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arXiv:2607.02280

Bockstein braiding statistics

Po-Shen Hsin, Yu-An Chen

quant-phcond-mat.str-elhep-thmath-phmath.QASubmitted Jul 2, 2026• Announced Jul 3, 2026• 0 citations
v1 · Jul 2, 2026v2 · Jul 9, 2026v3 · Jul 16, 2026
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Abstract

Braiding statistics, from the Aharonov-Bohm phase to anyons in fractional quantum Hall systems, play a central role in quantum physics. For $p$- and $q$-dimensional excitations in $d$ spatial dimensions, ordinary braiding requires $p+q=d-2$. In a field-theoretic description of $\mathbb Z_N$ excitations, ordinary braiding is described by the linking response $(2πi/N)\int A_{d-p}\cup B_{d-q}$, where $A_{d-p}$ and $B_{d-q}$ are background fields coupled to the two excitation types. In this work, we identify new mutual statistics in the adjacent case $p+q=d-1$. For two invertible excitations obeying $\mathbb Z_N$ fusion, one can choose local creation operators $X$ and $Y$ whose supports have a staggered one-dimensional overlap. The closed unitary process $W_N(X,Y)=(Y^{-1}X^{-1})^N(YX)^N$ measures the resulting mutual statistic. Its field-theory description is $(2πi/N)\int A_{d-p}\cupβ_N B_{d-q}$, where $β_N$ is the Bockstein operation; we therefore call the invariant Bockstein braiding statistics. The construction yields particle-particle statistics in one dimension, particle-loop statistics in two dimensions, and loop-loop or particle-membrane statistics in three dimensions. Nontrivial Bockstein braiding statistics obstructs simultaneous condensation of the two $\mathbb Z_N$ excitations. It also rules out a fully symmetric gapped phase for systems with the corresponding mixed anomaly and implies symmetry fractionalization when one of the $\mathbb Z_N$ symmetries is broken.

Comments
23 pages, 5 figures

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