# Research

My research focuses on the statistical physics of real and emergent Coulomb liquids. One application of my work is superconductivity. This is the state of zero electrical resistance achieved by certain materials at low temperature, which behave as emergent planar Coulomb liquids in the case of superconducting films. I recently discovered the symmetry-breaking mechanism that provokes the superconducting state, building on my model for topological ergodicity breaking in the planar Coulomb liquid. Experimentalists had previously used this as a base explanation for strongly correlated resistance fluctuations at the superconducting transition in films of lanthanum strontium copper oxide. I am now applying my symmetry-breaking framework to develop a full explanation for the strongly correlated dynamics near the transition.

I also model real electrostatics in biological and soft-matter physics, which are key to understanding a broad range of physical phenomena from protein folding in biological cells to ionic fluids in battery technology. We recently designed an event-chain Monte Carlo algorithm for simulating electrical interactions in an atomistic water model. This sampling method is based on piecewise deterministic Markov processes (PDMPs) which usually mix faster than the stochastic dynamics of standard Markov-chain Monte Carlo, and also guarantee numerical stability, unlike molecular-dynamics simulations. We therefore expect our accompanying open-source software application JeLLyFysh to outperform other modern methods when applied to these electrically charged systems.

The above work straddles the boundary between statistical physics and Bayesian computation in statistics. I also collaborate with statisticians to develop Markov-chain Monte Carlo algorithms for computational data science. More recently, my colleague and I wrote an in-depth paper on statistical physics and its sampling algorithms, but in the language of statistics and machine learning. This will accelerate innovation and cross-pollination of knowledge between the fields, by strengthening understanding within each discipline of the goals and nomenclature of the other field. We now plan to use this basis to explore correlated dynamics at phase transitions across statistical science.

My key đź”‘ scientific achievements split between my three interconnected specialisms:

### Planar materials

- Discovered general symmetry breaking at the Berezinskii-Kosterlitz-Thouless (BKT) transition. This resolved the paradox of symmetry breaking being observed in many BKT experiments in spite of a predicted absence of spontaneous symmetry breaking. Examples include superconducting films of lanthanum strontium copper oxide (LSCO). The result provides a model for directional mixing (or memory) timescales in a wide array of experimental systems.
- Defined the above new concept â€” general symmetry breaking â€” which encompasses both spontaneous symmetry breaking and the experimental anomalies.
- Discovered topological ergodicity breaking at the BKT transition in the 2D lattice-field Coulomb liquid (with Steve Bramwell and Peter Holdsworth). This was cited as a possible explanation for strongly correlated dynamics at the superconducting transition in the LSCO film and proved to induce the above general symmetry breaking.
- Developed the grand-canonical analogue of the Maggs lattice-field model of Coulomb liquids and showed its equivalence to the Villain model of planar magnets (see Section II and Appendix B of the paper on topological ergodicity breaking). We thenâ€¦
- â€¦presented an electric-field representation of the harmonic XY model â€” a more realistic model of planar magnets, superfluids and superconductors â€” which mapped the topological nonergodicity to the ergodic exclusion of global phase twists in the magnetic spins / condensate wavefunction.

### Molecular simulation and event-chain Monte Carlo

- Designed an event-chain algorithm for numerically stable all-atom molecular Coulomb simulations in soft matter (with Liang Qin, Tony Maggs and Werner Krauth). This is the only molecular simulation algorithm that mixes (equilibrates from a random initial configuration) Coulomb-based models in
*O(N log(N))*computations, where*N*is the number of particles. It also achieves machine precision and is the basis ofâ€¦ - â€¦our mediator-based Python-C application JeLLyFysh, which we set out in detail here with Philipp HĂ¶llmer.

### Sampling algorithms and interface with Bayesian computational statistics

- Presented an in-depth paper on statistical physics and its sampling algorithms, but in the language of statistics and machine learning (with statistician Sam Livingstone). We took a particular interest in phase transitions and event-chain Monte Carlo, presenting the latter in the language of PDMPs in Bayesian computations. This project used super-aLby and xy-type-models to simulate the models presented. We now plan to use this basis to explore correlated dynamics at phase transitions across statistical science â€” as we identified analogies with the emergent planar Coulomb liquid described above.
- Designed super-relativistic Monte Carlo for high-stability simulation of probability models in Bayesian computation (with statisticians Sam Livingstone and Gareth Roberts â€” see section 5.2 of the linked paper for details). By slowing down the Newtonian dynamics in high-gradient regions of probability space, this new simulation algorithm circumvents the numerical instabilities of Hamiltonian Monte Carlo when applied to light-tailed probability distributions. It also achieves machine precision and is the basis of our Python application super-aLby.