A theoretical study of static, time-periodic and quasi-periodic mixed dimensional topological phases of matter

Abstract

In the last decade, artificially designed systems have emerged as a promising path to the realisation of topological phases of matter. A particular focus has been on chains of magnetic impurities placed on superconductors which realise mixed dimensional topological phases. To better understand these phases we provide a comprehensive exploration of their topological properties. We demonstrate the challenges of mixed dimensional systems and provide a proof of the conditions of validity of a classification method based on an effective lower-dimensional topological Hamiltonian. The applicability is illustrated through an analytically tractable model of a spiral magnetic interface embedded into a superconductor. We also show that this method provides correct results in regimes where we can demonstrate that other methods proposed for mixed-dimensional models fail.

A limitation arises for models with multiple interfaces though, where we find that zeros of Green’s functions can act as false phase boundaries that make the classification by the topological Hamiltonian generally unreliable. Nonetheless, the latter remains applicable for a model of two spiral magnetic interfaces embedded into a superconductor, where in combination with a low-energy
effective Hamiltonian, it can probe the local structure of the topological phase.

Finally we extend the classification to models with time-periodic and quasi-periodic modulations. We exactly solve a model of a periodically precessing magnetic interface in a superconductor, demonstrating that the driving reduces the superconductor gap. Considering then quasi-periodically evolving magnetic interface, we propose a classification by an extension of the topological Hamiltonian, the topological quasi-energy operator. For a purely one-dimensional model, we provide evidence that quasi-periodic driving tends to destabilise a topological phase. But for a mixed dimensional model, we demonstrate the existence of an enhanced localising potential causing a larger regime of the topological phase.

Date of Award	1 Dec 2026
Original language	English
Awarding Institution	University of St Andrews
Supervisor	Bernd Braunecker (Supervisor)