Bockstein braiding statistics

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.