The asymmetric valence-bond-solid states in quantum spin chains: The difference between 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 $Ĥ=−∑jŜj(z)$⁠, which describes independent spins in a uniform magnetic field. The ground state |Φ_up⟩ = ⊗_j|S⟩_j is obviously unique, accompanied by a gap, and has vanishing truncated correlations. We, thus, call |Φ_up⟩ a disordered ground state although all the spins are pointing in the same direction.

A preprint version of this paper appeared in arXiv in 2009 slightly after.¹⁶ 

We say that a state is disordered if all truncated correlation functions decay exponentially.

An example is the sequence 2, 4, … of even numbers.

To be precise, for each i, we define the ground state $|ΦGS(i)〉$ on the finite chain {−(L_i/2) + 1, …, L_i/2} by shifting the original ground state on {1, …, L_i}. Then, the state ω is defined by $ω(Â)=limi↑∞〈ΦGS(i)|Â|ΦGS(i)〉$ for any local operator $Â$⁠. See, e.g., Refs. 5 and 33 for more details.

Recall that the total spin S_tot 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 $|σ〉=⊗j=1L|σj〉j$ with σ_j = ±1, 0 such that σ_k ≠ 0 for at least one k. Then, there is at least one pair j, j + 1 of neighboring sites such that the spin configuration (σ_j, σ_j+1) differs from the four configurations listed above. This means $〈σ|Ĥ0|σ〉≥1$⁠. Since |σ⟩ is orthogonal to the ground state |Φ_zero⟩, we see that there is a gap, which is equal to 1.

See Ref. 39.

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.

Bachman and Nachtergale also constructed corresponding Hamiltonians.^29,30

The material in this appendix is primarily due to Katsura.