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Symmetries of the periodic Fredkin chain

Andrei G. Pronko

TL;DR

The paper investigates the periodic Fredkin spin-1/2 chain and shows that two nonlocal operators $\Sigma^{\pm}$ commute with the Hamiltonian and act as raising/lowering operators on the degenerate ground-state manifold, linking ground states labeled by $S^z$ to a Lie-algebraic structure. These operators generate a $B_n$- or $C_n$-type symmetry algebra, with rank $n=\lceil N/2\rceil$, depending on the parity of $N$, and there exists a central element $p$ that combines $S^z$ and Cartan elements; for even $N$ this central relation mirrors a known Motzkin-chain formula, hinting at representation-independent features. The authors provide a rigorous proof of the raising/lowering properties (Theorem ['Th2']) and present four conjectures about the full symmetry, its expression in terms of total-spin components, the precise Lie algebra type, and the central extension. The work highlights deep algebraic structure in a nonintegrable-looking spin chain under periodic boundaries and suggests connections to Motzkin chains and potential generalizations to tensor-network formulations. Overall, the results advance understanding of nonlocal symmetries in quantum spin chains and point to broader applicability of B- and C-type algebras in half-integer and integer spin systems.

Abstract

The Fredkin chain is a spin-$1/2$ model with interaction of three nearest neighbors. In the case of periodic boundary conditions, the ground state is degenerate and can be described in terms of equivalence classes of Dyck paths. We introduce two operators commuting with the Hamiltonian which play the roles of lowering and raising operators when acting on the ground states. These operators generate the $B$- or $C$-type Lie algebras, depending on whether the number of sites $N$ is odd or even, respectively, with rank $n=\lceil N/2\rceil$. The third component of the total spin operator can be represented as a sum of the Cartan subalgebra elements and some central element. In the $C$-type Lie algebra case (even number of sites), this representation coincides with a similar formula previously conjectured for spin-$1$ operators, in the context of the periodic Motzkin chain.

Symmetries of the periodic Fredkin chain

TL;DR

The paper investigates the periodic Fredkin spin-1/2 chain and shows that two nonlocal operators commute with the Hamiltonian and act as raising/lowering operators on the degenerate ground-state manifold, linking ground states labeled by to a Lie-algebraic structure. These operators generate a - or -type symmetry algebra, with rank , depending on the parity of , and there exists a central element that combines and Cartan elements; for even this central relation mirrors a known Motzkin-chain formula, hinting at representation-independent features. The authors provide a rigorous proof of the raising/lowering properties (Theorem ['Th2']) and present four conjectures about the full symmetry, its expression in terms of total-spin components, the precise Lie algebra type, and the central extension. The work highlights deep algebraic structure in a nonintegrable-looking spin chain under periodic boundaries and suggests connections to Motzkin chains and potential generalizations to tensor-network formulations. Overall, the results advance understanding of nonlocal symmetries in quantum spin chains and point to broader applicability of B- and C-type algebras in half-integer and integer spin systems.

Abstract

The Fredkin chain is a spin- model with interaction of three nearest neighbors. In the case of periodic boundary conditions, the ground state is degenerate and can be described in terms of equivalence classes of Dyck paths. We introduce two operators commuting with the Hamiltonian which play the roles of lowering and raising operators when acting on the ground states. These operators generate the - or -type Lie algebras, depending on whether the number of sites is odd or even, respectively, with rank . The third component of the total spin operator can be represented as a sum of the Cartan subalgebra elements and some central element. In the -type Lie algebra case (even number of sites), this representation coincides with a similar formula previously conjectured for spin- operators, in the context of the periodic Motzkin chain.

Paper Structure

This paper contains 14 sections, 4 theorems, 42 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Hamiltonian, Dyck paths, and the ground states
  3. The Hamiltonian
  4. Dyck paths
  5. Ground states
  6. Lowering and raising operators
  7. Symmetry generators
  8. A relation to the total spin operator
  9. Proof of Theorem \ref{['Th2']}
  10. Symmetry algebra
  11. Preliminaries
  12. The Lie algebra of raising and lowering operators
  13. A central element
  14. Conclusion

Key Result

Theorem 1

For each value of $S^z\in \{-\frac{N}{2},-\frac{N}{2}+1,\dots, \frac{N}{2}-1,\frac{N}{2}\}$ there exists one and only one ground state $\left\lvert v_{S^z} \right\rangle$ of the periodic Fredkin chain Hamiltonian Hpbc with the zero eigenvalue, which corresponds to the eigenvalue $C=1$ of the cyclic In the case of $N$ even, $N=2n$, and $S^z=0$ there exists one more ground state which corresponds t

Figures (3)

  • Figure 1: A Dyck path belonging to the class $C_{a,b}(N)$, with $a=2$, $b=3$, and $N=9$
  • Figure 2: Equivalence classes of Dyck paths in the case $N=3$; the remaining can be obtained by a mirror transformation
  • Figure 3: Three classes $C_{a,a}(N)$ of the Dyck paths for $N=4$; the first line (even $a$) gives rise to the state $\left\lvert v_0^\text{even} \right\rangle$, and the second line (odd $a$) to the state $\left\lvert v_0^\text{odd} \right\rangle$

Theorems & Definitions (9)

  • Theorem 1: Salberger, Korepin SK-17SK-18
  • Theorem 2
  • Conjecture 1
  • Conjecture 2
  • Lemma 1
  • Proposition 1
  • proof
  • Conjecture 3
  • Conjecture 4