Fredkin Chain Quantum Spin Model

• Fredkin chain is a one-dimensional quantum spin model characterized by controlled-SWAP dynamics and constrained Dyck-path configurations that yield frustration-free ground states.
• The model features tunable entanglement scaling and a rich phase structure through deformations, colored variants, and emergent non-thermal eigenstates.
• It demonstrates atypical gap scaling and subdiffusive transport, with a rigorous proof of nonintegrability established by the absence of nontrivial local conserved charges.

Searching arXiv for recent and foundational papers on the Fredkin chain. The Fredkin chain is a family of one-dimensional quantum spin models with local three-site interactions whose defining structural feature is a constrained “Fredkin” or controlled-SWAP dynamics. In its canonical spin-$1/2$ open-chain form, the model is frustration-free and has an exactly known ground state given by the equal-weight superposition of Dyck-path spin configurations, equivalently balanced parenthesis words encoded by up and down spins (Salberger et al., 2016). Subsequent work introduced colored, deformed, and generalized variants with weighted Dyck-path ground states, tunable entanglement scaling, slow dynamics, Hilbert-space fragmentation, and exact non-thermal eigenstates (Salberger et al., 2016, Zhang et al., 2017, Langlett et al., 2021, Causer et al., 2024). A central recent development is the rigorous demonstration that the periodic and open spin-$1/2$ Fredkin chain lacks nontrivial local conserved charges, establishing nonintegrability in the Yang–Baxter local-charge sense (Fan et al., 5 Sep 2025).

1. Canonical spin-$1/2$ model and Hamiltonian

The standard Fredkin chain is a spin-$1/2$ chain with local basis $\{|\uparrow\rangle,|\downarrow\rangle\}$, typically studied on an open chain of even length with boundary projectors enforcing Dyck-compatible ends (Salberger et al., 2016, Movassagh, 2016). In one widely used projector form, the Hamiltonian is

$$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$

with

$$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$

where

$$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$

and the boundary term may be written as

$H_{\partial}=P_1^-+P_N^+.$

This formulation makes explicit that the model is frustration-free: the ground state is annihilated by every local projector (Salberger et al., 2016).

An equivalent description uses the three-qubit Fredkin gate $F_{j,k,l}$, or controlled-SWAP gate, acting as

$1/2$0

with

$1/2$1

In this language, the bulk density can be written as a sum of projectors generated by local Fredkin moves (Fan et al., 5 Sep 2025).

A closely related open-boundary Hamiltonian, used in the rigorous nonintegrability analysis, is

$1/2$2

with

$1/2$3

and

$1/2$4

again exhibiting three-site constrained exchange (Fan et al., 5 Sep 2025).

Under periodic boundary conditions, after discarding constants and identities, the bulk may be rewritten as a sum of nearest-neighbor and genuine three-site terms,

$1/2$5

where

$1/2$6

and

$1/2$7

This decomposition is important for integrability diagnostics, because the model combines an XXX-like nearest-neighbor piece with explicitly three-site interactions (Fan et al., 5 Sep 2025).

2. Dyck-path structure and exact ground states

The canonical combinatorial representation identifies $1/2$8 with an up-step and $1/2$9 with a down-step of a lattice walk. A Dyck path of length $1/2$0 is a walk from $1/2$1 to $1/2$2 with steps $1/2$3 that never goes below the axis. Equivalently, it is a balanced parenthesis word in which no prefix has more closing than opening parentheses (Salberger et al., 2016, Movassagh, 2016).

For the undeformed open chain, the unique ground state is the uniform superposition of all Dyck paths,

$1/2$4

where $1/2$5 is the Catalan number (Salberger et al., 2016). In the colored generalization, each up/down step carries one of $1/2$6 colors, with matched up/down pairs required to have the same color, and the ground state becomes the uniform superposition over all properly colored Dyck paths (Movassagh, 2016).

The open-chain ground state is frustration-free and unique once boundary projectors select the Dyck sector (Salberger et al., 2016, Movassagh, 2016). Under periodic boundary conditions, by contrast, the ground-state structure changes qualitatively. The periodic chain has a degenerate ground-state manifold organized by Dyck-path equivalence classes and by total $1/2$7, with degeneracy $1/2$8 for odd $1/2$9 and $1/2$0 for even $1/2$1 (Pronko, 24 Jul 2025). In the cyclic-invariant sector, the periodic ground states are

$1/2$2

and for even $1/2$3 there is an additional $1/2$4 state

$1/2$5

where $1/2$6 denotes the homogeneous sum over a Dyck-path equivalence class (Pronko, 24 Jul 2025).

The open-chain model also admits an exact matrix-product-state representation in terms of height-raising and height-lowering matrices,

$1/2$7

with boundary vector $1/2$8, which enforces the Dyck constraint by keeping the virtual height nonnegative and returning it to its initial value at the end (Salberger et al., 2016). This representation is exact with bond dimension $1/2$9 and can be truncated because typical heights scale as $\{|\uparrow\rangle,|\downarrow\rangle\}$0 (Salberger et al., 2016).

3. Deformations, colored variants, and entanglement structure

A major branch of Fredkin-chain research concerns deformations in which the amplitudes of local Fredkin moves are weighted by a parameter $\{|\uparrow\rangle,|\downarrow\rangle\}$1, producing ground states that are no longer uniform but weighted by the path area. In a standard uniform deformation,

$\{|\uparrow\rangle,|\downarrow\rangle\}$2

where $\{|\uparrow\rangle,|\downarrow\rangle\}$3 is the area under the path (Salberger et al., 2016, Udagawa et al., 2017, Zhang et al., 2017). The Hamiltonian remains frustration-free because the local projector amplitudes are chosen so that configurations related by a Fredkin move appear with relative amplitude $\{|\uparrow\rangle,|\downarrow\rangle\}$4 (Salberger et al., 2016, Zhang et al., 2017).

The deformed Hamiltonian may be written as

$\{|\uparrow\rangle,|\downarrow\rangle\}$5

where $\{|\uparrow\rangle,|\downarrow\rangle\}$6 is a sum of three-body projectors implementing weighted Fredkin moves, $\{|\uparrow\rangle,|\downarrow\rangle\}$7 imposes local color matching and equal color weights, and $\{|\uparrow\rangle,|\downarrow\rangle\}$8 enforces Dyck boundary conditions (Salberger et al., 2016, Udagawa et al., 2017). A multi-parameter deformation replaces $\{|\uparrow\rangle,|\downarrow\rangle\}$9 by site-dependent $$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$0, while preserving frustration-freeness under the constraint $$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$1 (Zhang et al., 2017).

The entanglement structure depends sharply on both the deformation and the number of colors. For the undeformed chain at $$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$2, the colorless case $$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$3 has logarithmic half-chain entanglement,

$$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$4

and the model is gapless (Fan et al., 5 Sep 2025, Zhang et al., 2017). In colored variants, the leading entanglement term is proportional to $$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$5 (Salberger et al., 2016, Salberger et al., 2016).

For $$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$6, area-weighting favors high-area paths. In the deformed colorless chain, the half-chain entanglement becomes bounded, i.e. it obeys an area law (Salberger et al., 2016, Zhang et al., 2017). In the colored case $$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$7, the entropy instead becomes extensive: $$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$8 or more precisely

$$H = H_{\mathrm{bulk}} + H_{\partial}, \qquad H_{\mathrm{bulk}} = \sum_{j=1}^{N-2} h_{j,j+1,j+2},$$9

depending on notation and cut location (Udagawa et al., 2017, Zhang et al., 2017). This phase is often described as a rainbow phase because the dominant high-area paths pair sites across the center into long entangled arcs (Salberger et al., 2016, Alexander et al., 2018).

For $$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$0, low-area paths dominate and entanglement is bounded for all $$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$1 (Salberger et al., 2016, Zhang et al., 2017). This yields a phase diagram with qualitatively distinct logarithmic, bounded, and volume-law regimes, all within frustration-free, strictly local one-dimensional Hamiltonians (Salberger et al., 2016, Udagawa et al., 2017, Zhang et al., 2017).

The same path structure underlies tensor-network constructions. A bulk “holographic rainbow” tensor network exactly represents the ground states of area-deformed Motzkin and Fredkin chains; in the Fredkin case, the network is obtained by removing the tiles associated with flat Motzkin moves (Alexander et al., 2018). This suggests that the path combinatorics are not merely a basis representation but an organizing principle for exact many-body constructions.

4. Spectral gap, transport, and anomalously slow dynamics

The Fredkin chain is gapless in the thermodynamic limit, but the scaling of its gap is atypical. For the undeformed open chain, rigorous results show that the spectral gap is polynomially small: $$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$2 with lower bound

$$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$3

and upper bound

$$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$4

derived by combining a Markov-chain mapping, comparison theorems, and Brownian-excursion scaling (Movassagh, 2016). Earlier analyses described the gap as closing as a high power of the system size (Salberger et al., 2016).

Numerical and analytical studies of the low-energy dynamics identify an unusually large dynamical exponent. Direct simulations of the open Fredkin chain found

$$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$5

from lifetime scaling, and

$$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$6

from survival-gap scaling, while a mean-field random-walk picture gave

$$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$7

as a lower-bound approximation (Adhikari et al., 2020). The physical picture is that low-energy excitations behave as extended “canted bonds” moving through a Dyck-path background under severe kinetic constraints, producing subdiffusive transport and heavy-tailed first-passage kinetics (Adhikari et al., 2020).

A later development sharpened the hydrodynamic understanding by identifying a pseudolocal conserved quantity, the matched-parentheses charge, in Fredkin- and Motzkin-type constrained chains. This leads to an exact variational spin-wave bound

$$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$8

and therefore

$$h_{j,j+1,j+2} = P_j^{+} P^{\mathrm{sing}}_{j+1,j+2} + P^{\mathrm{sing}}_{j,j+1} P_{j+2}^{-},$$9

establishing a distinct Fredkin dynamical universality class with subdiffusive transport (McCarthy et al., 2024). The key hydrodynamic input is that the matched-parentheses density is locally conserved yet pseudolocal, with algebraically decaying density tails governed by Dyck combinatorics (McCarthy et al., 2024).

Out of equilibrium, the chain exhibits strong observable dependence. After a local longitudinal-field quench, longitudinal connected correlators display a light-cone-like spreading, whereas transverse correlators respond essentially immediately across the chain (Dell'Anna et al., 2019). This contrast is tied to the ground-state correlation structure: the colorless Fredkin chain has clustering $$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$0-correlations but non-decaying transverse correlations. For example,

$$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$1

and deep in the bulk

$$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$2

while $$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$3 at large distances (Dell'Anna et al., 2019). The cluster decomposition property therefore fails in the transverse sector even though the Hamiltonian is short-ranged and frustration-free (Dell'Anna et al., 2019).

The deformed chain also provides an example of anomalous universal adiabatic dynamics. When the deformation parameter is ramped linearly across the critical point $$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$4, the defect density can scale with an exponent different from the standard Kibble–Zurek prediction. In the $$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$5 sector, finite-size scaling gives

$$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$6

and the anomalous defect exponent becomes

$$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$7

yielding $$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$8, in agreement with numerical results $$P_j^{\pm}=\frac{1\pm \sigma_j^z}{2}, \qquad P^{\mathrm{sing}}_{i,i+1}=\frac{1}{4}\left(1-\vec{\sigma}_i\cdot \vec{\sigma}_{i+1}\right),$$9 (Francica et al., 2022). This suggests that the Fredkin chain is a useful counterexample to naive universality assumptions based solely on conformal criticality.

5. Symmetries, fragmentation, non-thermal eigenstates, and variants

Although the canonical Fredkin chain conserves total $H_{\partial}=P_1^-+P_N^+.$0, its broader constrained-dynamics generalizations exhibit richer structures. In the periodic spin-$H_{\partial}=P_1^-+P_N^+.$1 chain, nonlocal operators $H_{\partial}=P_1^-+P_N^+.$2 commute with the Hamiltonian and act as raising and lowering operators on the cyclic ground-state manifold: $H_{\partial}=P_1^-+P_N^+.$3 These operators generate a Lie algebra of type $H_{\partial}=P_1^-+P_N^+.$4 for odd $H_{\partial}=P_1^-+P_N^+.$5 and $H_{\partial}=P_1^-+P_N^+.$6 for even $H_{\partial}=P_1^-+P_N^+.$7, with $H_{\partial}=P_1^-+P_N^+.$8 (Pronko, 24 Jul 2025). The total $H_{\partial}=P_1^-+P_N^+.$9 can be represented as a sum of Cartan generators plus a central element,

$F_{j,k,l}$0

and in the even-$F_{j,k,l}$1 case this reproduces a structure previously conjectured for the periodic Motzkin chain (Pronko, 24 Jul 2025). These symmetries organize the ground-state degeneracies under periodic boundaries, but, as later shown, they do not imply a tower of local conserved charges (Fan et al., 5 Sep 2025).

A separate line of work generalized the Fredkin chain by modifying the sign structure of the local terms so that the frustration-free mechanism is embedded in the middle of the spectrum rather than the ground state. This produces Hilbert-space fragmentation and exact quantum many-body scars (Langlett et al., 2021). In these generalized chains, the Hilbert space splits into exponentially many disconnected subspaces labelled by fixed “color sequences” under an effective particle-hole mapping, with sector dimension

$F_{j,k,l}$2

Each fractured sub-sector contains an exact zero-energy eigenstate

$F_{j,k,l}$3

with entanglement at most logarithmic and often area-law (Langlett et al., 2021). These states are mid-spectrum for generic chaotic sectors and therefore constitute exact scars rather than ground states (Langlett et al., 2021).

Related deformations of the quantum Fredkin chain away from its stochastic point interpolate between fast thermalizing and slow metastable regimes (Causer et al., 2024). For the two-parameter Hamiltonian

$F_{j,k,l}$4

increasing $F_{j,k,l}$5 suppresses off-resonant moves and yields a folded effective model with stronger emergent constraints, strong fragmentation, and non-thermal eigenstates closely related to simple product states $F_{j,k,l}$6 (Causer et al., 2024). These scar-like eigenstates persist to large sizes in variational MPS calculations and have area-law entanglement throughout the spectrum (Causer et al., 2024).

The semigroup Fredkin chain extends the local Hilbert space by “arrow indices” partly associated with the symmetric inverse semigroup $F_{j,k,l}$7, giving a six-state local basis $F_{j,k,l}$8 with $F_{j,k,l}$9 (1804.00978). This produces three distinct ground-state phases: one with logarithmic area-law violation, and two area-law phases, together with exactly localized excited states arising from disconnections of arrow indices (1804.00978). Inference should be cautious here: the mechanism is not randomness-induced localization but exact fragmentation by local conserved projectors. A plausible implication is that the Fredkin construction can serve as a template for clean disorder-free localization phenomena once the local degrees of freedom are enriched (1804.00978).

Finally, the deformed Fredkin structure has appeared in an applied condensed-matter setting as an effective quasi-one-dimensional model for the $1/2$00 Moore–Read state on thin cylinders. There the Hamiltonian

$1/2$01

with

$1/2$02

provides a frustration-free qubit-chain approximation whose ground state has high overlap with the Moore–Read wavefunction for $1/2$03, reproducing both entanglement and geometric-quench graviton dynamics (Voinea et al., 2023). This does not redefine the Fredkin chain itself, but it demonstrates that Fredkin-type constrained projectors can arise as effective descriptions of topological matter.

6. Integrability question and rigorous nonintegrability

The integrability status of the Fredkin chain remained open for several years because the model has an exactly known ground state, frustration-free structure, and nontrivial symmetries, features often associated with solvable systems. The decisive result is that the spin-$1/2$04 Fredkin chain is nonintegrable in the local-charge sense relevant to Yang–Baxter solvability (Fan et al., 5 Sep 2025).

The criterion adopted is the existence of nontrivial local conserved charges. A $1/2$05-local charge is an operator

$1/2$06

with each density $1/2$07 supported on at most $1/2$08 consecutive sites and satisfying $1/2$09. For translation-invariant periodic chains,

$1/2$10

expanded in Pauli strings, and the commutator with the three-local Hamiltonian has range at most $1/2$11 (Fan et al., 5 Sep 2025).

The proof of nonintegrability proceeds by systematic Pauli-string enumeration and a “column expression” bookkeeping device for commutators (Fan et al., 5 Sep 2025). Representative relations such as

$1/2$12

yield linear constraints among coefficients, for example

$1/2$13

and under the condition $1/2$14, the corresponding $1/2$15-coefficients vanish (Fan et al., 5 Sep 2025).

The main periodic-boundary results are two theorems:

• the Fredkin chain has no 4- or 5-local conserved charges;
• more generally, it has no nontrivial $1/2$16-local conserved charges for all

$1/2$17

under periodic boundary conditions (Fan et al., 5 Sep 2025).

The proof begins with exhaustive case analyses for 4-local and 5-local densities, showing that most Pauli strings generate unique $1/2$18-local outputs that cannot be canceled by any other candidate term, hence their coefficients must vanish (Fan et al., 5 Sep 2025). It then generalizes to arbitrary $1/2$19 by classifying “valid boundary sequences” of candidate densities and proving connectivity lemmas that propagate the vanishing of one coefficient to all others (Fan et al., 5 Sep 2025).

For open boundaries, the paper proves that with the standard boundary term

$1/2$20

the chain has no $1/2$21-local boundary conserved charges with $1/2$22 (Fan et al., 5 Sep 2025). The argument classifies boundary-supported Pauli strings and shows that commutation with the bulk generates unique boundary outputs, again preventing cancellation (Fan et al., 5 Sep 2025).

The study also examines truncated Fredkin chains obtained by removing one or more of the three-site terms in $1/2$23. The conclusion is precise:

The truncated Fredkin spin chains possess local conserved charges if and only if all three-site interaction terms are removed (Fan et al., 5 Sep 2025).

When all three-site terms are removed, the model reduces to the nearest-neighbor XXX chain, which is Yang–Baxter integrable and possesses a tower of local conserved quantities (Fan et al., 5 Sep 2025). If any three-site term remains, the same no-pairing obstructions eliminate all candidate local charges (Fan et al., 5 Sep 2025).

This result settles the long-standing integrability question in a strong sense. The identity, the Hamiltonian itself, and total $1/2$24 remain conserved; quasi-local charges with exponentially decaying tails are not excluded, because the analysis concerns strictly local finite-range densities (Fan et al., 5 Sep 2025). Still, the absence of any nontrivial local-charge tower rules out standard Yang–Baxter-type integrability (Fan et al., 5 Sep 2025).

A plausible implication is that the Fredkin chain should be viewed not as a hidden integrable model but as a constrained, frustration-free, nonintegrable system whose exotic behavior derives from path combinatorics, kinetic constraints, and nonlocal symmetry structures rather than from solvability in the conventional sense. That interpretation is consistent with the coexistence of slow dynamics, non-thermal eigenstates, and symmetry-enforced structures across the broader Fredkin-family literature (Langlett et al., 2021, Causer et al., 2024, Fan et al., 5 Sep 2025).