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arXiv:2503.04510 · Physical review. B./Physical review. B

Detection of two-dimensional symmetry protected topological order with partial symmetries

Alex Turzillo, Naren Manjunath, José Garre-Rubio

Published Jun 20, 2025• 0 citations
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Abstract

A method of using partial symmetries to distinguish two dimensional symmetry protected topological (SPT) phases of on-site, unitary symmetries is proposed. This novel order parameter takes a wavefunction, such as a ground state of a lattice model, and detects its SPT invariants as expectation values of finitely supported operators, without the need for flux insertion. The construction exploits the rotational symmetry of the lattice to extract on-site SPT invariants, building upon prior work on probing crystalline SPT phases with partial rotations. The method is demonstrated by computing the order parameter analytically on group cohomology models and numerically on a family of states interpolating between the CZX state and a trivial state. Its robustness is suggested by interpreting partial symmetries as generating the topological partition functions of lens spaces.

DOI
10.1103/mfwg-svcp

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A method of using partial symmetries to distinguish two dimensional symmetry protected topological (SPT) phases of on-site, unitary symmetries is proposed. This novel order parameter takes a wavefunction, such as a ground state of a lattice model, and detects its SPT invariants as expectation values of finitely supported operators, without the need for flux insertion. The construction exploits the rotational symmetry of the lattice to extract on-site SPT invariants, building upon prior work on probing crystalline SPT phases with partial rotations. The method is demonstrated by computing the order parameter analytically on group cohomology models and numerically on a family of states interpolating between the CZX state and a trivial state. Its robustness is suggested by interpreting partial symmetries as generating the topological partition functions of lens spaces.
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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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Symmetry-protected topological states cannot be deformed to a trivial state so long as the symmetry is preserved, yet there is no local order parameter that can distinguish them from a trivial state. We demonstrate how to detect whether a two-dimensional ground state has symmetry-protected topological order; the measurements play a similar role as the topological entanglement entropy does for detecting anyons. For any finite Abelian onsite symmetry, the measurement completely determines the third cohomology class that characterizes the order. The proposed measurement is validated numerically using the infinite density matrix renormalization group for a model with Z2{\mathbb{Z}}_{2}Z2​ symmetry-protected order.
According to a recent theoretical conjecture for a classification of symmetry protected topological (SPT) phases which can arise in a collection of lattice boson models called group cohomology models, each model belongs to a distinct SPT phase, and each SPT phase can be realized by a model. Here the authors show that for SPT phases with symmetry groups that are finite, Abelian, and unitary, each group cohomology model can be characterized by a unique set of topological invariants and therefore belongs to a distinct SPT phase. In two dimensions it is further shown that the complete set of group cohomology models accounts for all SPT phases, while in three dimensions evidence pointing toward such a proof is found.
We propose an order parameter for the symmetry-protected topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the SPT entanglement, is defined as the entanglement between AAA and BBB, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region CCC is measured and known, where CCC is a connected region surrounded by A,BA, BA,B, and the boundaries of the system. In the case of one-dimensional systems we prove that in the limit where AAA and BBB are large and far from each other compared to the correlation length, the SPT entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the nontrivial phases. Moreover, we show that the SPT entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Also, we show that there is an intriguing connection between SPT entanglement and the Fourier transform of the string order parameters, which are the traditional tool for detecting SPT phases. This leads to an algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, we discuss implications of our results in the context of measurement-based quantum computation.