Controlled bond expansion for DMRG ground state search at single-site costs

We present a controlled bond expansion (CBE) approach for ground state searches based on the density matrix renormalization group (DMRG) employing matrix product states (MPS). The main idea of CBE-DMRG is as follows: Before optimizing the center tensor of an MPS in the site canonical form, the isometry on the next site is expanded by the most important contributions of its orthogonal complement. These contributions are determined by considering how the Hamiltonian acts on the current state. The accuracy of CBE-DMRG is comparable to two-site DMRG, but its numerical costs scale like those of one-site DMRG. Moreover, in contrast to previous one-site methods involving bond expansion, it does not rely on any ad-hoc mixing parameters. Starting from an initial state with low bond-dimension, the CBE prescription for increasing the bond dimension is controlled, automated and more economical than the manual adjustment typically used for two-site DMRG, leading to an additional significant speedup for reaching convergence. We illustrate the numerical performance of CBE-DMRG with several examples of fermionic models, including free fermions on a chain for benchmarking and a Kondo lattice on a cylinder as a challenging application.