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Rational Yang–Baxter transfer matrices and Heisenberg charges — Examples

1 · Prerequisites

2 · Summary

This companion computes the one-down-spin sector of the periodic exchange Hamiltonian. Each basis state is fixed by the $N - 2$ bonds not touching the down spin, while the two incident bonds move it to a neighboring site. The discrete Fourier vectors therefore have eigenvalues $J (N - 2 + 2 \cos (2 π m / N))$ . This finite calculation checks the periodic shift orientation and coupling normalization; it is a mathematical example, not a reported measurement.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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

One down-spin Fourier dispersion

Example

Fix $N \geq 3$ , $J \in R$ , and periodic bonds $P_{N, N + 1} : = P_{N, 1}$ on $V_{N} = (C^{2})^{\otimes N}$ . Let $∣ j ⟩$ be the basis vector with $e_{↓}$ at site $j$ and $e_{↑}$ at every other site. Set $H_{N} : = J \sum_{j = 1}^{N} P_{j, j + 1}$ . For each integer $m$ with $0 \leq m < N$ , put $k_{m} : = 2 π m / N$ and $∣ k_{m} ⟩ : = \sum_{j = 1}^{N} e^{i k_{m} j} ∣ j ⟩ .$ Then $∣ k_{m} ⟩$ is a nonzero eigenvector of $H_{N}$ with eigenvalue $E_{m} = J (N - 2 + 2 \cos k_{m}) .$ If interpreted as an energy, $J$ and $E_{m}$ are in joules.

Example references

Logarithmic derivative is periodic exchange

Classical and Quantum Integrable Systems, Lecture 10, §§10.2–10.3

Faddeev, §4, equations (108)–(112); source normalization $H_{s o u r c e} = \frac{1}{2} \sum_{j} (P_{j, j + 1} - I)$

Facts & Assumptions

Given: $N \geq 3$ , $J \in R$ , and the finite tensor basis and periodic bond convention specified above.

[F1]

Logarithmic derivative is periodic exchange: for real $J$ , the periodic exchange operator is $H_{N} = J \sum_{j = 1}^{N} P_{j, j + 1}$ when its additive scalar offset is $c = 0$ .

Verification

1.1givenF1algebra

For a fixed $j$ , exactly two of the $N$ periodic bonds touch site $j$ ; they are distinct because $N \geq 3$ . Each of the other $N - 2$ swaps exchanges two up spins and fixes $∣ j ⟩$ , while the two touching swaps move the down spin to $j - 1$ and $j + 1$ . Therefore $H_{N} ∣ j ⟩ = J ((N - 2) ∣ j ⟩ + ∣ j - 1 ⟩ + ∣ j + 1 ⟩),$ with indices interpreted modulo $N$ .

2.1givenF1step 1.1algebra∎

The vectors $∣ j ⟩$ are linearly independent and every coefficient $e^{i k_{m} j}$ has modulus one, so $∣ k_{m} ⟩ \neq 0$ . Since $e^{i k_{m} N} = 1$ , periodic reindexing gives $\sum_{j} e^{i k_{m} j} ∣ j - 1 ⟩ = e^{i k_{m}} ∣ k_{m} ⟩$ and $\sum_{j} e^{i k_{m} j} ∣ j + 1 ⟩ = e^{- i k_{m}} ∣ k_{m} ⟩$ . Applying step 1.1 linearly now yields $H_{N} ∣ k_{m} ⟩ = J (N - 2 + e^{i k_{m}} + e^{- i k_{m}}) ∣ k_{m} ⟩$ ; using $e^{i k_{m}} + e^{- i k_{m}} = 2 \cos k_{m}$ proves the claimed eigenvalue.