Topological invariants of rotationally symmetric crystals

Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the classification proves their existence, it does not indicate a way of calculating the values of those invariants. Here we show that a specific set of concentric Wilson loops and line invariants yields the values of all topological invariants in two-dimensional systems with pure rotation symmetry in class AII. Examples of this analysis are given for specific models with twofold and threefold rotational symmetry. We find new invariants that relate to the presence of higher-order topology and corner charges

Topological invariants of rotationally symmetric crystals

Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the classification proves their existence, it does not indicate a way of calculating the values of those invariants. Here we show that a specific set of concentric Wilson loops and line invariants yields the values of all topological invariants in two-dimensional systems with pure rotation symmetry in class AII. Examples of this analysis are given for specific models with twofold and threefold rotational symmetry. We find new invariants that relate to the presence of higher-order topology and corner charges

Topological invariants of rotationally symmetric crystals

Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the classification proves their existence, it does not indicate a way of calculating the values of those invariants. Here we show that a specific set of concentric Wilson loops and line invariants yields the values of all topological invariants in two-dimensional systems with pure rotation symmetry in class AII. Examples of this analysis are given for specific models with twofold and threefold rotational symmetry. We find new invariants that relate to the presence of higher-order topology and corner charges

Ferromagnetism and skyrmions in the Hofstadter–Fermi–Hubbard model

Strongly interacting fermionic systems host a variety of interesting quantum many-body states with exotic excitations. For instance, the interplay of strong interactions and the Pauli exclusion principle can lead to Stoner ferromagnetism, but the fate of this state remains unclear when kinetic terms are added. While in many lattice models the fermions’ dispersion results in delocalization and destabilization of the ferromagnet, flat bands can restore strong interaction effects and ferromagnetic correlations. To reveal this interplay, here we propose to study the Hofstadter–Fermi–Hubbard model using ultracold atoms. We demonstrate, by performing large-scale density-matrix renormalization group simulations, that this model exhibits a lattice analog of the quantum Hall (QH) ferromagnet at magnetic filling factor ν  = 1. We reveal the nature of the low energy spin-singlet states around ν  ≈ 1 and find that they host quasi-particles and quasi-holes exhibiting spin-spin correlations reminiscent of skyrmions. Finally, we predict the breakdown of flat-band ferromagnetism at large fields. Our work paves the way towards experimental studies of lattice QH ferromagnetism, including prospects to study many-body states of interacting skyrmions and explore the relation to high- $T_\mathrm{c}$ superconductivity