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Periodic Motzkin chain: Ground states and symmetries

Andrei G. Pronko

TL;DR

This work extends the Motzkin spin chain to periodic boundary conditions and conjectures a highly structured ground-state sector: a $(2N+1)$-fold degeneracy indexed by $S^z$, with ground states expressible as sums of unconstrained Motzkin-like paths. It introduces nonlocal raising/lowering operators $\Sigma^{\pm}$ that commute with the periodic Hamiltonian and generate a rank-$N$ Lie algebra of type $C_N$ (i.e., $\mathfrak{sp}_{2N}$), supplemented by a central element that along with a cyclic shift enlarges the full symmetry to $\mathfrak{gl}_1 \oplus \mathfrak{gl}_1 \oplus \mathfrak{sp}_{2N}$. The paper provides explicit constructions in small systems ($N=2,3,4$) showing Cartan data of type $C_N$ and central extensions, thereby illustrating the conjectured integrable-like structure and offering a concrete path toward proving the conjectures via Yang–Baxter-based approaches. If established, these symmetries would enable powerful algebraic techniques for computing correlations and other observables in the periodic Motzkin chain and related models. The results also invite combinatorial interpretations of path counts and coefficients, potentially linking ground-state structure to Dyck-type path classifications and representation-theoretic data.

Abstract

Motzkin chain is a model of nearest-neighbor interacting quantum $s=1$ spins with open boundary conditions. It is known that it has a unique ground state which can be viewed as a sum of Motzkin paths. We consider the case of periodic boundary conditions and provide several conjectures about structure of the ground state space and symmetries of the Hamiltonian. We conjecture that the ground state is degenerate and independent states are distinguished by eigenvalues of the third component of total spin operator. Each of these states can be described as a sum of paths, similar to the Motzkin paths. Moreover, there exist two operators commuting with the Hamiltonian, which play the roles of lowering and raising operators when acting at these states. We conjecture also that these operators generate a $C$-type Lie algebra, with rank equal to the number of sites. The symmetry algebra of the Hamiltonian is actually wider, and extended, besides the cyclic shift operator, by a central element contained in the third component of total spin operator.

Periodic Motzkin chain: Ground states and symmetries

TL;DR

This work extends the Motzkin spin chain to periodic boundary conditions and conjectures a highly structured ground-state sector: a -fold degeneracy indexed by , with ground states expressible as sums of unconstrained Motzkin-like paths. It introduces nonlocal raising/lowering operators that commute with the periodic Hamiltonian and generate a rank- Lie algebra of type (i.e., ), supplemented by a central element that along with a cyclic shift enlarges the full symmetry to . The paper provides explicit constructions in small systems () showing Cartan data of type and central extensions, thereby illustrating the conjectured integrable-like structure and offering a concrete path toward proving the conjectures via Yang–Baxter-based approaches. If established, these symmetries would enable powerful algebraic techniques for computing correlations and other observables in the periodic Motzkin chain and related models. The results also invite combinatorial interpretations of path counts and coefficients, potentially linking ground-state structure to Dyck-type path classifications and representation-theoretic data.

Abstract

Motzkin chain is a model of nearest-neighbor interacting quantum spins with open boundary conditions. It is known that it has a unique ground state which can be viewed as a sum of Motzkin paths. We consider the case of periodic boundary conditions and provide several conjectures about structure of the ground state space and symmetries of the Hamiltonian. We conjecture that the ground state is degenerate and independent states are distinguished by eigenvalues of the third component of total spin operator. Each of these states can be described as a sum of paths, similar to the Motzkin paths. Moreover, there exist two operators commuting with the Hamiltonian, which play the roles of lowering and raising operators when acting at these states. We conjecture also that these operators generate a -type Lie algebra, with rank equal to the number of sites. The symmetry algebra of the Hamiltonian is actually wider, and extended, besides the cyclic shift operator, by a central element contained in the third component of total spin operator.

Paper Structure

This paper contains 8 sections, 1 theorem, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motzkin paths, Motzkin state, and open chain
  3. Periodic Motzkin chain
  4. Examples
  5. Two-site chain
  6. Three-site chain
  7. Four-site chain
  8. Conclusion

Key Result

Theorem 1

The Motzkin state $\lvert \mathcal{M}_N \rangle$ is a unique ground state with zero eigenvalue of the spin chain Hamiltonian where subscripts indicate sites where operators act.

Figures (4)

  • Figure 1: A Motzkin path, $N=7$
  • Figure 2: Motzkin paths for $N=2,3,4$
  • Figure 3: Paths connecting points $(x,y)=(0,0)$ and $(x,y)=(N,S^z)$ corresponding to the five ground states of the periodic two-site chain
  • Figure 4: Paths connecting points $(x,y)=(0,0)$ and $(x,y)=(N,S^z)$ corresponding to the seven ground states of the periodic three-site chain

Theorems & Definitions (5)

  • Theorem 1: Bravyi, Caha, Movassagh, Nagaj, Shor BCMNS-12
  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4