  Two-dimensional Euler insulators are novel kind of systems that host
multi-gap topological phases, quantified by a quantised first Euler number in
their bulk. Recently, these phases have been experimentally realised in
suitable two-dimensional synthetic matter setups. Here we introduce the second
Euler invariant, a familiar invariant in both differential topology
(Chern-Gauss-Bonnet theorem) and in four-dimensional Euclidean gravity, whose
existence has not been explored in condensed matter systems. Specifically, we
firstly define 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. In particular, we
derive from homotopy arguments the minimal Bloch Hamiltonian form from which
the tight-binding models of any second Euler phase can be generated.
Considering more concretely the gapped Euler phase associated with the tangent
bundle of the four-sphere, we show that the bulk band structure of the
nontrivial 4D Euler phase necessarily exhibits triplets of linked nodal
surfaces (where the three types of nodal surfaces are formed by the crossing of
the three successive pairs of bands within one four-band subspace). 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 and open doors for implementing such
unexplored higher-dimensional phases in artificial engineered systems, ranging
from ultracold atoms to photonics and electric circuits.

Mesoscale and Nanoscale Physics;Quantum Gases;General Relativity and Quantum Cosmology;High Energy Physics - Theory