bibkey: pronko2025motzkin
authors: A. G. Pronko
year: 2025
title: "Periodic Motzkin chain: Ground states and symmetries"
doi: 10.48550/arXiv.2504.00835
url: https://arxiv.org/abs/2504.00835v3
claim: "The paper studies the periodic Motzkin spin-1 chain H^periodic = sum_{i=1}^{N-1} Pi_{i,i+1} + Pi_{N,1}, conjectures (Conjecture 1) that its ground states are the 2N+1 path sums v_{S^z}, and conjectures (Conjecture 2) that the operators Sigma^{+-}, sums of products s_1^{r_1} ... s_N^{r_N} over r_i in {-2,...,2} with r_1 + ... + r_N = +-1, commute with the Hamiltonian and send v_{S^z} to nonzero multiples of v_{S^z+-1}, and v_{+-N} to 0."
strata_touched:
  - D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering
license: citation-only
triage: anchor

Periodic Motzkin chain: Ground states and symmetries

Pronko represents the steps of Motzkin paths by the basis vectors $|\mathrm{u}⟩,|\mathrm{f}⟩,|\mathrm{d}⟩$ of ${\mathbb{C}}^3$ and uses the spin-1 matrices (eq. spin1rep)

$s^{+}=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right)$, $s^{-}=\left(\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)$,

with $s_j^{\pm }$ acting on the $j$-th factor of $({\mathbb{C}}^3{)}^{\otimes N}$. The local projector is $\Pi =\mathrm{U}+\mathrm{D}+\mathrm{F}$ with (eq. UDF) $\mathrm{U}=\frac{1}{2}(|\mathrm{uf}⟩-|\mathrm{fu}⟩)(⟨\mathrm{uf}|-⟨\mathrm{fu}|)$, $\mathrm{D}=\frac{1}{2}(|\mathrm{df}⟩-|\mathrm{fd}⟩)(⟨\mathrm{df}|-⟨\mathrm{fd}|)$, $\mathrm{F}=\frac{1}{2}(|\mathrm{ud}⟩-|\mathrm{ff}⟩)(⟨\mathrm{ud}|-⟨\mathrm{ff}|)$, and the periodic Hamiltonian is (eq. Hpbc) ${\mathrm{H}}^{\text{Periodic}}={\sum }_{i=1}^{N-1}{\Pi }_{i,i+1}+{\Pi }_{N,1}$.

Conjecture 1 describes the ground states $|v_{S^z}⟩$ as “sums of paths connecting points $(x,y)=(0,0)$ and $(x,y)=(N,S^z)$, having at each step $\Delta x=1$ and $\Delta y\in \{-1,0,1\}$”. The raising and lowering operators should satisfy (eqs. Sigmavec, SpmH)

${\Sigma }^{\pm }|v_{S^z}⟩=c_{\pm }(S^z)|v_{S^z\pm 1}⟩$ for $S^z\ne \pm N$, $0$ for $S^z=\pm N$, where $c_{\pm }(S^z)\ne 0$ are some constants; $[{\Sigma }^{\pm },{\mathrm{H}}^{\text{periodic}}]=0$,

and Conjecture 2 reads

There exist raising and lowering operators satisfying \eqref{Sigmavec} and \eqref{SpmH}, and they are given by ${\Sigma }^{\pm }={\sum }_{r_1,\dots ,r_N\in \{-2,-1,0,1,2\},r_1+\cdots +r_N=\pm 1}{s}_1^{r_1}\cdots s_N^{r_N}$ with the following notation: $s_i^0\equiv I$, $s_i^{\pm 1}\equiv s_i^{\pm }$, $s_i^{\pm 2}\equiv (s_i^{\pm }{)}^2$.

The paper reports symbolic verification up to $N=6$.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2504.00835
• URL: https://arxiv.org/abs/2504.00835v3
• Journal: Nuclear Physics B 1017 (2025) 116963, https://doi.org/10.1016/j.nuclphysb.2025.116963 (Crossref record retrieved 2026-09-27).
• Version and location: arXiv:2504.00835v3, source file sympmot_3a.tex: eq. spin1rep (line 261), eq. UDF (line 367), eq. Hpbc (line 430), Conjecture 1 (lines 493–502), eqs. Sigmavec and SpmH (lines 553–568), Conjecture 2 (lines 570–593); retrieved 2026-09-27.