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The nonlocal charge algebra of the periodic Motzkin chain

Yu Li, Qihang Wang

Published Oct 23, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.23585434

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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.
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.
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.
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.
Local frustration-free constraints can transport amplitudes consistently on every bond yet fail around a ring. We solve this obstruction for an interacting periodic spin-one Motzkin Hamiltonian with arbitrary nonzero complex ordered-bond parameters. The complete finite-size ground-space degeneracy and surviving-sector pattern depend only on the complex product of the bond parameters: for every ring size N ≥ 2, the dimension is 2N + 1 at trivial holonomy and exactly 3 otherwise. On the unit circle, the principal total boundary flux removes every nonextreme charged exact ground state and shifts the two charge-conjugate one-flat bands oppositely. Because those bands are invariant sectors of the full Hamiltonian, a fixed nonzero total flux gives a rigorous many-body gap upper bound of order N⁻². Moreover, every lifted charged sector contains a positive level at or below the squared principal flux divided by 4N, yielding at least 2N − 2 soft eigenvalues, counting multiplicity. Boundary flux therefore collapses the growing exact ground manifold without opening a thermodynamic gap. No gap lower bound, bulk transition, topology, or quantized transport is claimed. This v1 record contains the eight-page combined preprint and its minimal four-file LaTeX source archive. This is a preprint and has not undergone journal peer review.
We present exact results on the exactly solvable spin chain of Bravyi et al. [Phys. Rev. Lett. 109, 207202 (2012)]. This model is a spin one chain and has a Hamiltonian that is local and translationally invariant in the bulk. It has a unique (frustration free) ground state with an energy gap that is polynomially small in the system’s size (2n). The half-chain entanglement entropy of the ground state is 12logn+const.[Bravyi et al., Phys. Rev. Lett. 109, 207202 (2012)]. Here we first write the Hamiltonian in the standard spin-basis representation. We prove that at zero temperature, the magnetization is along the z-direction, i.e., ⟨sx⟩=⟨sy⟩=0 (everywhere on the chain). We then analytically calculate ⟨sz⟩ and the two-point correlation functions of sz. By analytically diagonalizing the reduced density matrices, we calculate the Schmidt rank, von Neumann, and Rényi entanglement entropies for the following: 1. Any partition of the chain into two pieces (not necessarily in the middle) and 2. L consecutive spins centered in the middle. Further, we identify entanglement Hamiltonians (Eqs. (49) and (59)). We prove a small lemma (Lemma (1)) on the combinatorics of lattice paths using the reflection principle to relate and calculate the Motzkin walk “height” to spin expected values. We also calculate the, closely related (scaled), correlation functions of Brownian excursions. The known features of this model are summarized in a table in Sec. I.
The Motzkin spin chain is a spin-1 model introduced in \cite{shor} as an example of a system exhibiting a high degree of quantum fluctuations whose ground state can be mapped to Motzkin paths that are generated with local equivalence moves. This model is difficult to solve in general but keeping just the height preserving local equivalence moves we show that the model becomes integrable which when projected to certain subspaces of the full Hilbert space is isomorphic to the spin-12\frac{1}{2}21​ XXX chain. In fact in the full Hilbert space the system is akin to two non-interacting spin-12\frac{1}{2}21​ XXX chains making the spectrum the same as the latter with the change coming in the degeneracy of the states. We then show that including the height-changing local-equivalence move is the same as introducing interactions in the above system.