PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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An exact local DMRG ground solve does not increase Rayleigh energy

Statement

Let $ψ_{0} = E x_{0}$ be a normalized finite MPS in mixed canonical gauge about a one-site or two-site center, with $∥ x_{0} ∥_{2} = 1$ . Let $H_{e f f} = E^{*} H E$ be its effective Hermitian matrix, and replace the center by the deterministic unit ground vector $x_{*}$ of $H_{e f f}$ , giving $ψ_{*} = E x_{*}$ . Then $ψ_{*}$ is normalized and $E_{H} (ψ_{*}) \leq E_{H} (ψ_{0}) .$ This claim is before any bond truncation.

Facts & Assumptions

Given: The finite-state and Hermitian-Hamiltonian conventions, an isometric one-site or two-site center embedding $E$ , and the current unit center vector $x_{0}$ .

[F1]

Canonical MPS environments are isometric and the effective Hamiltonian is Hermitian: $E^{*} E = I$ , $H_{e f f} = E^{*} H E$ is Hermitian, and $⟨ E x, H E x ⟩_{F} = x^{*} H_{e f f} x$ .

[F2]

Finite Hermitian ground vectors and a deterministic tie rule: the selected unit vector $x_{*}$ minimizes the Rayleigh quotient of the finite Hermitian matrix $H_{e f f}$ .

[F3]

Finite matrix-product states and Hermitian Hamiltonians: a normalized finite state has Frobenius norm one and $E_{H} (ψ) = ⟨ ψ, H ψ ⟩_{F}$ .

Proof

1.1givenF1F3algebra

Since $E^{*} E = I$ by [F1], the current center state $ψ_{0} = E x_{0}$ has norm $∥ x_{0} ∥_{2} = 1$ , and the updated state $ψ_{*} = E x_{*}$ has norm $∥ x_{*} ∥_{2} = 1$ . Thus both are admissible normalized states under [F3].

2.1F1F2F3step 1.1algebra∎

By [F2], $x_{*}^{*} H_{e f f} x_{*} \leq x_{0}^{*} H_{e f f} x_{0}$ . Use the pullback identity in [F1] on both sides and then [F3] to obtain $E_{H} (ψ_{*}) \leq E_{H} (ψ_{0})$ .

Depends on

Used by

TT discarded weights bound the post-sweep DMRG energy increase Theorem

Dependency tree · two levels

12 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

Ulrich Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326 (2011), §§6.3–6.4 (standard reference, not scraped)
Grégoire Misguich, Introduction to matrix-product states and tensor networks, complete lecture notes (2026), §2.7 (standard reference, not scraped)