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The open XXZ local 4-by-4 matrix satisfies the Temperley–Lieb relations

Statement

For every $q = e^{i γ} \neq 0$ , put $δ = q + q^{- 1}$ and let $U (q)$ be the local matrix defined in Open XXZ spin-half generators and Hamiltonian convention. Its embeddings $e_{i}$ on $(C^{2})^{\otimes L}$ satisfy $e_{i}^{2} = δ e_{i}$ , $e_{i} e_{i + 1} e_{i} = e_{i} = e_{i + 1} e_{i} e_{i + 1}$ , and $e_{i} e_{j} = e_{j} e_{i}$ when $∣ i - j ∣ > 1$ . Thus they define a representation of the presented open Temperley–Lieb algebra, including at values where $δ = 0$ .

Facts & Assumptions

Given: $q = e^{i γ}$ and the ordered spin basis and matrix $U (q)$ from Open XXZ spin-half generators and Hamiltonian convention.

[F1]

On the two mixed-spin basis vectors, $U (q)$ has block $A = (\begin{array}{cc} q^{- 1} & - 1 \\ - 1 & q \end{array})$ and is zero on $↑ ↑, ↓ ↓$ . (Open XXZ spin-half generators and Hamiltonian convention)

[F2]

The presented Temperley–Lieb relations are the square relation with loop parameter $δ$ , the two adjacent triple relations, and distant commutation. (The abstract Temperley–Lieb generator presentation)

Proof

Given: The matrix and relations in [F1]–[F2].

1.1F1algebra

Multiplying the nonzero block gives $A^{2} = (\begin{array}{cc} q^{- 2} + 1 & - (q + q^{- 1}) \\ - (q + q^{- 1}) & q^{2} + 1 \end{array}) = (q + q^{- 1}) A = δ A$ , because $q^{- 2} + 1 = δ q^{- 1}$ and $q^{2} + 1 = δ q$ . The zero subspace remains zero, so $U (q)^{2} = δ U (q)$ , including when $δ = 0$ .

1.2F1algebra

On the three-spin subspace with one down spin, in the ordered basis $↓ ↑ ↑, ↑ ↓ ↑, ↑ ↑ ↓$ , the two embedded operators are $A_{1} = (\begin{array}{ccc} q & - 1 & 0 \\ - 1 & q^{- 1} & 0 \\ 0 & 0 & 0 \end{array})$ and $A_{2} = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & q & - 1 \\ 0 & - 1 & q^{- 1} \end{array})$ . Multiplication gives $A_{1} A_{2} = (\begin{array}{ccc} 0 & - q & 1 \\ 0 & 1 & - q^{- 1} \\ 0 & 0 & 0 \end{array})$ and $(A_{1} A_{2}) A_{1} = A_{1}$ ; likewise $A_{2} A_{1} = (\begin{array}{ccc} 0 & 0 & 0 \\ - q & 1 & 0 \\ 1 & - q^{- 1} & 0 \end{array})$ and $(A_{2} A_{1}) A_{2} = A_{2}$ . The two-down-spin subspace has the same pair of matrices in reverse order, so both triple identities hold there as well. On the all-up and all-down vectors both operators vanish. These four invariant subspaces exhaust the three-spin space, proving both adjacent relations.

2.1F1F2step 1.1step 1.2algebra∎

If $∣ i - j ∣ > 1$ , the operators act on disjoint tensor factors. For elementary tensors, applying the two operators in either order changes separate tensor coordinates and yields the same vector; linearity gives $e_{i} e_{j} = e_{j} e_{i}$ . This verifies all relations in [F2] for every chain length.

Depends on

Used by

The q = 1 three-site open XXZ chain has loop value 2 Example

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Sources

A. Nichols, The Temperley-Lieb Algebra and its Generalizations in the Potts and XXZ Models (standard reference, not scraped)
Vaughan F. R. Jones, Planar Algebras, I (standard reference, not scraped)