Symmetry-enforced minimal entanglement and correlation in quantum spin chains

The interplay between symmetry, entanglement and correlation is an interesting and important topic in quantum many-body physics. Within the framework of matrix product states, in this paper we study the minimal entanglement and correlation enforced by the $SO(3)$ spin rotation symmetry and lattice translation symmetry in a quantum spin-$J$ chain, with $J$ a positive integer. When neither symmetry is spontaneously broken, for a sufficiently long segment in a sufficiently large closed chain, we find that the minimal Rényi-$α$ entropy compatible with these symmetries is $\min\{ -\frac{2}{α-1}\ln(\frac{1}{2^α}({1+\frac{1}{(2J+1)^{α-1}}})), 2\ln(J+1) \}$, for any $α\in\mathbb{R}^+$. In an infinitely long open chain with such symmetries, for any $α\in\mathbb{R}^+$ the minimal Rényi-$α$ entropy of half of the system is $\min\{ -\frac{1}{α-1}\ln(\frac{1}{2^α}({1+\frac{1}{(2J+1)^{α-1}}})), \ln(J+1) \}$. When $α\rightarrow 1$, these lower bounds give the symmetry-enforced minimal von Neumann entropies in these setups. Moreover, we show that no state in a quantum spin-$J$ chain with these symmetries can have a vanishing correlation length. Interestingly, the states with the minimal entanglement may not be a state with the minimal correlation length.