Identifying quantum phases from the injectivity of symmetric matrix product states

Given a local gapped Hamiltonian with a global symmetry on a one-dimensional lattice we describe a method to identify whether the Hamiltonian belongs to a quantum phase in which the symmetry is spontaneously broken in the ground states or to a specific symmetry-protected phase, without using local or string order parameters. We obtain different matrix product state (MPS) descriptions of the symmetric ground state(s) of the Hamiltonian by restricting the MPS matrices to transform under different equivalence classes of projective representations of the symmetry. The phase of the Hamiltonian is identified by examining which MPS descriptions, if any, are injective, namely, whether the largest eigenvalue of the transfer matrix obtained from the MPS is unique. We demonstrate the method for translationally invariant Hamiltonians with a global SO(3), Z2, and Z2×Z2 symmetry on an infinite chain.