Symmetry-twisted multi-entropies: Order parameters for 2D SPT phases
Abstract
Although symmetry-protected topological phases (SPTs) can be distinguished by their entanglement properties, it has been unclear how to extract this information directly from expectation values beyond the 1D case. Here, we close this gap and propose a pair of nonlocal order parameters that can detect and distinguish all bosonic SPTs in 2D protected by internal discrete, Abelian unitary symmetries. The desired topological invariants are extracted by these quantities by effectively simulating the SPT path integral on topologically non-trivial spacetime manifolds. Our order parameters are defined in terms of expectation values of partial symmetry and permutation operations acting on fixed numbers of replicas of the system in finite spatial regions. These expectation values correspond to symmetry-twisted versions of multipartite entanglement quantities known as multi-entropies. We show explicitly that our two order parameters detect symmetry-protected four-party and six-party entanglement, respectively, and we constrain possible "spurious" contributions. We analytically test our proposal in fixed-point lattice models. Our results suggest multipartite entanglement to be a defining feature of SPTs; indeed, we expect our methods to generalize to fermionic and higher-dimensional systems.
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BibTeX
@misc{sohal2026symmetrytwisted,
title = {{Symmetry-twisted multi-entropies: Order parameters for 2D SPT phases}},
author = {Ramanjit Sohal and Michael Levin and Ruben Verresen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.12023},
eprint = {2607.12023},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.12023},
}
RIS
TY - UNPB
TI - Symmetry-twisted multi-entropies: Order parameters for 2D SPT phases
AU - Ramanjit Sohal
AU - Michael Levin
AU - Ruben Verresen
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2607.12023
UR - https://arxiv.org/abs/2607.12023
AB - Although symmetry-protected topological phases (SPTs) can be distinguished by their entanglement properties, it has been unclear how to extract this information directly from expectation values beyond the 1D case. Here, we close this gap and propose a pair of nonlocal order parameters that can detect and distinguish all bosonic SPTs in 2D protected by internal discrete, Abelian unitary symmetries. The desired topological invariants are extracted by these quantities by effectively simulating the SPT path integral on topologically non-trivial spacetime manifolds. Our order parameters are defined in terms of expectation values of partial symmetry and permutation operations acting on fixed numbers of replicas of the system in finite spatial regions. These expectation values correspond to symmetry-twisted versions of multipartite entanglement quantities known as multi-entropies. We show explicitly that our two order parameters detect symmetry-protected four-party and six-party entanglement, respectively, and we constrain possible "spurious" contributions. We analytically test our proposal in fixed-point lattice models. Our results suggest multipartite entanglement to be a defining feature of SPTs; indeed, we expect our methods to generalize to fermionic and higher-dimensional systems.
ER -