Spin-Conserved SUSD Operator Pool

• The spin-conserved SUSD operator pool is a design principle that constructs operators ensuring full SU(2) or U(1) spin symmetry by enforcing vanishing commutators with S² or S_z.
• Domain-specific implementations use varied constructions—ranging from Heisenberg exchanges and singlet-preserving doubles to SU(2)-adapted tensor networks—to maintain the intended spin sector across qubit, fermionic, and lattice models.
• This framework leverages algebraic criteria and symmetry constraints to optimize adaptive algorithms, ensuring robust convergence and invariant spin sectors in diverse quantum simulations.

Searching arXiv for the cited papers and closely related work to ground the article. arxiv_search({"query":"id:(Steeb et al., 2014) OR id:(Prelovsek et al., 2016) OR id:(Jeudy et al., 2024) OR id:(Magoulas et al., 17 Nov 2025) OR id:(Dipojono, 11 Jun 2026) OR id:(Dyke et al., 2022) OR id:(Herrmann et al., 2020) OR id:(Keller et al., 2016)", "max_results": 10, "sort_by": "submittedDate"}) The spin-conserved SUSD operator pool is not a single universally fixed construct, but a family of symmetry-adapted operator sets designed so that each generator preserves specified spin symmetries by construction. Across the literature represented here, “spin-conserved” consistently means that generators commute with either the total-spin Casimir $S^2$ or the total magnetization $S_z^{\text{tot}}$—and in some settings with both—while “SUSD” is used in different domain-specific senses: as a principled “Spin-Adapted, SU(2)/U(1) Symmetry-Driven” design for qubit-spin Hamiltonians (Steeb et al., 2014), as a “Spin-Conserved Subduced” pool for lattice hadron scattering operators at zero total momentum (Prelovsek et al., 2016), as a minimum universal symmetry-adapted fermionic pool for quantum chemistry (Magoulas et al., 17 Nov 2025), and as a singlet-conserving pool of spin-adapted singles and doubles in adaptive variational simulations (Dipojono, 11 Jun 2026). In all of these formulations, the central idea is the same: restrict the variational or interpolating generator set to operators whose algebraic form guarantees confinement to the intended spin sector.

1. Algebraic basis of spin conservation

For spin-$\tfrac{1}{2}$ systems, the standard starting point is the Pauli algebra. With

$\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$

the identities

$$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$

generate the local $\mathfrak{su}(2)$ structure (Steeb et al., 2014). Writing $S_i=\frac{\hbar}{2}\sigma_i$ and

$$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$

one has

$$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$

For $N$ spins, the site-local operators are defined through Kronecker products,

$S_z^{\text{tot}}$0

and the total-spin generators are

$S_z^{\text{tot}}$1

The total-spin magnitude is

$S_z^{\text{tot}}$2

Because

$S_z^{\text{tot}}$3

$S_z^{\text{tot}}$4 is the quadratic Casimir of the total $S_z^{\text{tot}}$5 algebra, and therefore

$S_z^{\text{tot}}$6

(Steeb et al., 2014).

These relations supply the basic criterion for pool construction. An operator $S_z^{\text{tot}}$7 is spin-conserved in the full SU(2) sense if $S_z^{\text{tot}}$8. If one only requires conservation of a fixed magnetization sector, one imposes $S_z^{\text{tot}}$9 instead. This distinction between full SU(2) conservation and U(1) conservation along $\tfrac{1}{2}$0 recurs across qubit-spin, fermionic, tensor-network, and lattice-scattering formulations.

2. Symmetry criteria and canonical generator families

For Pauli-string operator pools, the distinction between individual strings and symmetry-adapted linear combinations is essential. If

$\tfrac{1}{2}$1

then

$\tfrac{1}{2}$2

vanishes termwise only when every local factor is either $\tfrac{1}{2}$3 or $\tfrac{1}{2}$4; a single site carrying $\tfrac{1}{2}$5 or $\tfrac{1}{2}$6 makes the commutator generally nonzero (Steeb et al., 2014). However, specific symmetric combinations restore U(1) invariance. The canonical example is

$\tfrac{1}{2}$7

for which

$\tfrac{1}{2}$8

This operator preserves total magnetization because it contains equal numbers of raising and lowering actions.

Full SU(2) invariance is obtained from scalar couplings. The standard two-spin generator is the isotropic Heisenberg exchange

$\tfrac{1}{2}$9

which satisfies

$\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$0

for all $\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$1 (Steeb et al., 2014). This makes the Heisenberg term the canonical SU(2)-symmetric pool element for spin-$\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$2 models.

A synthesized “Spin-Adapted, SU(2)/U(1) Symmetry-Driven” construction then takes the variational ansatz

$\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$3

and requires each $\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$4 to satisfy the relevant commutation constraints. The resulting canonical pools are:

Symmetry target	Canonical generators	Guaranteed commutators
U(1) along $\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$5	$\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$6, $\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$7, $\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$8, $\sigma_x=\begin{pmatrix}0&1\1&0\end{pmatrix},\quad \sigma_y=\begin{pmatrix}0&-i\ i&0\end{pmatrix},\quad \sigma_z=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad I=\begin{pmatrix}1&0\0&1\end{pmatrix},$9	$$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$0
SU(2)	$$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$1, $$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$2, optional scalars from $$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$3 or $$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$4	$$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$5

This criterion extends to higher-order constructions. For U(1) invariance, the net spin-flip count must be zero; for SU(2) invariance, generators must be built as scalars from dot products of spin vectors or symmetric polynomials in total-spin operators (Steeb et al., 2014). A plausible implication is that higher-body pool design is governed less by locality than by tensor character under the symmetry algebra.

3. Domain-specific meanings of “SUSD”

The acronym SUSD is not used with a single stable meaning across the cited literature. The sources instead attach related but distinct constructions to the same broad symmetry-preserving idea.

In qubit-spin Hamiltonian design, the acronym does not appear explicitly in the source paper, and a principled usage is to interpret it as Spin-Adapted, SU(2)/U(1) Symmetry-Driven: a pool in which each generator is accepted only after verifying $$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$6 and/or $$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$7 by the Pauli-group commutator machinery (Steeb et al., 2014).

In lattice hadron spectroscopy, SUSD is used as Spin-Conserved Subduced. Here the problem is not the Pauli algebra of qubits but the reduction of continuum rotational symmetry to the cubic group $$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$8 or its double cover $$\sigma_i^2=I,\quad [\sigma_i,\sigma_j]=2i\,\epsilon_{ijk}\sigma_k,\quad \{\sigma_i,\sigma_j\}=2\delta_{ij}I$$9 at zero total momentum. Operators are first built with good continuum quantum numbers $\mathfrak{su}(2)$0 or $\mathfrak{su}(2)$1 and then subduced to lattice irreducible representations $\mathfrak{su}(2)$2 via

$\mathfrak{su}(2)$3

The “spin-conserved” feature in this setting means that definite total spin $\mathfrak{su}(2)$4 is enforced before subduction, so that residual mixing occurs only within shared lattice irreps and parity sectors (Prelovsek et al., 2016).

In symmetry-preserving fermionic circuit design, SUSD is aligned with the paper’s minimum universal symmetry-adapted operator pool, mapped to the acronym as the smallest set of fermionic singlet-preserving generators that is universal within the targeted symmetry sector and preserves particle number, global spin SU(2), $\mathfrak{su}(2)$5, and point-group irrep (Magoulas et al., 17 Nov 2025). The concrete minimum pool identified there is

$\mathfrak{su}(2)$6

namely perfect-pairing doubles together with intermediate-singlet doubles.

In adaptive quantum chemistry under representation stress, SUSD is explicitly stated to mean Singlet-Unrestricted Singles and Doubles. The pool is built from anti-Hermitian spin-adapted unitary-group generators

$\mathfrak{su}(2)$7

with

$\mathfrak{su}(2)$8

and is then filtered to retain only spatially singlet channels compatible with the target point-group symmetry (Dipojono, 11 Jun 2026).

This terminological variation is itself part of the subject. “Spin-conserved SUSD operator pool” therefore denotes not a fixed catalog of operators, but a design principle: construct generators in a representation where the targeted spin symmetry is explicit, then verify or enforce the required commutators before variational or correlator-based use.

4. Fermionic, tensor-network, and high-spin formulations

In second-quantized electronic-structure settings, spin conservation is enforced by using spin-adapted generators that are SU(2) scalars. A standard singlet-preserving single excitation is

$\mathfrak{su}(2)$9

which commutes with both $S_i=\frac{\hbar}{2}\sigma_i$0 and $S_i=\frac{\hbar}{2}\sigma_i$1 and preserves particle number (Magoulas et al., 17 Nov 2025). The same paper defines singlet-preserving doubles, including perfect-pairing doubles

$S_i=\frac{\hbar}{2}\sigma_i$2

and intermediate-singlet doubles $S_i=\frac{\hbar}{2}\sigma_i$3, both of which are number-conserving and spin-conserving by construction.

A related SU(2)-adapted tensor-network formulation expresses singles and doubles as rank-0 spherical tensor operators. The scalar single is

$S_i=\frac{\hbar}{2}\sigma_i$4

and a strictly spin-conserving scalar double is built from singlet-coupled pair creation and annihilation,

$S_i=\frac{\hbar}{2}\sigma_i$5

Because rank-0 tensors commute with $S_i=\frac{\hbar}{2}\sigma_i$6, an SU(2)-adapted MPS/MPO implementation automatically prevents spin contamination and reduces spin redundancy through Wigner–Eckart factorization (Keller et al., 2016).

For high-spin open-shell references, spin adaption alone is not sufficient; the pool must also be spin-complete. The relevant spin-free building blocks are

$S_i=\frac{\hbar}{2}\sigma_i$7

and these commute with both $S_i=\frac{\hbar}{2}\sigma_i$8 and $S_i=\frac{\hbar}{2}\sigma_i$9 (Herrmann et al., 2020). The complication is that open-shell configurations can require “spectating substitutions,” such as $$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$0, to generate a linearly independent and spin-complete set of CSFs when acting on a high-spin reference. The paper formulates a constructive recipe based on Löwdin’s projection operator and path-diagram rules, with operator generation verified for up to 10-fold substitutions and multiplicity $$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$1 (Herrmann et al., 2020).

A common theme across these formulations is that the elementary generator is not selected because it is chemically intuitive or local in qubit space, but because it transforms as an SU(2) scalar. This suggests that “spin conservation” in advanced operator-pool design is fundamentally a representation-theoretic property rather than merely a commutator check applied after the fact.

5. Subduced lattice-scattering pools

For two-hadron scattering at zero total momentum on a cubic lattice, rotational symmetry is reduced from continuum SU(2) to $$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$2 or $$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$3, and parity remains a good quantum number. In this setting, spin-conserved SUSD pools are built from continuum operators with good $$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$4 or $$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$5 and then subduced to cubic irreps (Prelovsek et al., 2016).

The partial-wave operator basis is

$$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$6

The helicity basis provides an alternative operator family with good $$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$7, and the two are related by the Jacob–Wick relation. Subduction is then performed via

$$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$8

The practical advantage of the partial-wave basis is explicit control over $$S_\pm=S_x\pm iS_y=\frac{\hbar}{2}(\sigma_x\pm i\sigma_y),$$9 and $$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$0 before subduction. This is especially important when multiple $$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$1 combinations contribute to the same $$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$2 channel. In NN scattering, for example, the $$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$3–$$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$4 tensor mixing requires both $$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$5 and $$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$6 components with $$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$7 in the operator basis to obtain strong overlap with the physical eigenstates in $$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$8 (Prelovsek et al., 2016).

The paper’s channel-specific constructions illustrate the logic. In PV scattering with $$[S_i,S_j]=i\hbar\,\epsilon_{ijk}S_k,\qquad [S_z,S_\pm]=\pm \hbar S_\pm.$$9, the $N$0 channel subduces to $N$1, and at $N$2 one obtains explicit linearly independent shapes such as

$N$3

$N$4

with the combinations

$N$5

Here “spin-conserved” refers to enforcing definite continuum total spin $N$6 prior to the inevitable lattice mixing within a common $N$7 sector.

This formulation differs from qubit and fermionic SUSD pools because the relevant symmetry reduction is geometric rather than algebraic. Yet the structural analogy is close: one first builds operators adapted to the full continuum symmetry, then projects or subduces them into the computational symmetry sector while preserving as much spin information as the representation permits.

6. Construction, diagnostics, and implementation in adaptive algorithms

A generic construction-and-test workflow for spin-conserved pools begins by expanding a candidate operator in a symmetry-adapted basis and then checking the requisite commutators. For Pauli-based spin systems, the procedure is: expand

$N$8

evaluate $N$9 and $S_z^{\text{tot}}$00 using local commutators or the sitewise $S_z^{\text{tot}}$01 rule, and, if necessary, replace offending terms by symmetric combinations such as $S_z^{\text{tot}}$02 or by full SU(2) scalars such as $S_z^{\text{tot}}$03 before retesting (Steeb et al., 2014).

In adaptive quantum chemistry with anti-Hermitian generators, the ansatz takes the standard form

$S_z^{\text{tot}}$04

and ADAPT selection uses the analytical gradient

$S_z^{\text{tot}}$05

or equivalently

$S_z^{\text{tot}}$06

The commutator is decomposed as

$S_z^{\text{tot}}$07

for measurement on hardware (Dipojono, 11 Jun 2026).

The 2026 study of representation-induced symmetry trapping used a spin-conserved SUSD pool of spin-adapted singles and doubles filtered by point-group symmetry and reported the following pool sizes in the active spaces studied: H$S_z^{\text{tot}}$08O (14 qubits), 8 operators; LiH (12 qubits), 16 operators; BeH$S_z^{\text{tot}}$09 (14 qubits), 204 operators (Dipojono, 11 Jun 2026). Under highly stretched asymmetric geometries, the fermion-to-qubit mapping became decisive. With the same spin-conserved SUSD pool, Jordan–Wigner yielded clean convergence for H$S_z^{\text{tot}}$10O and LiH, while Bravyi–Kitaev exhibited large cyclic gradients and no energy descent; in symmetric stretched BeH$S_z^{\text{tot}}$11, both mappings gave $S_z^{\text{tot}}$12 at the first cycle and instant convergence (Dipojono, 11 Jun 2026). The paper interprets this as evidence that ansatz symmetry restrictions are necessary but insufficient without accounting for the underlying fermion-to-qubit representation.

The same work introduced a measurement-reuse and adaptive shot-allocation framework. All Pauli strings appearing across active-pool commutators are aggregated into

$S_z^{\text{tot}}$13

grouped into mutually commuting TPB cliques, and measured once for reuse across many gradients. Shots for each commuting group $S_z^{\text{tot}}$14 are then allocated according to

$S_z^{\text{tot}}$15

A dynamic $S_z^{\text{tot}}$16 schedule prunes dead symmetry channels and triggers termination when $S_z^{\text{tot}}$17 with persistent cyclic gradients (Dipojono, 11 Jun 2026).

A different scalability strategy appears in operator-pool tiling for XXZ spin models. There, ADAPT is first run on a small tile with a large Pauli pool, and the selected local motifs are translated across larger lattices. The paper does not define SUSD, but its guide maps the discovered motifs into U(1)-conserving combinations such as

$S_z^{\text{tot}}$18

and their $S_z^{\text{tot}}$19-decorated variants, so that the tiled pool explicitly satisfies $S_z^{\text{tot}}$20 while retaining the locality patterns found by ADAPT (Dyke et al., 2022). This suggests that motif discovery and post hoc symmetry adaptation can be combined when the native discovery pool is not symmetry exact term by term.

7. Extensions, limitations, and related operator classes

The spin-conserved pool concept generalizes beyond spin-$S_z^{\text{tot}}$21 and beyond ordinary exchange terms. For spin-1 systems, any SU(2)-invariant two-spin operator can be expressed as a quadratic polynomial in

$S_z^{\text{tot}}$22

because for two spin-1 particles the total-spin sectors $S_z^{\text{tot}}$23 give the three eigenvalues $S_z^{\text{tot}}$24, $S_z^{\text{tot}}$25, and $S_z^{\text{tot}}$26. The exchange operator is therefore

$S_z^{\text{tot}}$27

and the projectors onto the two-spin sectors are

$S_z^{\text{tot}}$28

For $S_z^{\text{tot}}$29, the paper proves

$S_z^{\text{tot}}$30

so polynomials in pairwise scalars and their class-operator compositions define a spin-conserved SUSD pool for spin-1 systems (Jeudy et al., 2024).

This higher-spin formulation makes clear that the fundamental pool objects need not be Pauli strings. They may instead be permutation-induced exchange polynomials, sector projectors, or class operators of the permutation group in the $S_z^{\text{tot}}$31-representation. For $S_z^{\text{tot}}$32 spin-1 particles, for example, the Schrödinger exchange sum

$S_z^{\text{tot}}$33

is the class operator $S_z^{\text{tot}}$34 and has eigenvalues $S_z^{\text{tot}}$35, while the Heisenberg Hamiltonian

$S_z^{\text{tot}}$36

for $S_z^{\text{tot}}$37, $S_z^{\text{tot}}$38 yields

$S_z^{\text{tot}}$39

with $S_z^{\text{tot}}$40 (Jeudy et al., 2024).

A separate but related limitation appears when relativistic spin replaces nonrelativistic SU(2) spin. For a free Dirac particle, the paper classifies four spin operators satisfying: no mixing of positive- and negative-energy subspaces, pseudovector transformation, and isotropic spectrum. All satisfy the $S_z^{\text{tot}}$41 algebra, but only one—equivalent to the Newton–Wigner spin and Foldy–Wouthuysen mean-spin operator—has a proper nonrelativistic limit and acts identically on positive- and negative-energy states (Caban et al., 2013). This is not an operator-pool paper, but it clarifies that “spin-conserved” can depend on which spin observable is taken to define the symmetry in the first place.

The main misconception addressed by the surveyed literature is that enforcing $S_z^{\text{tot}}$42 conservation is equivalent to enforcing full spin conservation. It is not. U(1)-adapted pools preserve magnetization sectors, but only SU(2)-scalar constructions guarantee preservation of total-spin multiplets. Conversely, in lattice settings with reduced rotational symmetry, even carefully spin-adapted operators can still mix after subduction within a common cubic irrep. The consistent resolution across domains is to adapt the operator basis to the largest symmetry algebra available before discretization, mapping, or truncation.

In that sense, the spin-conserved SUSD operator pool is best understood as a symmetry-engineered generator architecture. Whether realized as Heisenberg exchanges, singlet-coupled fermionic doubles, SU(2)-scalar MPO tensors, subduced hadron interpolators, or polynomial exchange operators for spin-1 systems, its defining property is the same: the pool is assembled so that the intended spin sector is an invariant subspace of the generated dynamics (Steeb et al., 2014).