Second Euler number in four-dimensional synthetic matter

Two-dimensional Euler insulators are novel kind of spinless fermionic systems that support topological phases, which exhibit a quantised first Euler number in their bulk. This topological invariant is protected by the spacetime inversion symmetry. Recently, these phases have been experimentally realised in suitable two-dimensional synthetic matter setups. Artificial engineered systems, ranging from ultracold atoms to photonics and electric circuits, offer the unique way to implement higher-dimensional phases that cannot exist in real quantum materials. Although the second Euler invariant is a familiar invariant in both differential topology (Gauss-Bonnet theorem) and in four-dimensional Euclidean gravity, its existence in synthetic matter has never been explored so far. In this talk, we firstly introduce and describe two specific novel models in four dimensions that support a non-zero second Euler number in the bulk together with peculiar gapless boundary states. Secondly, we discuss its robustness in general spacetime-inversion invariant phases and its role in the classification of topological degenerate real bands through real Grassmannians. Finally, we show how to engineer these new topological phases in a four-dimensional ultracold atom setup. Our results naturally generalize the second Chern and spin Chern numbers to the case of four-dimensional phases that are characterised by real Hamiltonians.