Nonlinear canonical transformations for studying spin-1/2 Hamiltonians

Strongly correlated quantum systems, including frustrated spin lattices, remain challenging for conventional coupled-cluster methods because their ground states often exhibit strong multireference character and symmetry breaking. Although multireference and symmetry-projected coupled-cluster theories offer promising avenues, they are not yet black-box methods and often require substantial system-specific development. Spin Hamiltonians such as the XXZ and J1-J2 models provide compact but demanding benchmarks for these challenges. In particular, spin-BCS-based coupled-cluster methods may require high excitation ranks to recover the exact ground-state physics in strongly correlated regimes. In this work, we introduce a nonlinear spin-fermion-spin (SFS) transformation designed to map a strongly correlated spin Hamiltonian to a new spin Hamiltonian whose ground state is more accurately described at the mean-field level. For a representative SFS transformation, we identify a compact Clifford circuit that maps transformed product states to circuit-dressed, generally entangled states in the original representation. The transformed Hamiltonian can then be treated using broken-symmetry mean-field theory and post-mean-field methods. Benchmarks for the XXZ and J1-J2 Hamiltonians show that spin mean-field theory in the transformed representation reduces the energy error relative to conventional spin-BCS mean field. The spin-BCS-based coupled-cluster calculations for the one-dimensional XXZ model provide substantially improved accuracy relative to the corresponding untransformed methods. In 2D, the accuracy of the transformed mean-field solutions depends on the site ordering and boundary conditions because nonlocal strings appear. These results show that exact transformations of operator representations can serve as many-body preconditioners for mean-field and low-rank coupled-cluster approximations to strongly correlated spin systems.