Generating and detecting topological phases with higher Chern number

Topological phases with broken time-reversal symmetry and Chern number $\left|\mathcal{C}\right|\ge 2$ are of fundamental interest, but it remains unclear how to engineer the desired topological Hamiltonian within the paradigm of spin-orbit-coupled particles hopping only between nearest neighbours of a static lattice. We show that phases with higher Chern number arise when the spin-orbit coupling satisfies a combination of spin and spatial rotation symmetries. We leverage this result both to construct minimal two-band tight-binding Hamiltonians that exhibit $\left|\mathcal{C}\right|=2,3$ phases and to show that the Chern number of one of the energy bands can be inferred from the particle spin polarization at the high-symmetry crystal momenta in the Brillouin zone. Using these insights, we provide a detailed experimental scheme for the specific realization of a time-reversal-breaking topological phase with $\left|\mathcal{C}\right|=2$ for ultracold atomic gases on a triangular lattice subject to spin-orbit coupling. The Chern number can be directly measured using Zeeman spectroscopy; for fermions the spin amplitudes can be measured directly via time of flight, while for bosons this is preceded by a short Bloch oscillation. Our results provide a pathway to the realization and detection of novel topological phases with higher Chern numbers in ultracold atomic gases.

Figure 1

Loop $\gamma$ in the BZ of a two-dimensional Bravais lattice with $C_4$ (left) and $C_6$ (right) symmetry. The area $A\left(\gamma \right)$ enclosed by $\gamma$ is shaded in each case.

Figure 2

Color map of spin polarization over the BZ for the trivial (left) and the topological (right) phase with $\mathcal{C}=-2$. The crystal momenta $k_x$ and $k_y$ are in units of $\pi /a$, with $a=|{\mathbf{a}}_q|$ denoting the lattice constant. The bright red (blue) color represents $+1$ ($-1$) value of the spin polarization ${\langle {\sigma }_z\rangle }_{\mathbf{k}}$. Parameters used were $t_{\mathrm{so}}=0.5,t_z=1, M_z=4$ (trivial phase), and $M_z=1$ (topological phase).

Figure 3

Proposed experimental setup for the realization and the detection of a $\left|\mathcal{C}\right|=2$ topological phase. In the figure, LA1 and LA2 denote two lasers with the same frequency ${\omega }_1$ polarized out of plane (along $z$ axis) and in plane (in $x-y$ plane), respectively. BS denotes a $50:50$ beam splitter, M1–M4 are polarization-preserving mirrors, and AOM1 and AOM2 are acousto-optic modulators. An external magnetic field of strength $B$ is applied out of plane.

Figure 4

The $\mathrm{\Lambda }$ configuration, responsible for the Raman transition, formed by $|\uparrow \rangle$ and $|\downarrow \rangle$ states with the excited state $|F=1,m_F=-1\rangle$ in the $2P_{1/2}$ multiplet. $F=2$ levels of the $2P_{1/2}$ multiplet are not shown for simplicity.