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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• Revised Jul 9, 2026 (v2)
v1 · Jul 2, 2026v2 · Jul 9, 2026v3 · Jul 16, 2026
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Abstract

Braiding statistics, familiar from anyons in fractional quantum Hall systems, are a central manifestation of topology in quantum physics. Ordinary braiding extends naturally to higher-dimensional excitations: a $p$-dimensional excitation and a $q$-dimensional excitation can braid in $d=p+q+2$ spatial dimensions. In this work, we identify a new type of mutual statistics that exists in one lower spatial dimension, $d=p+q+1$. This includes particle-particle statistics in one dimension, particle-loop statistics in two dimensions, and loop-loop or particle-membrane statistics in three dimensions. The corresponding field-theory response is governed by the Bockstein homomorphism, so we call the invariant Bockstein braiding statistics. On lattices, the Bockstein statistics is measured by the Berry phase accumulated in a universal microscopic unitary process built from local excitation operators. We further show that nontrivial Bockstein braiding is the statistical manifestation of a mixed anomaly of the corresponding symmetries. This anomaly rules out a fully symmetric gapped phase, obstructs simultaneous condensation of the two excitations, and implies fractionalization of higher-form symmetries. We illustrate these consequences in a $(1{+}1)$-dimensional spin-$\frac12$ chain, where Bockstein braiding statistics detects the mixed anomaly between $\prod_i X_i$ and $\prod_i \mathrm{CZ}_{i,i+1}$, and in strongly coupled $(3{+}1)$-dimensional continuum gauge theories.

Comments
28 pages, 6 figures

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2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations. In 4+1d, some topological phases contain only fractional loop excitations. What kind of loop statistics exist? We study the 4+1d gauge theory with 2-form Z2\mathbb{Z}_2Z2​ gauge field (the loop-only toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self "exchange" statistics are all trivial. In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their self-statistics. They tunnel into each other across 3+1d invertible domain walls which in turn give explicit unitary circuits that map the loop excitations into each other. The SL(2,Z2\mathbb{Z}_2Z2​) symmetry that permutes the loops, however, cannot be consistently gauged and we discuss the associated obstruction in the process. Moreover, we discuss a gapless boundary condition dubbed the "fractional Maxwell theory" and show how it can be Higgsed into gapped boundary conditions. We also discuss the generalization of these results from the Z2\mathbb{Z}_2Z2​ gauge group to ZN\mathbb{Z}_NZN​.
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