Symmetry-enforced minimal entanglement and correlation in quantum spin chains

Abstract

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-$\alpha$ entropy compatible with these symmetries is $\min \{-\frac{2}{\alpha -1}\ln (\frac{1}{2^{\alpha }}(1+\frac{1}{(2J+1{)}^{\alpha -1}})),2\ln (J+1)\}$, for any $\alpha \in {\mathbb{R}}^{+}$. In an infinitely long open chain with such symmetries, for any $\alpha \in {\mathbb{R}}^{+}$ the minimal Rényi-$\alpha$ entropy of half of the system is $\min \{-\frac{1}{\alpha -1}\ln (\frac{1}{2^{\alpha }}(1+\frac{1}{(2J+1{)}^{\alpha -1}})),\ln (J+1)\}$. When $\alpha \to 1$, 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.

17 pages + appendices + references

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