Study of long-range Ising model using irreducible representations of SU(2)*

Ising models dictate a broad class of natural spin-dynamics bridging integrability and chaos, of interest to quantum simulation and metrology. When they exhibit permutation symmetry by infinite-range interactions, they collapse to a single irreducible representation (irrep) of SU(2) of the collective spin (the symmetric subspace). Finite but long-range interactions tend to preserve much of the collective character, avoiding chaos when initialized in the largest SU(2)-irrep¹. Meanwhile irreducible spherical tensors form physically motivated orthonormal operator bases within irreps, and admit generalizations² across multiple irreps, but the generalized tensors' algebras beg exploration. We truncate a transverse Ising model with slow-decaying interactions, keeping first-order accessible subspaces from this symmetric irrep, and analytically decomposing the truncated Hamiltonian into generalized spherical tensors. We study the dynamics as the system leaks into lower-spin irreps via observable phenomena like dynamical quantum phase transitions. We develop Heisenberg equations of motion on these spherical tensors, and numerically probe the truncation's quality of approximation for finite-size systems.

¹Lerose, Alessio, "Theory of robust quantum many-body scars in long-range interacting systems." arXiv:2309.12504

²Klimov, A.B. and Romero, J.L., "A generalized Wigner function for quantum systems with the SU(2) dynamical symmetry group." 2008 J. Phys. A: Math. Theor. 41 055303