TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The XXZ generator Hamiltonian has the declared Pauli expansion and endpoint field

Statement

For $L \geq 2$ , $q = e^{i γ}$ and the open Hamiltonian $H_{L} = - \sum_{i = 1}^{L - 1} e_{i}$ from Open XXZ spin-half generators and Hamiltonian convention, $H_{L} = \frac{1}{2} \sum_{i = 1}^{L - 1} (X_{i} X_{i + 1} + Y_{i} Y_{i + 1} + \cos γ Z_{i} Z_{i + 1} - \cos γ I) + \frac{i \sin γ}{2} (Z_{1} - Z_{L}) .$ The sum has $L - 1$ bonds and contains no periodic bond from site $L$ to site $1$ .

Facts & Assumptions

Given: The Pauli expansion of $e_{i}$ and the open-chain range in Open XXZ spin-half generators and Hamiltonian convention.

[F1]

On bond $i, i + 1$ , $e_{i} = - \frac{1}{2} (X_{i} X_{i + 1} + Y_{i} Y_{i + 1} + \cos γ Z_{i} Z_{i + 1} - \cos γ I + i \sin γ (Z_{i} - Z_{i + 1}))$ , and $H_{L} = - \sum_{i = 1}^{L - 1} e_{i}$ . (Open XXZ spin-half generators and Hamiltonian convention)

Proof

Given: $L \geq 2$ , real $γ$ , and [F1].

1.1F1algebra

Negating the displayed formula for each $e_{i}$ changes its prefactor $- 1 / 2$ to $1 / 2$ . Separating the terms gives the bulk sum in the statement and the remaining term $\frac{i \sin γ}{2} \sum_{i = 1}^{L - 1} (Z_{i} - Z_{i + 1})$ .

2.1F1step 1.1algebra∎

In the endpoint sum, each interior $Z_{j}$ , $2 \leq j \leq L - 1$ , occurs once with coefficient $+ 1$ in $Z_{j}$ and once with coefficient $- 1$ in $- Z_{j}$ , hence cancels. Only $Z_{1} - Z_{L}$ remains. The original index range is $1 \leq i < L$ , so it contributes exactly the open bonds and no wrap-around term.

Depends on

Open XXZ spin-half generators and Hamiltonian convention

Used by

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

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)