# Transfer matrices as the free-energy of topological excitations

Lead Research Organisation:
University of Birmingham

Department Name: School of Physics and Astronomy

### Abstract

Transfer matrices allow us to deduce the thermodynamics of translationally invariant systems, which for large systems relates the free energy to the maximal eigenvalue. However, what do the excitations correspond to physically? The focus of my research is showing that, exactly in the case of the Ising model, these excitations correspond to topological objects in the underlying lattice. The idea is to generalise this to other, more complicated, models in statistical mechanics.

### Publications

Moodie J
(2020)

*Transition temperature scaling in weakly coupled two-dimensional Ising models*in Physica A: Statistical Mechanics and its Applications
Moodie J
(2021)

*An exact power series representation of the Baker-Campbell-Hausdorff formula*in Journal of Physics A: Mathematical and Theoretical### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509590/1 | 01/10/2016 | 30/09/2021 | |||

1820029 | Studentship | EP/N509590/1 | 01/10/2016 | 31/03/2020 | Jordan Moodie |

Description | I have developed a technique which allows one to probe models of statistical mechanics, that is mathematical constructs which approximate nature, with great accuracy. It adapts a well-known formalism, transfer matrices, but adds a physical interpretation which allows insights from quantum mechanics, giving cross-subject interest. Concretely this approach allows computation which far exceeds that of the current literature in accuracy. An example of this is in a paper I have submitted to Physical Review B, which uses this accuracy to find a missing piece in a puzzle concerning scaling theory close to an Ising phase transition. I have also developed a mathematical formula which solves an open problem dating back over a hundred years, namely constructing a power series of the Baker-Campbell-Hausdorff formula. I am hopeful that this will prove valuable in many different fields, from so-called Floquet dynamics found in quantum mechanics to transfer matrices in statistical mechanics |

Exploitation Route | I hope that other physicists and mathematicians can build on my work, using transfer matrices to define certain thermodynamic objects like phase transitions and transition temperatures. The numerical technique can be applied to a wide variety of models. The new representation of the Baker-Campbell-Hausdorff can find uses in any field involving products of exponentials, so quantum mechanics or statistical mechanics are natural homes. |

Sectors | Other |