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The strange correlator [Phys. Rev. Lett. 112, 247202 (2014)] has beenproposed as a measure of symmetry protected topological order in one- andtwo-dimensional systems. It takes the form of a spin-spin correlation function,computed as a mixed overlap between the state of interest and a trivial localproduct state. We demonstrate that it can be computed exactly (asymptotically,in the Monte Carlo sense) for various Affleck-Kennedy-Lieb-Tasaki states bydirect evaluation of the wave function within the valence bond loop gasframework. We present results for lattices with chain, square, honeycomb, cube,diamond, and hyperhoneycomb geometries. In each case, the spin quantum number Sis varied such that 2S (the number of valence bonds emerging from each site)achieves various integer multiples of the lattice coordination number. Weintroduce the concept of strange correlator loop winding number and point toits utility in testing for the presence of symmetry protected topologicalorder.

Detection of symmetry-protected topological order in AKLT states by exact evaluation of the strange correlator