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.

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生成和检测具有较高Chern数的拓扑阶段

具有逆时对称性和Chern数的拓扑阶段 $\left|\mathcal{C}\right|\ge \mathrm{2个}$有限元法是基本感兴趣的方法，但仍不清楚如何在仅在静态晶格的最近邻之间跳跃的自旋轨道耦合粒子范式内设计所需的拓扑哈密顿量。我们表明，当自旋轨道耦合满足自旋和空间旋转对称性的组合时，会出现具有较高Chern数的相位。我们利用这个结果来构造最小的两频带紧束缚哈密顿量，这些哈密顿量表现出$\left|\mathcal{C}\right|=\mathrm{2个}，3$并表明可以从布里渊区高对称晶体动量处的粒子自旋极化推断能带之一的切恩数。利用这些见解，我们提供了一个具体的实验方案，用于特定实现时间反转的拓扑阶段，其中$\left|\mathcal{C}\right|=\mathrm{2个}$用于自旋轨道耦合的三角晶格上的超冷原子气体。可使用塞曼光谱法直接测量Chern数；对于费米子，自旋幅度可以通过飞行时间直接测量，而对于玻色子，这之前是短暂的布洛赫振荡。我们的结果为超冷原子气体中具有较高Chern值的新型拓扑相的实现和检测提供了途径。