bibkey: pronko2025fredkin
authors: Andrei G. Pronko
year: 2025
title: 'Symmetries of the periodic Fredkin chain'
doi: 10.1088/1751-8121/ae1644
url: https://doi.org/10.1088/1751-8121/ae1644
claim: The paper defines the periodic Fredkin Hamiltonian and the operators Sigma plus and minus, states as Conjecture 1 that they annihilate every non-cyclic invariant eigenstate, and states as Conjecture 2 that they are finite anti-adjoint expansions of the total spin operators.
strata_touched:
  - D5/S3/Quantum/Dynamics/PronkoFredkinNonCyclicAnnihilation
  - D5/S3/Quantum/Dynamics/PronkoFredkinAntiAdjointExpansion
  - D5/S3/Quantum/Dynamics/PronkoFredkinXiCommutes
license: citation-only
triage: anchor

Symmetries of the Periodic Fredkin Chain

Pronko works on the state space (ℂ²)^{⊗N} with basis vectors indexed by spin words. In section 2.1, printed p. 3, the paper states:

We denote basis vectors in ℂ² by |↑⟩ = (1, 0)ᵀ, |↓⟩ = (0, 1)ᵀ. The basis in End(ℂ²) is provided by the Pauli matrices σ⁺ = [[0, 1], [0, 0]], σ⁻ = [[0, 0], [1, 0]], σᶻ = [[1, 0], [0, −1]]. We will also use projectors onto the spin-up and spin-down states: n↑ = ½(1 + σᶻ), n↓ = ½(1 − σᶻ).

The same section, printed p. 3, identifies the jth tensor factor with the jth site, defines P_{j,j+1} as the operator exchanging adjacent copies, and sets

Π_{j,j+1} = ½(1 − P_{j,j+1}).

It then gives the Hamiltonian density and periodic Hamiltonian:

F_{j,j+1,j+2} = n↑j Π{j+1,j+2} + Π_{j,j+1} n↓_{j+2}. (2.2)

H = Σ_{j=1}^{N} F_{j,j+1,j+2}. (2.3)

with site indices identified modulo N. Across printed pp. 3-4, the cyclic operator is given in ordered product form:

C = P_{1,2} · · · P_{N−2,N−1} P_{N−1,N}

and is described as acting on sites by i ↦ i + 1.

Theorem 2, equation (3.1), printed pp. 6-7, defines

Σ^± = Σ_{r₁,…,r_N ∈ {−1,0,1}, r₁+···+r_N = ±1} σ₁^{r₁} · · · σ_N^{r_N}. (3.1)

where σ_i^0 = 1 and σ_i^{±1} = σ_i^±. Conjecture 1, printed p. 7, is:

The operators Σ^± annihilate all non-cyclic invariant (C ≠ 1) eigenstates of the Hamiltonian.

The formal statement uses complex scalars, the convention 0 = up and 1 = down, sites modulo N, and N >= 3. An eigenstate is a nonzero vector that is an eigenvector of both H and C; non-cyclic means that its C-eigenvalue differs from one. The formal proof retains the Hamiltonian eigenvector hypothesis but establishes annihilation from the C-eigenvector hypothesis alone.

Section 2.1, printed p. 3, fixes in an unnumbered display the normalization of the total spin operators:

S^± = Σ_{j=1}^N σ_j^±, S^z = ½ Σ_{j=1}^N σ_j^z.

Conjecture 2, printed p. 7, is:

For the operators Σ^± there exists the representation Σ^± = Σ_{k=1}^{⌈N/2⌉} γ_k (ãd S^± ãd S^∓)^{k−1} S^±, where γ_k are some coefficients and ãd denotes the anti-adjoint action, (ãd a) b ≡ {a, b} = ab + ba.

The paper gives Σ^± = −¼ S^± + ⅛ {S^±, {S^∓, S^±}} for N = 3 and lists the coefficients γ_k for N ≤ 10 in Table 1, printed p. 8. The formal statement quantifies over every natural number N, takes complex coefficients, and requires one coefficient family to serve both signs; the printed index k corresponds to the formal index k − 1.

Section 2.2, printed pp. 4-5, reads a spin word as a lattice path (a letter ↑ is a step up, ↓ a step down) and groups the paths into classes C_{a,b}(N): start at height a, never go below height 0, touch height 0, end at height b. Section 4.1, printed pp. 9-10, defines for even N the operator

Ξ = Σ_{k=0}^{N/2} (−1)^k Σ_{ℓ₁ℓ₂…ℓ_N ∈ C_{k,k}(N)} n_1^{ℓ₁} n_2^{ℓ₂} ⋯ n_N^{ℓ_N} (4.1)

and states on printed p. 10 that it commutes with the Hamiltonian and anticommutes with the cyclic shift: “The first relation may require a proof though we find it satisfied in all checks”.

Verified locator

• DOI: 10.1088/1751-8121/ae1644
• URL: https://doi.org/10.1088/1751-8121/ae1644
• Source locations: section 2.1, printed pp. 3-4; section 2.2, printed pp. 4-5; Theorem 2, equation (3.1), printed pp. 6-7; Conjecture 1, printed p. 7; Conjecture 2, printed p. 7; Table 1, printed p. 8; section 4.1, equation (4.1), printed pp. 9-10.