Second Euler number in four dimensional synthetic matter

Abstract

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.

Tuning Second Chern Number in a Four-Dimensional Topological Insulator by High-Frequency Time-Periodic Driving

“…[21] The 𝑛th Chern numbers can characterize topological phases in 2𝑛 dimensions. [22][23][24][25][26][27] The 4D topological insulator (TI) exhibiting the 4D QHE supports fully gapped bulk and gapless three-dimensional (3D) boundary states, characterized by the second Chern number. [23,[28][29][30][31] The 4D TI is impossible to realize in real materials due to the limitation of spatial dimensions.…”

Quantum geometry beyond projective single bands

“…Multi-gap topologies have been seen in several meta-material studies [44][45][46][47][48][49], and the proposed unusual quench dynamical behavior of Euler class phases [50] has been observed in trapped ion insulators [48]. Similarly, recent proposals in phonon bands and electronic structures under strain have shown to provide feasible routes towards real materials [51][52][53][54][55][56] and novel anomalous phases [57].…”