Locality-Induced Hierarchical Backflow Wavefunctions for Correlated Fermions

We show that locality provides a natural principle to hierarchically organize backflow wavefunctions. This leads us to propose a family of variational fermionic states, termed hierarchical backflow (HB) wavefunctions. The expressive power of HB is systematically improvable, controlled by a path depth K which reflects the range of backflow correlations. At half-filling, the HB with K=1 already achieves high energy precision, with an accuracy around 0.5\% for system sizes from 4times 4 to 10times 10. At hole doping n_h=0.125, the method scales efficiently to 12times16 and 16times16 systems, and the energy systematically achieves higher accuracy with K increasing, yielding a clear stripe phase. The HB further enables a local-nonlocal decomposition, naturally bridging to neural quantum states, while featuring compact representations and efficient optimization. Our work reveals locality as a natural organizing principle of backflow wavefunctions, opening a new framework with systematic improvability and interpretability for large-scale simulations of correlated fermion systems.