Open quantum systems and boundary integrability
Open quantum systems describe quantum systems which are put in contact with an environment. They represent an ever-growing area in quantum science, at both fundamental and applied levels, with repercussions in quantum technological applications, ranging from quantum computing to quantum sensing. More broadly, they allow the study of all those cases where delicate quantum features such as entanglement and quantum coherence need to be protected against environment-induced disruptive effects. The typical approach to studying open quantum systems is to integrate out the environment to produce effective dynamics for the system of interest. This is usually hard to achieve mathematically, if not impossible in certain systems. However, in low dimensionality, often the possibility arises that the original system, comprising the environment, can be treated exactly. This is the case of integrable systems, whose eigenfunctions are known algebraically and where a host of results become available. This project proposes to leverage these advanced mathematical techniques to seek exact solutions that will greatly enhance our understanding of open quantum systems.
Start date
1 October 2026Duration
3.5 yearsApplication deadline
Funding source
Faculty of Engineering and Physical SciencesFunding information
Fully funded studentship opportunities covering home university fees, additional research training, travel funds and UKRI standard rate (£21,805 for 2026/27 academic year) – please check your project of interest for full funding offer details.
About
This project will analyse boundary integrable systems, where the boundary models the interaction between the sub-system and its environment, retaining the exact solvability.
The project will start by analysing a Caldeira-Leggett-type model corresponding to a single spin surrounded by a bath of one-dimensional bosonic oscillators. A series of fermionisation maps is available in the literature to transform this model into a fermionic boundary integrable theory, whose Bethe ansatz solution can be constructed. The theory describes an arbitrary number of solitons (phonons) propagating on the negative half line – the environment – bouncing against the boundary spin – the system – and communicating changes to it, like the waves of an ocean hitting the shore. This picture is envisaged to be generalisable to arbitrary configurations of boundary spins and more complicated environments.
Since this is a relativistic quantum field theory at least in its initial incarnation (relativistic invariance enforced by the choice of linear dispersion), then one can construct in- and out-scattering states compatible with the boundary state of the spin. This will be done using the technique of the integrable reflection matrix. The states so obtained are stationary states which diagonalise the quantum Hamiltonian and have definite energy and have the rough form of [plane wave] + [reflection matrix] × [reflected plane wave]. The number of phonons, and the absolute value of each of their individual momenta, are conserved in this model.
The project is expected to produce results in terms of computing observables, for instance identifying signatures of the quantum-classical transition in the language of the boundary picture and the associated boundary renormalisation group features. Subsequently, if time permits, different scenarios may also be explored (supersymmetry could be one possibility, relying on the primary supervisor’s expertise in supersymmetric spin-chains).
Eligibility criteria
You will need to meet the minimum entry requirements for our Mathematics PhD programme.
Requirement of a minimum 2:1 in a degree relevant to the research area.
Open to candidates who pay UK/home rate fees. See UKCISA for further information.
How to apply
Applications should be submitted via the Mathematics PhD programme page. In place of a research proposal, you should upload a document stating the title of the project that you wish to apply for and the name of the relevant supervisor.
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