Bockstein braiding statistics

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 $(2\pi i/N)\int A_{d-p}\cup B_{d-q}$, 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 $(2\pi i/N)\int A_{d-p}\cup\beta_N B_{d-q}$, where $\beta_N$ 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.