Periodic Reporting for period 2 - TMCS (Topological Matter and Crystal Symmetry: From Microscopic Structure to Phenomenology) Periodo di rendicontazione: 2020-07-01 al 2021-12-31 Sintesi del contesto e degli obiettivi generali del progetto This project will develop an understanding of Topological States of Matter, using Crystalline Symmetry as an organizing principle. Topological states of matter are characterised by subtle structures in the quantum entanglement between different degrees of freedom. While some can be understood from a weakly-interacting starting point, likely many more involve strong interactions. These are a challenge to traditional techniques, and new tools are required to analyse their properties. Besides fundamental interest, topological phases promise new applications: their nonlocal encoding of information could make them unusually robust quantum computing platforms, and they have unconventional electromagnetic and optical properties. This project will develop tools to identify, simulate and experimentally detect these phases. Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti Progress on Planned Projects:Fractons: We have clarified the role of translational symmetries in fracton models. Recently we identified a new set of models with fracton phenomenology in a classical setting, and are working on implementing quantum fluctuations. In related work with S. Gazit and others we explored simulations of random gauge theories as a warm-up to studying fractons.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.Work in Progress: The proposed “Parton inspired” mean field techniques appear to work well in one dimension but struggle in higher dimensions. We will address this challenge by collaboration and strategic recruitment. The linked project on classification is postponed until the numerics are more developed. Other projects -- on interacting topological crystalline insulators and on spin liquid classification -- have been deprioritized because competing groups have pre-empted our work.New developments: Since the proposal was funded, 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.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. We are also now collaborating with the materials group of Johannes Lischner (Imperial College) and the 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. “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 is now being pursued primarily by a postdoc who has also developed similar techniques in the random setting. 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 begun to develop theories of nonlinear response of correlated systems as these may provide a new route to probe fractionalization. Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora) Results beyond the state of the art include:(1) Microsocopic and long-wavelength understanding of domain walls in quantum Hall nematics and orbital Chern insulators; (2) Establishment of a new frontier in the study of dimer models by extending them to the quasiperiodic setting; (3) The first detailed understanding of strongly-interacting gapped surfaces of higher-order topological phases; (4) Several new directions in moiré systems, ranging from their unconventional exciton structure to the identification of a new type of order, the "incommensurate kekulé spiral", in twisted bilayer graphene; (5) Several contributions to the understanding of quantum oscillations in semimetals; (6) New approaches to identifying fractin-like physics in classical spin systems and to understanding their proximate orders; (7) Resolution of a decade-old puzzle in scattering experiments on the quasi-1d magnet CoNb2O6; (8) Exact approaches to nonlinear response in clean and disordered interacting magnets.Expected Future Results: (1) Further results on surface topological order for higher-order topological phases; (2) Links between the SYK models and frustrated magnetic systems; (3) Further results on exciton band topology, many-body exciton physics in moiré systems (4) Developments in the theory and numerics of moiré systems, and analysis of topological states in this setting -- especially tWTe2. (5) Theory of nonlinear spectroscopy of confined phases (6) Quasiparticle decay and dynamics in CoNb2O6 and other nonsymmorphic symmetry-breaking chains. (7) Further progress on classification of topological phases and numerical techniques. Hartree-Fock renormalized dispersion and the "band shifted" picture of Kekule spiral order. Different possible surface terminations for higher-order topological phases. New model based on glide symmetry can correctly capture quasiparticle breakdown in CoNb2O6.