 Magnetic Skyrmions in solvable chiral magnets as part of the Geometric Models of Nuclear Matter Digital Seminar Series.
January 2020 Magnetic Skyrmions from Gauged CP^1 Models at the CP^N model: recent developments and future directions workshop at Keio University.
May 2019 Magnetic Skyrmions at Critical Coupling as a condensed matter theory seminar at the University of Mainz.
April 2019 Magnetic Skyrmions at Critical Coupling at the Keio Topological Science Project Seminars.
February 2019 Magnetic Skyrmions at Critical Coupling at the DAMTP Mathematical Physics Seminars in Cambridge.
July 2018 Hyperbolic vortices and flat connections at the LEEDS workshop on topological solitons in Leeds.
January 2018 Vortices, Cartan connections and Magnetic Zero-Modes at the local EMPG Seminars.
June 2017 Magnetic Zero-Modes, Vortices and Cartan connections at the Sen Conjecture and Beyond workshop.
April 2017 Metric structure of the moduli space of solutions to the self-duality equations at the BIG Workshop on Higgs Bundles.
September 2016 Solutions of the Perturbed Freund Equations at the second Geometric Models of Nuclear Matter Conference.
March 2016 Topological Electromagnetism and the Seiberg-Witten Equations at the 46th NBMPS.