Spin models with critical ground space degeneracy from Lie algebra relations

Abstract

We introduce an $\mathrm{SO}(3)$-symmetric spin model derived from the algebraic structure of $\mathfrak{so}(3)$: It is obtained as the nearest-neighbor parent Hamiltonian of a Matrix Product State (MPS) built from the generators of the Lie algebra $\mathfrak{so}(3)$ and the identity matrix, and therefore encodes the quadratic relations of the Lie algebra. We characterize the ground space structure of the resulting model and show that for open boundary conditions (OBC), it exhibits a quadratic ground space degeneracy, given by one irreducible representation (irrep) of each odd dimension (integer spin). For periodic boundary conditions (PBC), it has a linear ground space degeneracy, consisting of a singlet --- namely, the MPS underlying the model --- and a ferromagnet (the irrep with maximal spin), and thus exhibits gapless excitations. We also generalize the construction and the analysis of the OBC ground space to $\mathrm{SO}(d)$ and other compact simple Lie groups. The MPS on which the model is based is an injective MPS, and therefore, its 3-site parent Hamiltonian has a unique ground space and is gapped. The observed critical behavior thus has its origin in the small parent Hamiltonian considered. The model's algebraic ground space scaling thus provides an example distinct from that reported in (Schuch et al., 2025, arXiv:2503.10767), where it was shown that small parent Hamiltonians of injective MPS generically have unique ground states, and examples with an exponential ground space degeneracy were given.