Progress on Planned Projects:
Fractons: We have clarified the role of translational symmetries in fracton models and their proximate orders. We identified a new set of models with fracton phenomenology in a classical setting, using the tools of cellular automata. In the future we may try to implement quantum fluctuations. In related work with S. Gazit and others we explored simulations of random gauge theories as a warm-up to studying fractons. Separately Dr. A. Prakash, a project postdoc, has worked extensively on classical fracton models.
Phenomenology of semimetals: we explained quantum oscillation experiments in a topological nodal line semimetal, and developed a theory of the semimetallic bands of twisted bilayer graphene in a magnetic field. We have also identified new mechanisms for quantum oscillations in semimetals.
Spectroscopy of topological matter: We have developed a theory of domain walls in quantum Hall nematics. We discovered a hidden non-symmorphic symmetry that explains experiments on the quantum critical magnet CoNb2O6. We identified new aspects of quantum oscillation phenomena in topological semimetals.
“Parton inspired” numerical methods: these appear to work well in one dimension but struggle in higher dimensions. Work with J. Haegeman's group has finally produced promising results towards the end of the project. The linked project on classification was postponed until the numerics were more developed; a specific aspect of this as applied to the trillium lattice was completed near the end of the project, but we hope to build and release a generalized version of the computer algebra system used to automate the classification in the future.
Other projects -- on interacting topological crystalline insulators, QMC for fractons and on U(1) spin liquid classification -- have been deprioritized because competing groups have pre-empted our work or initial results were not promising.
New developments: Since the original proposal, exciting new directions have appeared that fit naturally into the broad sweep of this proposal. Working on these problems is extremely important for any leading research group in the area, and so these have been incorporated into the ERC project alongside existing tasks as reported on previous reports.
Moiré systems: These unusual 2D materials, obtained by placing two atomically thin 2D sheets on top of each other with a small relative twist, host correlated phases whose properties are linked to the ideas on quantum Hall nematic states, spin liquid phases, and fractionalization. We have developed a Hartree-Fock code for twisted bilayer graphene and used it to identify (a) an unusual class of topological exciton band near a so-called orbital Chern insulator state; (b) a new "incommensurate kekulé spiral" order capable of explaining a multitude of puzzling experimental features, which was recently directly confirmed in experiment; we have also (c) explored many fundamental theoretical issues in the field, such as obstructions to ordered phases from band topology. We are collaborated with experimental group of Sanfeng Wu (Princeton) on understanding the twisted transition-metal dichalcogenide WTe2. Codes will be released publicly at the end of the project where feasible.
Note that the conference originally planned for the conference could not be organised: this is because the original timescale for the conference coincided with the COVID pandemic, which was very prolonged in impact in the UK (travel restrictions remained in place until 2022 in many cases). In 2023 and 2024, the "overflow" of cancelled conferences and general uncertainty about COVID travel meant that it was unfeasible to organize a conference.
“Higher order" topological phases: The existence of these new types of topological states existence relies on crystal symmetry. We have developed tools to understand the "surface topological order" of these states. These will help us understand the interplay of interactions, symmetry and topology.
Quasicrystalline Dimer models: We initiated the analysis of these models of frustrated magnets, that also have properties complementary to fractons. The issue of quasiperiodicity is also linked naturally to "moiré" physics. The statistical mechanical properties of quasicrystalline dimer models was pursued extensively through the project.
SYK Model: Given the challenge of numerics on heavy fermion systems and frustrated magnets we have used the "Sachdev-Ye-Kitaev" model to explore the link between ground state degeneracy, stability to fluctuations, and unconventional dynamics.
Nonlinear spectroscopy: We have developed theories of nonlinear response of correlated systems and used these to identify a new route to probe fractionalization in nonlinear optical response.
Results have been disseminated in publications and a wide range of presentations to academic audiences at institutional seminars and at conferences and workshops, including some of the most prestigious in the field (e.g. KITP, Gordon Conference, etc.)
