DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Open transverse Ising chain and ordered Majorana operators

Definition

Let $N \geq 1$ be an integer and let $H_{N} = (C^{2})^{\otimes N}$ with the product inner product linear in its first argument. At each factor use

X = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), Y = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array}), Z = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}), I_{2} = (\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}) .

For $1 \leq j \leq N$ , $X_{j}, Y_{j}, Z_{j}$ denote these matrices on factor $j$ and $I_{2}$ on every other factor. For real $h, J$ measured in joules define the finite open-chain matrix $H_{N} (h, J) = - h \sum_{j = 1}^{N} Z_{j} - J \sum_{j = 1}^{N - 1} X_{j} X_{j + 1} .$ The upper limit $N - 1$ is the open-boundary convention: no term couples site $N$ to site $1$ . Put $S_{0} = I_{H_{N}}$ and $S_{j - 1} = Z_{1} \dots Z_{j - 1}$ for $j > 1$ . The ordered Majorana operators are $γ_{2 j - 1} = S_{j - 1} X_{j}, γ_{2 j} = S_{j - 1} Y_{j} (1 \leq j \leq N) .$

Definition references

Pauli matrices and the two-qubit exchange operator

Facts & Assumptions

Given: An integer $N \geq 1$ , real energy parameters $h, J$ , and the displayed finite tensor-product space.

[F1]

Pauli matrices and the two-qubit exchange operator: $C^{2}$ has its standard ordered basis and product inner product; the Pauli matrices $X, Y, Z$ are Hermitian, square to $I_{2}$ , and act on tensor factors by the stated tensor-product convention.

Proof

1.1givenF1algebra

Each displayed Pauli matrix is an everywhere-defined endomorphism of $C^{2}$ . Its tensor embedding $X_{j}, Y_{j}, Z_{j}$ is therefore an everywhere-defined endomorphism of the finite-dimensional product space $H_{N}$ , using the fixed factor order in [F1].

2.1givenstep 1.1algebra∎

Every $S_{j - 1}$ and $γ_{a}$ is a finite product of such endomorphisms, and both sums defining $H_{N}$ are finite (the second is empty when $N = 1$ ). Thus all displayed operators are well-defined matrices on $H_{N}$ ; the only bond terms are exactly those indexed by $j < N$ .

Remarks

For $N = 4$ , the following path records the open bond set and the onsite field terms. A node is one tensor factor; a solid edge labeled $- J X_{j} X_{j + 1}$ is one interaction term, and there is no edge from site $4$ back to site $1$ . The diagram records the indexing convention only; the matrix identities and later proofs are independent of the drawing.

Depends on

Pauli matrices and the two-qubit exchange operator