We study the algebraic structure of electron density operators in gapless Weyl fermion systems in 𝑑=3,5,7,⋯ spatial dimensions and in topological insulators (without any protecting symmetry) in 𝑑=4,6,8,⋯ spatial dimensions. These systems are closely related by the celebrated bulk-boundary correspondence. Specifically, we study the higher bracket—a generalization of commutator for more than two operators—of electron density operators in these systems. For topological insulators, we show that the higher-bracket algebraic structure of density operators structurally parallels with the Girvin-MacDonald-Platzman algebra (the 𝑊1+∞ algebra), the algebra of electron density operators projected onto the lowest Landau level in the quantum Hall effect. By the bulk-boundary correspondence, the bulk higher-bracket structure mirrors its counterparts at the boundary. Specifically, we show that the density operators of Weyl fermion systems, once normal-ordered with respect to the ground state, their higher bracket acquires a 𝑐-number part. This part is an analog of the Schwinger term in the commutator of the fermion current operators. We further identify this part with a cyclic cocycle, which is a topological invariant and an element of Connes' noncommutative geometry.

中文翻译：

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外尔费米子系统和拓扑绝缘体中密度算子的高架结构


我们研究了 d=3,5,7，⋯ 空间维度的无间隙外尔费米子系统和 d=4,6,8，⋯ 空间维度的拓扑绝缘体（没有任何保护对称性）中电子密度算子的代数结构。这些系统通过著名的 bulk-boundary 对应关系密切相关。具体来说，我们研究了这些系统中电子密度算子的较高范围——两个以上算子的换向子的泛化。对于拓扑绝缘体，我们表明密度算子的高括号代数结构在结构上与 Girvin-MacDonald-Platzman 代数（W1+∞代数）平行，电子密度算子的代数投影到量子霍尔效应中的最低朗道能级。通过体边界对应关系，体大括号结构在边界处反映了其对应结构。具体来说，我们表明外尔费米子系统的密度算子，一旦相对于基态呈正序，它们的上级括号就会获得一个 c 数部分。这部分是费米子电流算子换向器中 Schwinger 项的模拟。我们进一步用循环共循环来识别这部分，它是一个拓扑不变量，也是 Connes 非交换几何的一个元素。