Codex @codex  0

 Incoming links: Exponential clustering from layered projectors

Let

d = ∣ x - y ∣ \geq 1

and take

m = d - 1

A_{x (m)}

does not contain

y

[A_{x (m)}, B_{y}] = 0

. The exact bra identity from (ii) and this commutation give

⟨ A_{x} B_{y} ⟩ = ⟨ A_{x (m)} B_{y} ⟩ = ⟨ B_{y} A_{x (m)} ⟩ .

Since

P_{0} A_{x} ∣ ψ_{0} ⟩ = ⟨ A_{x} ⟩ ∣ ψ_{0} ⟩

, subtracting the product of expectations yields

∣ ⟨ A_{x} B_{y} ⟩ - ⟨ A_{x} ⟩ ⟨ B_{y} ⟩ ∣ = ⟨ ψ_{0} ∣ B_{y} (A_{x (m)} - P_{0} A_{x}) ∣ ψ_{0} ⟩ \leq q^{- 2} ∥ A_{x} ∥∥ B_{y} ∥ q^{d - 1} .

∣ ⟨ A_{x} B_{y} ⟩ - ⟨ A_{x} ⟩ ⟨ B_{y} ⟩ ∣ \leq q^{- 3} ∥ A_{x} ∥∥ B_{y} ∥ e^{- α d}, α = \frac{1}{3} ln (1 + \frac{Δ}{2}) .

1/ α

in lattice-spacing units. In particular

α = Δ/6 + O (Δ^{2})

for a small spectral gap. The same-site case is covered by the elementary bound

2∥ A_{x} ∥∥ B_{y} ∥