Emanant and emergent symmetry-topological-order from low-energy spectrum

Low-energy emergent and emanant symmetries can be anomalous, higher-group, or non-invertible. Such symmetries are systematically captured by topological orders in one higher dimension, known as symmetry topological orders (symTOs). Consequently, identifying the emergent or emanant symmetry of a system is not simply a matter of determining its group structure, but rather of computing the corresponding symTO. In this work, we develop a method to compute the symTO of 1+1D systems by analyzing their low-energy spectra under closed boundary conditions with all possible symmetry twists. Applying this approach, we show that the gapless antiferromagnetic (AF) spin-1/2 Heisenberg model possesses an exact emanant symTO corresponding to the D8 quantum double, when restricted to the ℤ^x₂ ×ℤ^z₂ subgroup of the SO(3) spin-rotation symmetry and lattice translations. Moreover, the AF phase exhibits an emergent SO(4) symmetry, whose exact components are described jointly by the symTO and the SO(3) spin-rotations. Using condensable algebras in symTO, we further identify several neighboring phases accessible by modifying interactions among low-energy excitations: (1) a gapped dimer phase, connected to the AF phase via an SO(4) rotation, (2) a commensurate collinear ferromagnetic phase that breaks translation by one site with a ω ∼ k2 mode, (3) an incommensurate, translation-symmetric ferromagnetic phase featuring both ω ∼ k2 and ω ∼ k modes, connected to the previous phase by an SO(4) rotation, and (4) an incommensurate ferromagnetic phase that breaks translation by one site with both ω ∼ k2 and ω ∼ k modes.