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Nearest-neighbor Ising interaction satisfies the finite-range bounds

Example

Take $d = 1$ , $q = 2$ , $Γ = Z$ , and let $σ^{z} = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}) .$ For every $n \in Z$ , let $E_{n} = {n, n + 1}$ and set $Φ (E_{n}) = J σ^{z} \otimes σ^{z} \in A_{E_{n}}$ for a fixed $J \in R$ ; set $Φ (S) = 0$ for every other finite nonempty support $S$ . If interpreted as an energy coefficient, $J$ has units J, but no value or measurement is assumed. This is a nearest-neighbor diagonal spin interaction of the kind described by Uniformly bounded finite-range quantum spin interaction.

Facts & Assumptions

Given: The displayed Pauli matrix, $J \in R$ , and the local tensor-factor conventions of Local spin matrix algebras and their factor inclusions.

[F1]

Local spin matrix algebras and their factor inclusions: the two-site algebra is the finite matrix algebra $A_{{n, n + 1}}$ with the ordinary operator norm.

[F2]

Uniformly bounded finite-range quantum spin interaction: an interaction term must be self-adjoint, have uniformly bounded support size and diameter, and have finite per-site norm sum.

Verification

technique · direct finite-dimensional norm and support calculation

Given: [F1]–[F2] and the interaction in the Example.

1.1givenF1algebra

Since $(σ^{z})^{*} = σ^{z}$ and $(σ^{z})^{2} = I$ , the tensor product $σ^{z} \otimes σ^{z}$ is a self-adjoint unitary in $A_{E_{n}}$ and has norm one. Therefore every nonzero term $Φ (E_{n})$ is self-adjoint and has norm $∣ J ∣$ .

2.1givenF2step 1.1algebra∎

Each support $E_{n}$ has cardinality two and max-metric diameter one. Each site belongs to exactly two such edges, so $\sup_{x \in Z} \sum_{S ∋ x} ∥ Φ (S) ∥ = 2 ∣ J ∣$ . All other terms vanish. Hence the interaction satisfies the definition with $M = 2$ , $R = 1$ , and $κ = 2 ∣ J ∣$ , including the case $J = 0$ .

Nearest-neighbor Ising interaction satisfies the finite-range bounds

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources