Tensor Network States, Entanglement, and Anomalies of Topological Phases of Matters

Higher-form symmetries acting on extended objects have reshaped our understanding of quantum phases and topological order, yet explicit incorporation of 1-form symmetry in tensor-network states remains unexplored. Here, we introduce a pull-through framework to embed 1-form symmetries into Projected Entangled-Pair States (PEPS), providing an operator-level definition of gauge flux and algebraic constraints on virtual symmetry matrices that encode anomalous braiding phases and enable efficient optimization of topologically ordered states. Our framework offers diagnostic tools for anomalies in topological phases and paves the way for more efficient tensor-network simulations in a variety of models with higher-form symmetries. Quantum many-body systems with symmetries acting on extended objects host exotic phases that are difficult to simulate. We develop a pull-through tensor-network framework that embeds 1-form symmetries in projected entangled-pair states, enabling symmetry-resolved optimization and diagnostics of topologically ordered states.Higher-form symmetries acting on extended objects have reshaped our understanding of quantum phases and topological order, yet explicit incorporation of 1-form symmetry in tensor-network states remains unexplored. Here, we introduce a pull-through framework to embed 1-form symmetries into Projected Entangled-Pair States (PEPS), providing an operator-level definition of gauge flux and algebraic constraints on virtual symmetry matrices that encode anomalous braiding phases and enable efficient optimization of topologically ordered states. Our framework offers diagnostic tools for anomalies in topological phases and paves the way for more efficient tensor-network simulations in a variety of models with higher-form symmetries. Quantum many-body systems with symmetries acting on extended objects host exotic phases that are difficult to simulate. We develop a pull-through tensor-network framework that embeds 1-form symmetries in projected entangled-pair states, enabling symmetry-resolved optimization and diagnostics of topologically ordered states.

arXiv:2407.16531Communications PhysicsJul 14, 20260 citations

Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases

Ke Ding, Haoran Zhang, Bowei Liu +1 more

We generalize the topological response theory to detect the boundary anomalies of linear subsystem symmetries. This approach allows us to distinguish different subsystem symmetry-protected topological (SSPT) phases and uncover new ones. We focus on the cases where the mixed anomaly exists within the adjacent subsystems. Using numerical simulations, we demonstrate the power of this method by identifying strong and weak ${Z}_{2}^{\ensuremath{\tau}}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{\ensuremath{\sigma}}$ SSPT phases in a tunable tensor network state. Our analysis reveals an intrinsic ${Z}_{2}$ SSPT phase characterized by its degenerate entanglement spectrum. Furthermore, we extend the anomaly indicator to mixed-state density matrices and show that quantum anomalies of subsystem symmetry can persist under both uniform and alternating disorders. This finding establishes a connection between boundary quantum anomalies in pure and mixed states. Our work provides a numerical method to detect quantum anomalies of subsystem symmetries, offering new insights into the study of topological quantum phases.We generalize the topological response theory to detect the boundary anomalies of linear subsystem symmetries. This approach allows us to distinguish different subsystem symmetry-protected topological (SSPT) phases and uncover new ones. We focus on the cases where the mixed anomaly exists within the adjacent subsystems. Using numerical simulations, we demonstrate the power of this method by identifying strong and weak ${Z}_{2}^{\ensuremath{\tau}}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{\ensuremath{\sigma}}$ SSPT phases in a tunable tensor network state. Our analysis reveals an intrinsic ${Z}_{2}$ SSPT phase characterized by its degenerate entanglement spectrum. Furthermore, we extend the anomaly indicator to mixed-state density matrices and show that quantum anomalies of subsystem symmetry can persist under both uniform and alternating disorders. This finding establishes a connection between boundary quantum anomalies in pure and mixed states. Our work provides a numerical method to detect quantum anomalies of subsystem symmetries, offering new insights into the study of topological quantum phases.

arXiv:2412.07563Physical review. B./Physical review. BMay 19, 20253 citations

Fermionic projected entangled-pair states and topological phases

Nick Bultinck, Dominic J. Williamson, Jutho Haegeman +1 more

We study fermionic matrix product operator algebras and identify the associated algebraic data. Using this algebraic data we construct fermionic tensor network states in two dimensions that have non-trivial symmetry-protected or intrinsic topological order. The tensor network states allow us to relate physical properties of the topological phases to the underlying algebraic data. We illustrate this by calculating defect properties and modular matrices of supercohomology phases. Our formalism also captures Majorana defects as we show explicitly for a class of symmetry-protected and intrinsic topological phases. The tensor networks states presented here are well-suited for numerical applications and hence open up new possibilities for studying interacting fermionic topological phases.We study fermionic matrix product operator algebras and identify the associated algebraic data. Using this algebraic data we construct fermionic tensor network states in two dimensions that have non-trivial symmetry-protected or intrinsic topological order. The tensor network states allow us to relate physical properties of the topological phases to the underlying algebraic data. We illustrate this by calculating defect properties and modular matrices of supercohomology phases. Our formalism also captures Majorana defects as we show explicitly for a class of symmetry-protected and intrinsic topological phases. The tensor networks states presented here are well-suited for numerical applications and hence open up new possibilities for studying interacting fermionic topological phases.

arXiv:1707.00470Journal of Physics A Mathematical and TheoreticalNov 10, 201753 citations

Topological tensor products and quantum entanglement

Florian Sokoli

Jan 1, 20170 citations

{Z}_{2}

SSPT phase characterized by its degenerate entanglement spectrum. Furthermore, we extend the anomaly indicator to mixed-state density matrices and show that quantum anomalies of subsystem symmetry can persist under both uniform and alternating disorders. This finding establishes a connection between boundary quantum anomalies in pure and mixed states. Our work provides a numerical method to detect quantum anomalies of subsystem symmetries, offering new insights into the study of topological quantum phases.