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Schmidt truncation, transfer operators and parent Hamiltonians — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

GHZ transfer has persistent connected correlations

Example

Assume Countable Choice. Use the standard inner product on $C^{2}$ and its ordered basis $∣ 0 ⟩, ∣ 1 ⟩$ . For $i, j \in {0, 1}$ , let $E_{i j} \in M_{2} (C)$ have entry $1$ in row $i$ , column $j$ , and entry $0$ elsewhere, with rows and columns ordered $0, 1$ . For each integer $N \geq 2$ , index the factors cyclically modulo $N$ and let $A^{0} = ∣ 0 ⟩ ⟨ 0 ∣ = E_{00} = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array})$ and $A^{1} = ∣ 1 ⟩ ⟨ 1 ∣ = E_{11} = (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array})$ . The periodic trace contraction has amplitude $tr (A^{i_{1}} \dots A^{i_{N}})$ on basis vector $∣ i_{1} \dots i_{N} ⟩$ and equals $∣ 0^{N} ⟩ + ∣ 1^{N} ⟩$ ; after normalization it is the GHZ vector $(∣ 0^{N} ⟩ + ∣ 1^{N} ⟩) / \sqrt{2}$ . The transfer map fixes $E_{00}$ and $E_{11}$ and annihilates $E_{01}$ and $E_{10}$ . With $ρ = I_{2} / 2$ and $ℓ = I_{2}$ , the associated infinite transfer state has $ω (Z_{1}) = 0$ and $ω (Z_{1} I^{\otimes g} Z_{g + 2}) = 1$ for every integer gap $g \geq 0$ , where $Z = ∣ 0 ⟩ ⟨ 0 ∣ - ∣ 1 ⟩ ⟨ 1 ∣$ acts on one site. Thus its connected $Z$ correlation is exactly $1$ at every gap and does not decay.

Example references

Finite open MPS and injective translation invariant tensor

Transfer-generated infinite-chain state and correlations

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Facts & Assumptions

Given: Countable Choice, the two matrices $A^{0}, A^{1}$ , a ring length $N \geq 2$ , the one-site observable $Z$ , and the positive matrices $ρ = I_{2} / 2$ , $ℓ = I_{2}$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every sequence of nonempty sets has a choice function, as assumed by the infinite transfer-state construction.

[F1]

Finite open MPS and injective translation invariant tensor: the tensor matrices have finite dimension and their transfer map is $E (X) = \sum_{i = 0}^{1} A^{i} X A^{i *}$ .

[F2]

Transfer-generated infinite-chain state and correlations: the state is defined by $ω (O) = tr (ℓ E_{O} (ρ))$ , and insertions on separated blocks compose with $E^{g}$ for a gap of $g$ identity sites.

Verification

1.1givenF1algebra

Each $A^{i}$ is the diagonal projection $E_{i i}$ . A product $A^{i_{1}} \dots A^{i_{N}}$ is zero if the word contains both $0$ and $1$ , and is $E_{00}$ or $E_{11}$ for the two constant words. Its trace is therefore one exactly for those two words and zero otherwise, proving the unnormalized periodic vector formula. The two basis vectors $∣ 0^{N} ⟩$ and $∣ 1^{N} ⟩$ are orthonormal, so its squared norm is $2$ and the stated normalization follows.

2.1givenA1F1F2step 1.1algebra∎

On the basis ${E_{i j} : i, j \in {0, 1}}$ defined in the Statement, $E (E_{00}) = E_{00}$ , $E (E_{11}) = E_{11}$ , and $E (E_{01}) = E (E_{10}) = 0$ . Thus the fixed subspace ${X \in M_{2} (C) : E (X) = X}$ is exactly $span {E_{00}, E_{11}}$ and has dimension two. Also $E (ρ) = ρ$ , $E^{*} (ℓ) = ℓ$ , and $tr (ℓ ρ) = 1$ , so [F2] applies under [A1]. The insertion for $Z$ is $E_{Z} (X) = A^{0} X A^{0 *} - A^{1} X A^{1 *}$ , whence $E_{Z} (ρ) = (E_{00} - E_{11}) / 2$ , $E^{g} E_{Z} (ρ) = E_{Z} (ρ)$ , and $E_{Z} E^{g} E_{Z} (ρ) = I_{2} / 2$ . Therefore $ω (Z_{1}) = tr (ℓ E_{Z} (ρ)) = 0$ and $ω (Z_{1} I^{\otimes g} Z_{g + 2}) = tr (ℓ I_{2} / 2) = 1$ for every $g \geq 0$ . Subtracting the product of the one-point expectations gives connected correlation $1$ .