AI 中文总结
针对空间维度d中p维和q维激发，当p+q=d-1时，提出Bockstein编织统计，通过Bockstein操作描述互统计，阻碍同时凝聚并排除全对称能隙相。
AI 中文摘要
编织统计，从阿哈罗诺夫-玻姆相位到分数量子霍尔系统中的任意子，在量子物理中扮演核心角色。对于d空间维度中的p维和q维激发，普通编织要求p+q=d-2。在Z_N激发的场论描述中，普通编织由链接响应(2πi/N)∫A_{d-p}∪B_{d-q}描述，其中A_{d-p}和B_{d-q}是耦合到两种激发类型的背景场。在这项工作中，我们在相邻情况p+q=d-1中识别出新的互统计。对于两个服从Z_N融合的可逆激发，可以选择局部创建算子X和Y，其支撑具有交错的一维重叠。闭合幺正过程W_N(X,Y)=(Y^{-1}X^{-1})^N(YX)^N测量了由此产生的互统计。其场论描述为(2πi/N)∫A_{d-p}∪β_N B_{d-q}，其中β_N是Bockstein操作；因此我们将该不变量称为Bockstein编织统计。该构造产生了一维中的粒子-粒子统计、二维中的粒子-环统计以及三维中的环-环或粒子-膜统计。非平凡的Bockstein编织统计阻碍了两个Z_N激发的同时凝聚。它还排除了具有相应混合反常的系统的完全对称能隙相，并在其中一个Z_N对称性破缺时意味着对称性分数化。
英文摘要
Braiding phenomena, from the charge-flux Aharonov-Bohm effect to anyonic statistics in fractional quantum Hall systems, are paradigmatic manifestations of topology in quantum physics. Ordinary mutual braiding between $p$- and $q$-dimensional excitations occurs in $d=p+q+2$ spatial dimensions. In this work, we introduce a universal construction of mutual statistics in the adjacent dimension $d=p+q+1$, applicable to excitations obeying $\mathbb Z_N$ fusion for arbitrary $N$ and all excitation dimensions $p$ and $q$. The corresponding invariant is the Berry phase accumulated in a simple $4N$-step microscopic unitary process built from local excitation operators on lattices. This process measures the linking of one excitation with the $N$-fold fusion junction of the other, encompassing particle-particle statistics in one dimension, particle-loop statistics in two dimensions, and loop-loop or particle-membrane statistics in three dimensions. We establish the quantization and bilinearity of the invariant and show that its field-theory response is governed by the Bockstein homomorphism, motivating the name Bockstein braiding statistics. Interpreting the excitation operators as open symmetry operators turns the same invariant into a direct microscopic diagnostic of mixed anomalies between symmetries. We demonstrate this diagnostic in a (1+1)D spin chain, where the nontrivial Bockstein braiding phase proves the mixed anomaly between the spin-flip symmetry $\prod X$ and the nearest-neighbor controlled-$Z$ symmetry $\prod CZ$. We construct explicit (2+1)D and (3+1)D lattice analogs, yielding new anomalous symmetry pairs, and apply the framework to strongly coupled (3+1)D continuum gauge theories. Nontrivial Bockstein braiding rules out a fully symmetric gapped phase, obstructs simultaneous condensation of the two excitations, and implies fractionalization of higher-form symmetries.
Comments44 pages, 8 figures. Additional examples included
