高维参数空间的拓扑带理论
Topological Band Theory for High-Dimensional Parameter Spaces
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中文总结 AI 辅助
本研究探讨高维参数空间中量子霍尔系统的拓扑带理论，发现拓扑不变量涉及高阶曲率，并通过亏格二 Bolza 曲面的双曲带理论实例验证，为强关联物质研究提供新视角。
中文摘要 AI 辅助
我们研究了量子霍尔系统的拓扑带理论，其中外部规范势的绝热磁通参数形成一个高维流形。这种情况既出现在亏格 $g > 1$ 的空间黎曼曲面上的强关联物质中，也出现在与 $p$-形式规范势耦合的 $4\ell+2 = 2p$ 维紧致定向流形上物质的正式研究中。在这些情况下，定义在参数空间上的底层拓扑不变量比低维（即二维）情况丰富得多，并且涉及曲率（即第一陈类）以及更高阶的曲率不变量。这些高阶曲率不变量对应于拓扑保护的贡献，类似于由物理电流算符的关联函数构造的库伯公式。我们通过基于亏格二 Bolza 黎曼曲面的双曲带理论的一个明确例子来说明这些一般性考虑，该例子最近已在合成维度平台上进行了模拟。
英文摘要
We study the topological band theory of quantum Hall systems in which the adiabatic flux parameters of an external gauge potential form a high-dimensional manifold. This situation arises both for strongly correlated matter on spatial Riemann surfaces of genus $g > 1$, as well as in the formal study of matter on compact oriented manifolds of dimension $4\ell+2 = 2p$ coupled to $p$-form gauge potentials. In these cases, the underlying topological invariants defined over the parameter space are significantly richer than the low-dimensional (i.e., two-dimensional) case and involve both the curvature (i.e., first Chern class) as well as higher order curvature invariants. These higher curvature invariants correspond to topologically protected contributions to Kubo-like formulae, constructed from correlation functions of the physical current operators. We illustrate these general considerations with an explicit example from hyperbolic band theory based on the genus-two Bolza Riemann surface, a case which has recently been simulated on a synthetic-dimension platform.
发表机构
- University of Pennsylvania（宾夕法尼亚大学）
- University of Saskatchewan（萨斯喀彻温大学）
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