FIG. 1.

(a) Low-lying energy levels of the Haldane chain for even L. The entanglement structure of ground states is shown at the bottom. The ground states in the total S^z = 0, 1, 2 subspace are named |0〉, |1〉, |2〉 and have energies E₀, E₁, E₂. (b), (c) The mth largest value λ_m (m = 1, 2, …, 8) of the ground-state entanglement spectrum for H_α (b) and $H_{\alpha }^{\prime }$ (c) using finite-size MPS calculations with L = 200. We choose the |1〉 state to avoid extra entanglement between edge spins. For $H_{\alpha }^{\prime }$, the entanglement spectrum for 1.5 ⩽ α ⩽ 4 will exhibit a smooth crossover between the α = 1.5 and α = 4 cases due to the finite system size, but we expect a sharp transition at some α ≲ 3 in the thermodynamic limit. The exact pair degeneracies in {λ_m} are a result of the spatial-inversion symmetry protecting the topological phase [44,49].