Boundary T-Q Relations
Lead Research Organisation:
University of York
Department Name: Mathematics
Abstract
The aim will be to study Q operators for the open spin chain with sl_2 symmetry and arbitrary boundary conditions. This will involve constructing infinite dimensional representations of Bethe subalgebras of twisted Yangians based on symmetric pairs of sl_2. Then we would aim to extend these to sl_n and then to so_n and sp_2n.
Organisations
People |
ORCID iD |
| Niall MacKay (Primary Supervisor) | |
| Allan Gerrard (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509802/1 | 01/10/2016 | 31/03/2022 | |||
| 1792468 | Studentship | EP/N509802/1 | 01/10/2016 | 30/09/2019 | Allan Gerrard |
| Description | Spin chains are quantum mechanical models consisting of a line of particles, modelling electrons in a metal, which typically each interact only with their nearest neighbouring particle. In comparison to other quantum mechanical systems of multiple particles, spin chains are relatively simple. However, due to the mathematical simplicity of the model, spin chains are strongly linked to various other physical models in areas of statistical mechanics, to the extent that the models share the same physically relevant quantities. Furthermore, and most importantly, due to hidden underlying mathematical structures of the model, various solution techniques have been developed for studying spin chains. We have been studying one such technique, known as the algebraic Bethe ansatz. This technique is powerful as it not only finds the energy levels of the system, but also finds the energy eigenvectors, the wavefunctions corresponding to these energy levels, which are crucial when making predictions about the evolution of the system. A spin chain's possesses an underlying symmetry, which is typically one of four types, which are referred to as A, B, C or D. However, for spin chains with boundaries, the boundary conditions break the symmetry to a smaller one. The mathematical structure of these spin chains is the characterised by a pair of symmetry types: the original symmetry, and the smaller one. The simplest case is the A type, of which many results are known, while our research focusses on C or D. In our first paper, with Vidas Regelskis and Niall MacKay, we successfully applied the algebraic Bethe ansatz to a spin chain with 'soliton non-preserving boundary conditions', where type A is broken to type C or D. In our second paper, which is currently in progress with Vidas Regelskis and Curtis Wendlandt, we will apply the algebraic Bethe ansatz to a class of spin chains with C or D initial symmetry, and a large class of boundary conditions. We hope this work offers some insight into the limits of the algebraic Bethe ansatz technique, and how other systems with C or D symmetry may be solved. |
| Exploitation Route | This research has some relevance to areas of gauge and string theory, which involve D type symmetry, where spin chains appear as subsystems of a larger problem. Due to their mathematical simplicity, spin chains often may be reinterpreted in other contexts, such as 2D lattice models or Markov chains. It is yet unclear if our research will have applications in these fields. Additionally, spin chains have direct applications in quantum computing, although not those of type C or D. In general, our findings may tie together with previous work to allow a more generalised technique to be developed, which applies to a larger class of models. |
| Sectors | Other |