计算不含本征态的观测量：布洛赫哈密顿量的应用
Computing observables without eigenstates: applications to Bloch Hamiltonians
摘要：
计算哈密顿量的可观量需要在本征态基中取算子的矩阵元素。由于本征态仅定义为依赖于哈密顿参数的任意相位，因此观测量的解析表达式通常难以简化。在这项工作中，我们展示了如何为小希尔伯特空间维数N的所有可观测量可以表示的哈密顿量和它的特征值使用SU（N）代数的性质，我们得出明确的表达式N= 2，3，4。然后，我们提出了多个应用程序专门的情况下，布洛赫电子在晶体中，包括计算的Berry曲率，量子度规和轨道矩，以及一个更复杂的可观察到的非线性响应，线性光电效应（LPGE）。作为一个物理的例子，我们考虑多带哈密顿与节点退化，以显示第一，当从两个到三个波段的模型时，这些观测之间的约束是如何放松的，第二，二次色散如何在小频率下导致恒定的LPGE。
英文摘要：
Calculating the observables of a Hamiltonian requires taking matrix elements of operators in the eigenstate basis. Since eigenstates are only defined up to arbitrary phases that depend on Hamiltonian parameters, analytical expressions for observables are often difficult to simplify. In this work, we show how for small Hilbert space dimension N all observables can be expressed in terms of the Hamiltonian and its eigenvalues using the properties of the SU(N) algebra, and we derive explicit expressions for N=2,3,4. Then we present multiple applications specializing to the case of Bloch electrons in crystals, including the computation of Berry curvature, quantum metric and orbital moment, as well as a more complex observable in non-linear response, the linear photogalvanic effect (LPGE). As a physical example we consider multiband Hamiltonians with nodal degeneracies to show first how constraints between these observables are relaxed when going from two to three-band models, and second how quadratic dispersion can lead to constant LPGE at small frequencies.
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