We generalize Schrödinger’s factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach is the fact that the Hamiltonian is represented as a sum over factorizations in terms of coupled operators that depend on the coordinates and momenta in each Cartesian direction. We determine the eigenstates and energies, the wavefunctions in both coordinate and momentum space, and we also illustrate how this technique can be employed to develop the conventional confluent hypergeometric equation approach. The methodology developed here could potentially be employed for other Hamiltonians that can be represented as the sum over coupled Schrödinger factorizations.

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In this work, we study the two-dimensional S = 2 model that can lead to the well-known cubic singleion anisotropy along with a ferromagnetic Heisenberg interaction under a perpendicular magnetic field. When the Heisenberg term (J) is ignored, this model can be exactly solved with a reshuffled local Hilbert space. A three-fold (two-fold) eigenstate degeneracy is found at h = 0 (h = 2K), where h stands for the field strength and K is the strength of cubic anisotropy. Around the second degeneracy point there is a huge gap that separates two low-energy states from the rest three. As we turn on the J term, its leading-order contribution boils down to an effective hard-core bosonic model as K, h J. By using the two-dimensional tensor network ansatz we reveal that indeed various ground-state ansatz with different magnetic easy axes possess competing energies. Our finding demonstrates that the easy axes can be "softened" as we turn on the magnetic field for such quantum model with cubic anisotropy, which goes beyond the mean-field analysis. Such easy-axis softening might provide a new measure to control the magnetic orientation for spinful semiconductors.

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