Detecting Bogomolov-multiplier SPT phases

Develop a method to detect one-dimensional bosonic symmetry-protected topological phases whose cohomology classes lie in the Bogomolov multiplier and therefore induce no nontrivial defect-endpoint charges detectable by string order parameters or the twisted entropic order parameter.

Background

The paper studies the twisted entropic order parameter as a reduced-density-matrix diagnostic for one-dimensional bosonic SPT phases. The construction detects projective symmetry-defect endpoint charges, which are obtained from the slant-product characters associated with the residual SPT cohomology class.

The authors note that some nontrivial SPT phases do not induce any nontrivial charge detectable by string order parameters. In particular, when the Bogomolov multiplier is nontrivial, there are SPT phases that are invisible to both ordinary string-order probes and the proposed twisted entropic order parameter. Detecting these phases would require a probe sensitive to additional topological data, such as higher-genus partition functions or the full boundary projective representation.

References

Note that there exist nontrivial SPT phases for which no nontrivial charge is induced. Such phases cannot be detected even by string order parameters. In fact, when the Bogomolov multiplier is nontrivial, there exist SPT phases that are undetectable by either of these probes. See e.g. for further details. We leave the detection of this class of SPT phases for intriguing future work.

— Detecting one-dimensional bosonic SPT phases via twisted entropic order parameter  (2609.11089 - Fujiki et al., 10 Sep 2026) in Introduction, footnote following the discussion of one-dimensional bosonic SPT classification