The Asymmetric Valence-Bond-Solid States in Quantum Spin Chains: The Difference Between Odd and Even Spins

The qualitative difference in low-energy properties of spin $S$ quantum antiferromagnetic chains with integer $S$ and half-odd-integer $S$ discovered by Haldane can be intuitively understood in terms of the valence-bond picture proposed by Affleck, Kennedy, Lieb, and Tasaki. Here we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd $S$ and even $S$, which is at the heart of the theory of symmetry-protected topological (SPT) phases. More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck-Kennedy-Lieb-Tasaki (AKLT) state and the trivial zero state in quantum spin chains with $S=1$ and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the $\mathbb{Z}_2\times\mathbb{Z}_2$, and the bond-centered inversion symmetries for $S=2$, but not for $S=1$. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if $S=2$ and to a nontrivial SPT phase if $S=1$. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer $S$ and provides us with an intuitive explanation of the essential difference between models with odd and even spins.