Raising and lowering operators of the periodic Motzkin chain

Abstract

On the periodic Motzkin spin-1 chain with N at least 2 sites, the operators Sigma^+ and Sigma^-, the sums of the products of local powers s_i^{r_i} with r_1 + … + r_N equal to 1, respectively -1, commute with the periodic Hamiltonian, send each ground state v_m to a nonzero multiple of v_(m+1), respectively v_(m-1), and annihilate v_N, respectively v_(-N). This proves Conjecture 2 of Pronko.

Definition 1.1 (Heights of the letters).

$$\mathrm{ht}\left(0\right)=1\wedge \left(\mathrm{ht}\left(1\right)=0\wedge \mathrm{ht}\left(2\right)=-1\right)$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.ht (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

The letters u, f, d (basis vectors 0, 1, 2 of C^3) have heights 1, 0 and -1: the steps of a Motzkin path.

Definition 1.2 (The spin-1 raising matrix).

$$sp=\mathrm{matrix}\left((0,1,0),(0,0,1),(0,0,0)\right)$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.sp (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

The basis vectors 0, 1, 2 of C^3 are u, f and d (up, flat, down); s^+ is the 0/1 matrix of eq. spin1rep.

Definition 1.3 (The spin-1 lowering matrix).

$$sm=\mathrm{matrix}\left((0,0,0),(1,0,0),(0,1,0)\right)$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.sm (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

s^- is the 0/1 matrix of eq. spin1rep.

Definition 1.4 (Local powers).

$$(\mathrm{spow}\left(0\right)=s{m}^2)\wedge \left((\mathrm{spow}\left(1\right)=sm)\wedge \left((\mathrm{spow}\left(2\right)=1)\wedge \left((\mathrm{spow}\left(3\right)=sp)\wedge (\mathrm{spow}\left(4\right)=s{p}^2)\right)\right)\right)$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.spow (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

s^r for r = -2, …, 2, indexed by k = r + 2: s^0 is the identity, s^{±1} = s^±, s^{±2} = (s^±)^2.

Definition 1.5 (Exponents).

$$\mathrm{rv}\left(r\right)=r-2$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.rv (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

The index r = 0, …, 4 of the local powers stands for the exponent r - 2 = -2, …, 2.

Definition 1.6 (An operator on one site).

$$\mathrm{site}\left(i,A\right)\left(a,b\right)=\mathrm{if}\forall k\in \mathrm{Fin}\left(N\right),\mathrm{ne}\left(k,i\right)\Rightarrow (a\left(k\right)=b\left(k\right))\mathrm{then}A\left(a\left(i\right),b\left(i\right)\right)\mathrm{else}0$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.site (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

site(i, A) acts as the 3 x 3 matrix A on the i-th tensor factor and as the identity on the others (eq. spmsz).

Definition 1.7 (The operators Sigma^+ and Sigma^-).

$$\mathrm{Sig}\left(N,e\right)=\sum _{r\in \{r\in \mathrm{Fin}{\left(5\right)}^N\mid {\sum }_{i\in \mathrm{Fin}\left(N\right)}\mathrm{rv}\left(r\left(i\right)\right)=e\}}\mathrm{prod}\left(\mathrm{ofFn}\left((i\mapsto \mathrm{site}\left(i,\mathrm{spow}\left(r\left(i\right)\right)\right))\right)\right)$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.Sig (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

Sig(N, 1) is Sigma^+ and Sig(N, -1) is Sigma^- of eq. Sigmapmsum: the sum, over the exponent vectors r in {-2, …, 2}^N with r_1 + … + r_N = e, of the ordered product s_1^{r_1} … s_N^{r_N}.

Definition 1.8 (The height S^z of a word).

$$S\left(a\right)=\sum _{i\in \mathrm{Fin}\left(N\right)}\mathrm{ht}\left(a\left(i\right)\right)$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.S (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

The eigenvalue of the third component of the total spin on a basis word: the number of u minus the number of d, with ht(u) = 1, ht(f) = 0, ht(d) = -1.

Definition 1.9 (Two-site basis vectors).

$$\mathrm{ket}\left(x,y\right)=\mathrm{single}\left((x,y),1\right)$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.ket (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

ket(x, y) is the basis vector |x y> of C^3 (x) C^3: the function on pairs that is 1 at (x, y) and 0 elsewhere.

Definition 1.10 (Half projectors).

$$\mathrm{proj}\left(w\right)=\frac{1}{2}\cdot \mathrm{vecMulVec}\left(w,w\right)$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.proj (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

proj(w) = (1/2) w w^T, one half of the product of w with its transpose (Matrix.vecMulVec, no complex conjugation); for the three real vectors in piProj this is (1/2)|w><w|.

Definition 1.11 (The local projector Pi).

$$piProj=\mathrm{proj}\left(\mathrm{ket}\left(0,1\right)-\mathrm{ket}\left(1,0\right)\right)+\mathrm{proj}\left(\mathrm{ket}\left(2,1\right)-\mathrm{ket}\left(1,2\right)\right)+\mathrm{proj}\left(\mathrm{ket}\left(0,2\right)-\mathrm{ket}\left(1,1\right)\right)$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.piProj (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

piProj is Pi = U + D + F of eq. UDF, with U = (1/2)(|uf> - |fu>)(<uf| - <fu|), D = (1/2)(|df> - |fd>)(<df| - <fd|) and F = (1/2)(|ud> - |ff>)(<ud| - <ff|), where u, f, d are the basis vectors 0, 1, 2.

Definition 1.12 (An operator on two sites).

$$\mathrm{twoSite}\left(i,j,P\right)\left(a,b\right)=\mathrm{if}\forall k\in \mathrm{Fin}\left(N\right),\left(\mathrm{ne}\left(k,i\right)\wedge \mathrm{ne}\left(k,j\right)\right)\Rightarrow (a\left(k\right)=b\left(k\right))\mathrm{then}P\left((a\left(i\right),a\left(j\right)),(b\left(i\right),b\left(j\right))\right)\mathrm{else}0$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.twoSite (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

twoSite(i, j, P) acts as the 9 x 9 matrix P with its first factor on site i and its second on site j, and as the identity on the other sites.

Definition 1.13 (The periodic Hamiltonian).

$$H\left(N\right)=\sum _{i\in \mathrm{Fin}\left(N\right)}\mathrm{twoSite}\left(i,\mathrm{finRotate}\left(N,i\right),piProj\right)$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.H (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

H^periodic of eq. Hpbc: Pi on the sites i, i + 1 for i = 1, …, N - 1 and Pi_{N,1} on the sites N, 1; with the sites numbered 0, …, N - 1 these are the pairs i, finRotate(N, i) with finRotate the cyclic shift i -> i + 1 mod N.

Definition 1.14 (The ground states v_m).

$$v\left(N,m\right)\left(a\right)=\mathrm{if}S\left(a\right)=m\mathrm{then}1\mathrm{else}0$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.v (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

v_m is the sum of the paths from (0, 0) to (N, m) with steps in {-1, 0, 1} (Conjecture 1): the sum of the basis words of height m.

Definition 1.15 (Conjecture 2).

$$claim\iff (\forall N\in \mathbb{N},2\le N\Rightarrow (\left(\mathrm{Sig}\left(N,1\right)\cdot H\left(N\right)=H\left(N\right)\cdot \mathrm{Sig}\left(N,1\right)\wedge \mathrm{Sig}\left(N,-1\right)\cdot H\left(N\right)=H\left(N\right)\cdot \mathrm{Sig}\left(N,-1\right)\right)\wedge \left(\left(\forall m\in \mathbb{Z},\left(-N\le m\wedge m<N\right)\Rightarrow (\exists c\in \mathbb{C},\mathrm{ne}\left(c,0\right)\wedge \mathrm{mulVec}\left(\mathrm{Sig}\left(N,1\right),v\left(N,m\right)\right)=c\cdot v\left(N,m+1\right))\right)\wedge \left(\left(\forall m\in \mathbb{Z},\left(-N<m\wedge m\le N\right)\Rightarrow (\exists c\in \mathbb{C},\mathrm{ne}\left(c,0\right)\wedge \mathrm{mulVec}\left(\mathrm{Sig}\left(N,-1\right),v\left(N,m\right)\right)=c\cdot v\left(N,m-1\right))\right)\wedge \left(\mathrm{mulVec}\left(\mathrm{Sig}\left(N,1\right),v\left(N,N\right)\right)=0\wedge \mathrm{mulVec}\left(\mathrm{Sig}\left(N,-1\right),v\left(N,-N\right)\right)=0\right)\right)\right)))$$

Formalization. D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.claim (✓ std3).

Citation. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

For N at least 2: Sigma^+ and Sigma^- commute with H; for -N <= m < N, Sigma^+ v_m is a nonzero multiple of v_(m+1); for -N < m <= N, Sigma^- v_m is a nonzero multiple of v_(m-1); and Sigma^+ v_N = 0, Sigma^- v_(-N) = 0 (eqs. Sigmavec and SpmH).

Theorem 1.16 (Proof of Conjecture 2).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.result (✓ std3). ∎

Resolves. Problems/pronko-2025-motzkin-raising-lowering (proved) by D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.result.

Source. Repository-derived.

Acknowledgement. A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Commentary.

An ordered product of operators on distinct sites has as entry at a, b the product of the local entries. The entry of s^r at x, y is 1 when ht(x) = ht(y) + r and 0 otherwise, so for given basis words a and b exactly one exponent vector, r_i = ht(a_i) - ht(b_i), gives a nonzero product, and Sig(N, e) has entry 1 at a, b when S(a) = S(b) + e and 0 otherwise; that is, Sigma^± is the sum over m of |v_(m±1)><v_m|. Each of U, D, F pairs two basis states of the same height sum with opposite signs, so every row of Pi sums to zero over each height class; hence every two-site term, and H, annihilate every v_m, and H is symmetric, so v_m^T H = 0 as well. Therefore H Sigma^± = 0 = Sigma^± H. Finally Sigma^± v_m = T_m v_(m±1), where T_m, the number of words of height m, is positive for |m| <= N, while no word has height ±(N + 1).

References

• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.H
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.S
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.Sig
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.claim
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.ht
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.ket
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.piProj
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.proj
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.result
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.rv
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.site
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.sm
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.sp
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.spow
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.twoSite
• Truth anchor: D5/S3/Quantum/Dynamics/PronkoMotzkinRaisingLowering.v