Symmetry of Spin Systems as Automorphisms of Undirected Weighted Graphs: Realizability Criterion and Complete Taxonomy up to 14 Spins

Abstract

Exact simulation of high-resolution NMR spectra requires block diagonalization of the spin Hamiltonian, whose dimension grows exponentially with the number of spins $N$; symmetry is the principal tool for taming this growth, yet which permutation groups can occur as the full symmetry group of a scalar-coupled spin system has lacked an exhaustive treatment. Formulating the spin system as an undirected edge-weighted complete graph, we prove an exact realizability criterion: a subgroup of $S_N$ is realizable if and only if it coincides with its symmetrized (undirected) Wielandt 2-closure. In particular, purely rotational symmetry of a single spin ring is impossible, yet chiral spin systems do exist as multi-orbit twisted stacks, and we determine the minimal spin count $μ^{*}(C_n)$ for every cyclic group, including the counter-intuitive realizations $C_8$ and $C_9$ at $N = 12$. A sequential symmetrization algorithm, completed by an orbit-partition decomposition, yields a provably exhaustive enumeration of all realizable symmetry types up to $N = 14$: the apparently new sequence $a(N) = 1, 1, 3, 8, 11, 27, 36, 90, 131, 282, 394, 948, 1316, 2866$ with the tower law $a(N) = a(N-1) + f(N)$ - a catalogue of 6112 entries in all, organized by canonical identifiers and a structural grammar extending the Pople nomenclature. Finally, we present a hierarchical methodology for exact block diagonalization without physical approximations: factorization by the conserved total spin projection, Schur-Weyl contraction of magnetically equivalent composites, orbit-weight deduplication of the spin configurations, and isotypic projection over the representations of the factor group, with a uniform treatment of non-abelian groups and complex characters.