具有边界 Toeplitz 编织的三维分形子拓扑序
Three-dimensional fracton topological orders with boundary Toeplitz braiding
AI总结：
本文研究三维分形子拓扑序的边界Toeplitz编织现象，通过iCS场论和K-矩阵分析发现特定K-矩阵在开放边界下产生非零振荡的任意子编织相位，并证明该现象无需全局对称性保护。
摘要：
在本文中，我们从理论上研究了一类在热力学极限下展现出奇异边界现象的三维非液体状态。更具体地说，我们关注一类通过沿 z 方向堆叠二维扭曲的 \\(\mathbb{Z}_N\\) 拓扑序层形成的三维 fracton topological orders。相邻层之间相互耦合，同时保持沿 z 方向的平移对称性。其有效场论由无限分量 Chern-Simons (iCS) 场论给出，具有一个尺寸在热力学上极大的整数值对称块三对角 Toeplitz K-矩阵。在沿 z 方向开放边界条件 (OBC) 下，某些 K-矩阵的选择展现出奇异的边界“Toeplitz 编织”，其中位于相反边界的两个任意子之间的相互编织相位角发生振荡，并在热力学极限下保持非零。相比之下，在平凡情况下，相互编织相位角在热力学极限下指数衰减至零。作为必要条件，这种现象要求在 OBC 下的 K-矩阵谱中存在边界零模。我们将非平凡 K-矩阵分为两种不同类型。每个 type-I 具有两个边界零模，而每个 type-II 仅具有一个边界零模。有趣的是，熟悉的 1D “Su-Schrieffer-Heeger model” 的整数值哈密顿矩阵可以用作非平凡 K 矩阵。重要的是，由于 large-gauge-invariance 确保了整数化的 K-矩阵项，因此不需要全局对称性来保护这些零模。我们还提供了数值模拟以及有限尺寸标度，进一步证实了上述解析结果。文中还简要讨论了 iCS 场论中的对称性分数化。受当前场论工作的启发，探索……将是有趣的。
英文摘要：
In this paper, we theoretically study a class of 3D non-liquid states that show exotic boundary phenomena in the thermodynamical limit. More concretely, we focus on a class of 3D fracton topological orders formed via stacking 2D twisted \(\mathbb{Z}_N\) topologically ordered layers along \(z\)-direction. Nearby layers are coupled while maintaining translation symmetry along \(z\) direction. The effective field theory is given by the infinite-component Chern-Simons (iCS) field theory, with an integer-valued symmetric block-tridiagonal Toeplitz \(K\)-matrix whose size is thermodynamically large. With open boundary conditions (OBC) along \(z\), certain choice of \(K\)-matrices exhibits exotic boundary ``Toeplitz braiding'', where the mutual braiding phase angle between two anyons at opposite boundaries oscillates and remains non-zero in the thermodynamic limit. In contrast, in trivial case, the mutual braiding phase angle decays exponentially to zero in the thermodynamical limit. As a necessary condition, this phenomenon requires the existence of boundary zero modes in the \(K\)-matrix spectrum under OBC. We categorize nontrivial \(K\)-matrices into two distinct types. Each type-I possesses two boundary zero modes, whereas each type-II possesses only one boundary zero mode. Interestingly, the integer-valued Hamiltonian matrix of the familiar 1D ``Su-Schrieffer-Heeger model'' can be used as a non-trivial $K$ matrix. Importantly, since large-gauge-invariance ensures integer quantized \(K\)-matrix entries, global symmetries are not needed to protect these zero modes. We also present numerical simulation as well as finite size scaling, further confirming the above analytical results. Symmetry fractionalization in iCS field theory is also briefly discussed. Motivated by the present field-theoretical work, it will be interesting to ... ....
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