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arXiv:2507.18291 · Journal of Physics A Mathematical and Theoretical

Symmetries of the periodic Fredkin chain

A. G. Pronko

Published Oct 22, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ae1644

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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.
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.
We introduce a new model of interacting spin 1/2. It describes interactions of three nearest neighbors. The Hamiltonian can be expressed in terms of Fredkin gates. The Fredkin gate (also known as the controlled swap gate) is a computational circuit suitable for reversible computing. Our construction generalizes the model presented by Peter Shor and Ramis Movassagh to half-integer spins. Our model can be solved by means of Catalan combinatorics in the form of random walks on the upper half plane of a square lattice (Dyck walks). Each Dyck path can be mapped on a wave function of spins. The ground state is an equally weighted superposition of Dyck walks (instead of Motzkin walks). We can also express it as a matrix product state. We further construct a model of interacting spins 3/2 and greater half-integer spins. The models with higher spins require coloring of Dyck walks. We construct a [Formula: see text] symmetric model (where [Formula: see text] is the number of colors). The leading term of the entanglement entropy is then proportional to the square root of the length of the lattice (like in the Shor–Movassagh model). The gap closes as a high power of the length of the lattice [5, 11].
We introduce a new model of interacting spin 1/2. It describes interaction of three nearest neighbors. The Hamiltonian can be expressed in terms of Fredkin gates. The Fredkin gate (also known as the CSWAP gate) is a computational circuit suitable for reversible computing. Our construction generalizes the work of Ramis Movassagh and Peter Shor. Our model can be solved by means of Catalan combinatorics in the form of random walks on the upper half of a square lattice [Dyck walks]. Each Dyck path can be mapped to a wave function of the spins. The ground state is an equally weighted superposition of Dyck walks [instead of Motzkin walks]. We can also express it as a matrix product state. We further construct the model of interacting spins 3/2 and greater half-integer spins. The models with higher spins require coloring of Dyck walks. We construct SU(k) symmetric model [here k is the number of colors]. The leading term of the entanglement entropy is then proportional to the square root of the length of the lattice [like in Shor-Movassagh model]. The gap closes as a high power of the length of the lattice.
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.