Maxine McCarthy, University of Sheffield

Library, A.2.500, Staudtstr. 2

Location details

Abstract:

Many topological materials exhibit unusual disorder-resistant phenomena, such as boundary-localised gapless modes that may have exotic localisation and transport properties, for instance scatter-free directional propagation. Such physical behaviour arises as a consequence of a system having a non-trivial bulk index. Typically, this ensures the system has a gapped bulk terminated by a gapless boundary and may be predicted with the use of the bulk-boundary correspondence. Although properties predicted by the bulk-boundary correspondence converge exponentially as system size increases, even for strongly disordered systems [1], the bulk-boundary correspondence applies exactly only in the (infinite) thermodynamic limit. This means that for a finite system there exists a finite disorder where topological properties of boundary modes are effectively lost. Such modes may no longer be well localised to the boundary, and gapless modes can become gapped, even if a non-trivial bulk index is retained. A loss of the bulk-boundary correspondence is the most prevalent in small structures, or those with a random underlying connectivity. We present an approach using graph theoretic methods that allow us to study exact topological properties of strongly disordered finite tight binding models, instead of relating topological phenomena to the bulk. Using this approach, we find three classes of systems that exhibit distinct topological phases separated by exact, and unavoidable, energy gap closures in the spectrum of a real space tight binding Hamiltonian [2]. In each class, a structure has 2N distinct phases where the value of N depends on the connectivity of the underlying structure. We also give a definition for topologically protected states, which arise due to a combinatorial property of a structure, rather than bulk protection. Finally, we discuss experimental results that use coaxial cable networks to implement controllable tight binding systems. The results corroborate our classification [2], showing phase transitions, localisation effects and topological protection [3].

[1] Emil Prodan. “Disordered topological insulators: a non-commutative geometry perspective”. In: Journal of Physics A: Mathematical and Theoretical 44.11 (2011), p. 113001. doi: 10.1088/1751-8113/44/11/113001.

[2] Maxine M. McCarthy and D. M. Whittaker. A Topological Classification of Finite Chiral Structures using Complete Matchings. 2024. arXiv: 2405.16274.

[3] D. M. Whittaker, Maxine M. McCarthy, and Qingqing Duan. Observation of a Topological Phase Transition in Random Coaxial Cable Structures with Chiral Symmetry. 2023. arXiv: 2311.11040