The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. 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. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) 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 , 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.
We note that our use of the term “disordered” may sometimes not be consistent with one’s intuition. Consider the Hamiltonian , which describes independent spins in a uniform magnetic field. The ground state |Φup⟩ = ⊗j|S⟩j is obviously unique, accompanied by a gap, and has vanishing truncated correlations. We, thus, call |Φup⟩ a disordered ground state although all the spins are pointing in the same direction.
We say that a state is disordered if all truncated correlation functions decay exponentially.
An example is the sequence 2, 4, … of even numbers.
Recall that the total spin Stot of two spins with quantum number S takes the values 0, 1, …, 2S.
We here define valence-bond as an unnormalized state |↑⟩p|↓⟩q − |↓⟩p|↑⟩q.
We note this is a slight abuse of terminology since a valence-bond should not be asymmetric.
The string order in the x- or y-direction presents only for μ = 1.
The uniqueness of the ground state can indeed be proved for any μ ∈ [0, 1] by a straightforward extension of the method developed in Ref. 64 (see also Sec. 7.1.3 of Ref. 5). However, we do not use this general result here. It is also likely that the existence of a gap for any μ ∈ [0, 1] can be proved by extending the method of Knabe40 (see also Sec. 7.1.4 of Ref. 5).
We first prove the uniqueness. From (4.17), we see that any ground state should locally consist of |1⟩j|0⟩j+1, |1⟩j| − 1⟩j+1, |0⟩j|0⟩j+1, or |0⟩j| − 1⟩j+1. The only global solution is the zero state if we impose the periodic boundary condition. To prove the existence of a gap, consider a state of the form with σj = ±1, 0 such that σk ≠ 0 for at least one k. Then, there is at least one pair j, j + 1 of neighboring sites such that the spin configuration (σj, σj+1) differs from the four configurations listed above. This means . Since |σ⟩ is orthogonal to the ground state |Φzero⟩, we see that there is a gap, which is equal to 1.
We learned this method from Katsura.
This may be interpreted as a non-Hermitian skin effect in a spin chain.
We learned this fact from Katsura.
We note that no spin pumping takes place along this path.
The material in this appendix is primarily due to Katsura.