This is a representative case of the central spin problem in science, one selected spin interacting with multiple non-interacting spins of a different kind, or a localized spin coupled to a “spin bath,” and parallels can be found in many problems of condensed matter and chemical physics and quantum information processing. This procedure provides a way of seeing things that can only be solved numerically, giving a useful tool to gain insights that complement the numeric simulations usually inevitable here, and shows an intriguing connection to discrete Fourier transform and spectral properties of standard graphs.

We provide a simple geometrization of energy levels in this case: given n spin- 1 2 nuclei with arbitrary positive couplings a i, take an n-dimensional hyper-ellipsoid with semiaxes a i, stretch it by a factor of n + 1 along the spatial diagonal ( 1, 1, …, 1 ), read off the semiaxes of thus produced new hyper-ellipsoid q i, augment the set with q 0 = 0, and obtain the sought n + 1 energies as E k = − 1 2 q k 2 + 1 4 ∑ i a i. Systems described by such blocks are now physically realizable, e.g., as radicals or radical pairs with polarized nuclear spins, appear as closed subensembles in more general radical settings, and have numerous counterparts in related central spin problems. This is already the case for the simplest nontrivial ( K max − 1 ) block for an isotropic hyperfine Hamiltonian for a radical with spin- 1 2 nuclei, where n nuclei produce an n-th order algebraic equation with n independent parameters. However, the eigenvalue problems arising here lead to algebraic problems too complex to be analytically tractable.

Description of interacting spin systems relies on understanding the spectral properties of the corresponding spin Hamiltonians.

Eigensystem mathematica