Studying 2D quantum magnetism using fractal path DMRG

Numerical simulations of quantum magnetism in two spatial dimensions are often constrained by the area law of entanglement entropy, which heavily limits the accessible system sizes in tensor network methods. The Density Matrix Renormalization Group algorithm in two-dimensions requires the identification of a bijective mapping between the 2D lattice and a 1D path to invoke the Matrix Product State ansatz. We investigate how the a priori choice of such mappings affects the accuracy of the computed groundstate quantities. We systematically evaluate all mappings corresponding to a subset of the Hamiltonian paths of the N × N grid graphs up to N=8 and demonstrate that the fractal space-filling curves generally lead to improved convergence in ground state searches on the square lattice Heisenberg antiferromagnet. To explain this performance gain, we analyze various locality metrics and propose a scalable method for constructing high-performing paths on larger lattices by tiling smaller optimal paths. Our results show that such paths consistently improve simulation convergence, with the advantage increasing with system size.