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Collective symmetry algebras of the periodic Fredkin chain

Yu Li, Qihang Wang

Published Oct 22, 2026• 0 citations
Publisher

Abstract

We determine the nonlocal symmetry algebra of the periodic spin-1/2 Fredkin chain for every length N ≥ 3. A rank-one generating function shows that the conserved raising and lowering charges act only on the symmetric Dicke multiplet and vanish on its orthogonal complement. An auxiliary spin on operator space yields a closed formula for their collective-spin expansion. The generated algebra is symplectic at odd length and orthogonal at even length, correcting the type labels in the original conjecture while confirming its rank and central splitting. For ideal independent charge controls, we characterize pure-state orbits and optimal transfer fidelities. At even length, we prove the path-sign symmetry and determine how additional controls act on the full ground space. We also obtain the operator Schmidt spectrum of every charge and an exact matrix product representation with minimal bond dimension at every cut.

DOI
10.5281/zenodo.23566538

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We determine the nonlocal symmetry algebra of the periodic spin-1/2 Fredkin chain for every length N ≥ 3. A rank-one generating function shows that the conserved raising and lowering charges act only on the symmetric Dicke multiplet and vanish on its orthogonal complement. An auxiliary spin on operator space yields a closed formula for their collective-spin expansion. The generated algebra is symplectic at odd length and orthogonal at even length, correcting the type labels in the original conjecture while confirming its rank and central splitting. For ideal independent charge controls, we characterize pure-state orbits and optimal transfer fidelities. At even length, we prove the path-sign symmetry and determine how additional controls act on the full ground space. We also obtain the operator Schmidt spectrum of every charge and an exact matrix product representation with minimal bond dimension at every cut.
This preprint determines the complex Lie algebra generated by the original nonlocal ladder charges of the undeformed periodic spin-one Motzkin chain for every length N >= 2. A rank-one factorization shows that the charges act on the known (2N+1)-dimensional ground space and annihilate its orthogonal complement. Their algebra is so(2N+1,C) in its defining representation, establishing the conjectured rank and displayed Cartan matrix while correcting its type designation to B_N. An auxiliary sl(2) triple and a conditional-variance estimate yield a sufficient moment criterion for positive reflection-symmetric weighted ladders of either parity. For the periodic spin-1/2 Fredkin chain with N >= 3, the same criterion gives so(N+1,C) for even N and sp(N+1,C) for odd N. Central magnetization components and simple-coroot coefficients are obtained explicitly for both chains. Under independent signed control by the two Hermitian Motzkin charge components, the reachable group is SO(2N+1); the exact optimal transfer fidelity between pure ground states is derived. Adding quadratic collective magnetization gives full unitary control, and degree two is minimal among polynomial magnetization additions.
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 = ⌈ N / 2 ⌉ . 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.
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.
Conservation laws serve as the hallmark of integrability. The absence of conserved charges typically implies that the model is nonintegrable. The recently proposed Fredkin spin chain exhibits rich structures, and its ground state is analytically known. However, whether the Fredkin spin chain is integrable remains an open question. In this work, through rigorous analytical calculations, we demonstrate that the Fredkin spin chain, under both periodic and open boundary conditions, lacks local conserved charges, thereby confirming its nonintegrable nature. Furthermore, we find that when one or a portion of the Hamiltonian terms are removed (referred to as the truncated Fredkin spin chain), local conserved charges are still absent. Our findings suggest that in models involving three-site interactions, integrable models are generally rare.
We study a nearest-neighbor non-Hermitian spin chain obtained from one member of an exact non-braided scalene Yang--Baxter triple. Its local Hamiltonian density violates both the difference-form Reshetikhin condition and its general non-difference counterpart, obstructing its realization by a differentiable homogeneous regular solution of the ordinary Yang--Baxter equation. The transfer matrices constructed from the regular member do not commute among themselves at distinct spectral parameters. Nevertheless, the scalene Yang--Baxter relation implies cross-commutativity with another transfer matrix constructed from the third member of the scalene triple. We evaluate the latter for arbitrary chain length and show that, on even periodic chains, it is a finite generating function of a non-obvious staggered nilpotent symmetry. The resulting conserved hierarchy belongs entirely to the algebra generated by this single symmetry and hence does not constitute an extensive family of algebraically independent charges. Nevertheless, this example demonstrates that scalene Yang--Baxter triples can act as an algebraic symmetry-discovery mechanism beyond the ordinary self-commuting transfer-matrix framework.
We prove an exact finite-volume symmetry formula for two-point functions in the periodic NNN-state superintegrable chiral Potts spin chain. We show that, for every chain length LLL and every simultaneous eigenvector of the Hamiltonian and the one-site translation operator, the correlations satisfy ⟨Z0rZR†r⟩∗=⟨Z0rZL−R†r⟩\langle Z_0^r Z_R^{\dagger r}\rangle^*=\langle Z_0^r Z_{L-R}^{\dagger r}\rangle⟨Z0r​ZR†r​⟩∗=⟨Z0r​ZL−R†r​⟩ for 1⩽r⩽N−11\leqslant r\leqslant N-11⩽r⩽N−1. Hence, whenever LLL is even, the midpoint correlation ⟨Z0rZL/2†r⟩\langle Z_0^r Z_{L/2}^{\dagger r}\rangle⟨Z0r​ZL/2†r​⟩ is real. Then we generalise the three-state chain case to arbitrary NNN and to every translation eigensector. This resolves a conjecture of Fabricius and McCoy.