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

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A periodic finite MPS has an exact open-boundary TT conversion

Statement

Let a periodic MPS presentation have order $d \geq 2$ , positive cyclic bond sizes $r_{0}, \dots, r_{d}$ with $r_{d} = r_{0}$ , and matrices $A_{k}^{(i_{k})} \in F^{r_{k - 1} \times r_{k}}$ . It has an exactly equal open-boundary TT presentation with internal bond sizes $s_{k} = r_{k} r_{0}$ for $1 \leq k < d$ . In the pair index convention $(a_{k}, a_{0})$ , define $G_{1} (1, i_{1}, (a_{1}, a_{0})) = (A_{1}^{(i_{1})})_{a_{0}, a_{1}},$ $G_{k} ((a_{k - 1}, b_{0}), i_{k}, (a_{k}, c_{0})) = δ_{b_{0} c_{0}} (A_{k}^{(i_{k})})_{a_{k - 1}, a_{k}} (2 \leq k < d, 1 \leq b_{0}, c_{0} \leq r_{0}),$ where $δ_{b_{0} c_{0}} = 1$ for $b_{0} = c_{0}$ and $0$ otherwise, so every middle-core entry with unequal carried indices is zero, and $G_{d} ((a_{d - 1}, a_{0}), i_{d}, 1) = (A_{d}^{(i_{d})})_{a_{d - 1}, a_{0}} .$ For $d = 2$ the middle-core clause is empty.

Facts & Assumptions

Given: A periodic trace MPS presentation over $F \in {R, C}$ with $d \geq 2$ and positive cyclic bond sizes.

[F1]

Finite matrix-product states and Hermitian Hamiltonians: the periodic entry is $ψ_{i_{1}, \dots, i_{d}} = tr (A_{1}^{(i_{1})} \dots A_{d}^{(i_{d})})$ , with $A_{k}^{(i_{k})}$ of size $r_{k - 1} \times r_{k}$ and $r_{d} = r_{0}$ .

[F2]

Tensor train cores, bond ranks, and canonical gauges: an open-boundary TT entry is the sum of the product of its cores over every shared bond index, with boundary ranks one.

Proof

1.1givenF1algebra

Expanding the matrix trace and the matrix products in [F1] gives $ψ_{i_{1}, \dots, i_{d}} = \sum_{a_{0} = 1}^{r_{0}} \sum_{a_{1} = 1}^{r_{1}} \dots \sum_{a_{d - 1} = 1}^{r_{d - 1}} (A_{1}^{(i_{1})})_{a_{0}, a_{1}} \dots (A_{d}^{(i_{d})})_{a_{d - 1}, a_{0}} .$ This is a finite sum, and remains the same for $d = 2$ with no intermediate factors.

2.1givenF2step 1.1algebra∎

Contract the cores defined in the Statement according to [F2]. Each internal TT bond is a pair $(a_{k}, a_{0})$ , so its summation matches the adjacent physical matrix index $a_{k}$ while carrying the same $a_{0}$ to the next core. The resulting contraction sums once over $a_{0}, a_{1}, \dots, a_{d - 1}$ and has exactly the product in step 1.1. Hence every array entry is unchanged, and the TT bond at cut $k$ has $r_{k} r_{0}$ possible pair indices.

Depends on

Used by

Finite alternating tensor optimization with effective eigenproblems Algorithm

Dependency tree · two levels

3 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), §§4.1, 4.6 (standard reference, not scraped)
Grégoire Misguich, Introduction to matrix-product states and tensor networks, complete lecture notes (2026), §§2.1, 2.4.3 (standard reference, not scraped)