Our work on creating these novel forms of matter included studying how precursors of fractional topological insulators (two-dimensional electronic matter arising from interactions, topology, and time-reversal symmetry) can arise in fermionic ladder systems and showing that such precursors may serve as ingredients for creating full-fledged two-dimensional fractional topological insulators. We also showed how geometrical twists in other fermionic ladder systems can lead to robust dynamical features linked to generalisations of Majorana fermions at the boundaries. We also proposed ways to create topological boundary dynamics directly. In particular, for high-complexity boundary dynamics corresponding to the so-called Sachdev-Ye-Kitaev (SYK) model (of great recent interest due to links to quantum information scrambling and gravity), we devised a realisation based on a Majorana topological quantum computer. We also described novel, intrinsically non-equilibrium, forms of topological quantum matter and showed how intermediate-term quantum computers may be used to create them.
The work on theoretical characterisation included studying how topological states, despite possessing certain non-local defining features, may be captured using tensor networks, an inherently local framework for describing and simulating quantum matter on classical computers. We have also developed a framework for uniting topological states with the inherently local phenomenon of many-body localisation (insulating behaviour arising from interactions and quantum interference). Our framework characterises such topological many-body localised systems in terms of weakly interacting local objects emerging from ingredients in topological quantum error correction. This allowed us to study topological order at high energies (in contrast to the usual setting of the lowest achievable energy), and to combine it with generalised forms of symmetries to describe and characterise the above mentioned intrinsically non-equilibrium topological systems. We also showed how this framework can be used to characterise topological many-body localised systems in computer simulations. Another approach to characterise topological states is via their boundary. Our work included establishing how boundary features link to bulk characteristics in topological superconductors with rotation symmetry. We also characterised the high-complexity boundary systems corresponding to the SYK model. Our results include establishing the symmetry classification of these systems and the striking finding of boundary supersymmetry.
Our work on detection included showing that the fractional topological insulator precursors mentioned above, despite living on a quasi-one-dimensional ladder, already display certain quantised signatures that one would normally expect only in full-fledged two-dimensional fractional topological insulators. We also established transport results on interacting devices supporting Majorana fermions. Such devices are leading candidates for demonstrating Majorana fermion topological qubits for quantum computing. Our results include the nonequilibrium conductance, the prediction of fractional charge quantisation in current fluctuations, and the study of the spatial structure of the sea of conduction electrons. We also studied signatures of topological boundaries linked to the SYK model, both in transport measurements and for probing the dynamics in our proposed Majorana fermion SYK realisation.
These and further works from this project have led to nearly 30 journal publications (with several more in preparation), and a similar number of presentations at seminars, summer schools, workshops, and conferences.
