Topological Euler Class as a Dynamical Observable in Optical Lattices

The last years have witnessed rapid progress in the topological characterization of out-of-equilibrium systems. We report on robust signatures of a new type of topology—the Euler class—in such a dynamical setting. The enigmatic invariant ($\xi$) falls outside conventional symmetry-eigenvalue indicated phases and, in simplest incarnation, is described by triples of bands that comprise a gapless pair featuring $2\xi$ stable band nodes, and a gapped band. These nodes host non-Abelian charges and can be further undone by converting their charge upon intricate braiding mechanisms, revealing that Euler class is a fragile topology. We theoretically demonstrate that quenching with nontrivial Euler Hamiltonian results in stable monopole-antimonopole pairs, which in turn induce a linking of momentum-time trajectories under the first Hopf map, making the invariant experimentally observable. Detailing explicit tomography protocols in a variety of cold-atom setups, our results provide a basis for exploring new topologies and their interplay with crystalline symmetries in optical lattices beyond paradigmatic Chern insulators.

Figure 1

Although the Chern number of the spectator band $\mathit{n}(\mathit{k})$ is zero, nontrivial Euler class can be related to the wrapping of the sphere $S^2$ by $\mathit{n}(\mathit{k})$ when considered as a Bloch vector (left panel), inducing monopole charge, see Eq. (4). The Hamiltonian induces a $\pi$ rotation around $\mathit{n}(\mathit{k})$. When acted upon initial state ${\mathrm{\Psi }}_0$, the resulting vector $\mathit{a}(\mathit{k})=H(\mathit{k}){\mathrm{\Psi }}_0$, covers the sphere twice within the BZ, clockwise or anticlockwise depending on the orientation of $\mathit{n}(\mathit{k})$ with respect to ${\mathrm{\Psi }}_0$, (red or blue vector on the right). As a result, each time $\mathit{n}(\mathit{k})$ wraps $S^2$, $\mathit{a}(\mathit{k})$ wraps and unwraps, giving rise to a monopole-antimonopole pair.

Figure 2

Hopf-linking structure of Euler class. (a) For Euler number $\xi =2$, the inverse images of $\pm \widehat{x}$ trace a link and antilink on two patches in the BZ. The gap is rescaled so that the time circle runs through $t\in [0,2\pi )$. (b) $\xi =4$ with two pairs of oppositely oriented links. Four different patches of the BZ clearly visible in the top view of the azimuthal angle $\varphi$ of the Bloch sphere.

Figure 3

Proposed experimental schemes. (a) Inverse image of a circle on $S^2$ corresponds to a torus in $(k_x,k_y,t)$ space, which can be measured as polarization $P_z=\cos (\theta )$. For $\xi =2$, different latitudinal circles ($\theta$) form two nested Hopf tori within the BZ (dashed line) due to the monopole-antimonopole signature of Euler class. (b) Snapshots of $P_z$ at different quench times $t\in [0,2\pi )$. (c) The linking of the inverse images can be also detected via a state tomography as detailed in the text. Phase and amplitude of the oscillations in the momentum density coming from TOF reveal the Bloch sphere angles, here illustrated for two representative $\mathit{k}$ values and $t=\pi$.