Irreducible Representation of Long-Range Interacting Many-Body Systems

Long-range interacting many-body systems dictate a broad class of natural dynamics bridging integrability and chaos, of interest to quantum simulation and metrology. When spin ensembles of this type exhibit permutation symmetry by infinite-range interactions, they collapse to a single irreducible representation ("irrep") of SU(2) on the collective spin (the symmetric subspace). Weakly permuting asymmetric interactions can preserve the collective character, avoiding thermalization when initialized in the largest SU(2)-irrep thanks to quantum many-body scars (QMBS) overlapping with the symmetric subspace. We truncate a transverse Ising Hamiltonian with long-range interactions and QMBS, keeping first-order accessible subspaces from this symmetric irrep by optimizing over the degrees-of-freedom of irrep degeneracies, and analytically decomposing the truncated Hamiltonian into generalized spherical tensors. This truncated Hamiltonian includes the minimal matrix elements necessary for a first-order perturbation theory of the QMBS from the symmetric subspace, and avails an interpretation as the Hamiltonian of two bodies: a "major" and "minor" spin. We study the dynamics of the truncated Hamiltonian at system sizes too large for exact simulation. We check the consistency of the approximation using Loschmidt echoes and confirm the existences of quantum phase transitions (QPTs) into the long-range regime.