通过 Středa 公式将多体 Chern 数与 Luttinger 定理联系起来
Connecting the Many-Body Chern Number to Luttinger's Theorem through Středa's Formula
AI总结：
本文利用 Streda 公式推导出多体 Chern 数的表达式，将其分解为 Ishikawa-Matsuyama 不变量与 Green 函数及自能导数项之和，揭示了其与 Luttinger 定理的联系，阐明了关联拓扑绝缘体量子化霍尔电导率的决定条件。
摘要：
将关联绝缘体的量子化霍尔响应与多体拓扑不变量联系起来是拓扑量子物质领域的一个关键挑战。在此，我们使用 Streda 公式推导出一个多体 Chern 数的表达式，该表达式由单粒子相互作用 Green 函数及其对磁场的导数表示。在这种方法中，我们发现该多体拓扑不变量可以分解为两项贡献之和，即 $N_3[G] + ΔN_3[G]$，其中 $N_3[G]$ 被称为 Ishikawa-Matsuyama 不变量，第二项涉及 Green 函数和自能对磁微扰的导数。作为副产品，我们证明了不变量 $N_3[G]$ 源于 Luttinger 定理对探测磁场的导数。这些结果揭示了在何种条件下，关联拓扑绝缘体的量子化霍尔电导率仅由不变量 $N_3[G]$ 决定，为强关联拓扑相中分数化的起源提供了新见解。
英文摘要：
Relating the quantized Hall response of correlated insulators to many-body topological invariants is a key challenge in topological quantum matter. Here, we use Streda's formula to derive an expression for the many-body Chern number in terms of the single-particle interacting Green's function and its derivative with respect to a magnetic field. In this approach, we find that this many-body topological invariant can be decomposed in terms of two contributions, $N_3[G] + ΔN_3[G]$, where $N_3[G]$ is known as the Ishikawa-Matsuyama invariant and where the second term involves derivatives of Green's function and the self energy with respect to the magnetic perturbation. As a by-product, the invariant $N_3[G]$ is shown to stem from the derivative of Luttinger's theorem with respect to the probe magnetic field. These results reveal under which conditions the quantized Hall conductivity of correlated topological insulators is solely dictated by the invariant $N_3[G]$, providing new insight on the origin of fractionalization in strongly-correlated topological phases.
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