Time-dependent variation principle with controlled bond expansion for matrix product states

We present a controlled bond expansion (CBE) approach to simulate quantum dynamics based on the time-dependent variation principle (TDVP) for matrix product states (MPS). While the original single-site TDVP integrator assumes a fixed bond dimension, CBE-TDVP starts with a small one and increases it on the fly as the entanglement entropy grows with time. To this end, CBE systematically introduces new subspaces, based on the isometry orthogonal to the TDVP tangent space which carries most weight in the projection error. CBE increases the bond dimension in an economical manner, and the truncation error sets in only once the bond dimension has reached a specified maximal value. Even beyond that time, the numerical accuracy remains well-controlled, being governed by the (growing) truncation error. CBE-TDVP is able to reach time scales comparable to any standard two-site algorithm, but without resorting to two-site update. Moreover, being based on TDVP, it can be used for long-ranged Hamiltonians. We illustrate its performance with several examples, including the one-axis twisting model and the Haldane-Shastry model for benchmark purposes, and polaron dynamics in the Peierls-Hubbard model.