Periodic Reporting for period 1 - ExcitingTopology (Topological order beyond the equilibrium ground state: driven quantum matter and magnon excitation spectra) Reporting period: 2019-12-01 to 2021-11-30 Summary of the context and overall objectives of the project This project deals with topological band theory, which has become a rather prominent research area in the field of condensed matter physics. In a nutshell, this discipline revolves around utilizing principles from the mathematical domain of topology, which studies properties of objects that are preserved under smooth deformations and cannot be altered without cutting or tearing such as the number of holes, and use them to classify phases of matter. That is, unlike spins aligning in a magnet, topological phases are not classified by symmetry breaking but instead require nonlocal invariants that in essence count generalized “knots or holes”. This has been particularly influential in the characterization of electron systems. A monumental triumph of quantum mechanics and its wave interpretation of particles is that it can shed light on material behaviors. These insights have been reinvigorated in the past years due to mentioned unanticipated connections with topology. Namely, it was found that the wave functions can tie distinctive collective knots when specific symmetries are present, topologically distinguishing different classes of insulators and metals. These topological materials are not only appealing from a theoretical point of view, but have in fact seen several material realizations. Moreover, due their proposed illustrious properties, topological insulators and metals exhibit remarkable phenomena such as protected metallic edge states that could shape next-generation power-efficient electronics. Moreover, excitations in topological materials can in certain scenarios even store and process quantum information, making them a key component in fault-tolerant quantum computing platforms. These fundamental insights are therefore anticipated to be of societal impact in the more distant future.Turning more concretely to the objectives of this action, we can summarize these as finding novel topological phases in out-of-equilibrium settings and excitations spectra. While, the structure of categorizing electronic topological phases in equilibrium has been discovered over the past years, specific evidence pinpoints that other phases outside this modern 'Mendeleev table' exist. In particular, we aim to take up this challenge and unearth new topological phases in systems that are driven out of equilibrium due to periodic matter-light interactions and bosonic excitation spectra, where the latter are effective wave theories that arise for example in crystalline spin structures [magnons] or the elastic arrangement of atoms [phonons]. As in the case of the electronic counterparts, symmetries are anticipated to play a role in providing for the necessary conditions for these topological phases to exist. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The project has received dedicated input and been rather successful. The themes, being the search of novel symmetry-protected topological phases in excitation spectra and out-of-equilibrium context have delivered many results that have been disseminated in 19 publications and, even though covid-19 severely restricted travelling during this period, in 5 on-line theory seminars, a research visit, via the hosting of colleagues/collaborators and an invited talk at the Material Research Society meeting in 2021. On top of delivering on the proposed objectives we have also extended our research programme into new areas such as topological characterizations of quasi-crystalline order.Turning to the out-of-equilibrium results, we highlight that via two publications in Physical Review Letters our efforts indeed resulted in the discovery of novel effects and phases. In the first we found that when two-fold rotations and time-reversal symmetry are present out-of-equilibrium phases can exhibit new quench signatures that can be measured via standard techniques. In the second, we found that Floquet engineering using bicircular light is a versatile way to control magnetic symmetries and topology in materials and showed that in Weyl materials, such as Cd3As2 in particular, this leads to novel topological phases that can be observed in the laboratory using current techniques. Finally, in an Editor’s suggested Physical Review Research publication, we introduced a new class of two-dimensional topological materials known as optical N-insulators that possess obstructions to constructing localized molecular polarizabilities. That is, we found a manner to characterize electromagnetic linear response theory in terms of topological winding numbers. Turning to the second theme, we have also found substantial results. Firstly, we found in a series of two Physical Review B papers, one of which was chosen as an Editor’s suggestion, that there exists a general class of electronic configurations within a set of anti-ferromagnetic-compatible space groups that must be topological, thereby leading to novel effect such as specific boundary states. In addition, we worked with experimentalist to characterize and observe new topological states in specifically designed acoustic metamaterals, being a striking example of an excitation spectrum. This study was published in Nature Physics. In addition, we contributed, using our expertise on the interplay of topology and magnetism, to a rather prominent discovery published in Nature that experimentally observed a novel splitting effect in magnetic topological materials. Finally, we also addressed new typed of topological charges in phonic spectra, using new topological insights. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) As outlined in the above section, the project has been rather successful, delivering a high number of publications, 19 in total, that moreover featured in top-tier journals. Here we mention in particular, a Nature, Nature Physics, two Nature Commutations and two Physical Review Letters publications. These top-tier publications will ensure a high readership and therefore high impact. More importantly, apart from delivering on the proposed topics, we also been productive in setting new directions. Indeed, during the course of the action, we found that a subset of the novel topological phases can be related to a new concept called multi-gap topology. This has set a rather rich new research direction. Similarly, we have also started new pursuits in quasi-periodic systems. Most importantly, some of our findings are now actively pursued in the laboratory, showing that our fundamental results in fact are also becoming of increasing impact outside the direct field of theoretical condensed matter physics. Given the fundamental nature of our results, it is a bit early to tell if these finding will find applications in the more distant future, but given the swift response outside our direct circle of colleagues, we are hopeful that our results will in fact be of significant as well as enduring impact. Using circular (CL) of bi-circular (BCL) light magnetic topologyies and edge states can be tuned.