1-Form Symmetric Projected Entangled-Pair States

The 1-form symmetry, manifesting as loop-like symmetries, has gained prominence in the study of quantum phases, deepening our understanding of symmetry. However, the role of 1-form symmetries in Projected Entangled-Pair States (PEPS), two-dimensional tensor network states, remains largely underexplored. We present a novel framework for understanding 1-form symmetries within tensor networks, specifically focusing on the derivation of algebraic relations for symmetry matrices on the PEPS virtual legs. Our results reveal that 1-form symmetries impose stringent constraints on tensor network representations, leading to distinct anomalous braiding phases carried by symmetry matrices. We demonstrate how these symmetries influence the ground state and tangent space in PEPS, providing new insights into their physical implications for enhancing ground state optimization efficiency and characterizing the 1-form symmetry structure in excited states.