Unusual Energy Spectra of Matrix Product States (Precision Many Body Physics Focus session)

Many methods for obtaining ground states are based on imaginary time evolution, where high energy components of an initial wavefunction are projected out. The approximate ground state one obtains after incomplete projection is a combination of low-lying states, and its spectrum--its decomposition into exact energy eigenstates-- falls off exponentially with the energy. The ground states from DMRG, for matrix product states in general, and probably all similar "compressed'" wavefunctions, are different. They tend to have a large amplitude in the exact ground state and many small amplitude components at quite high energies, with a nearly flat spectrum out to nearly the maximum possible energies. The flatness and broad energy range persist even as the approximate ground state is made more accurate. We show evidence for these generic unusual spectra from small system calculations for matrix product states and neural network states. I will present a rough counting argument for why this occurs. We discovered the unusual spectra in developing sampling methods for calculating the variance of the wavefunction, which can be used for extrapolation to infinite bond dimension in cases where traditional truncation error extrapolation is not available, such as single-site DMRG. However, we found large fluctuations in the local energy, due to a fat tailed distribution. The fat tails make traditional variance estimation very inefficient. However, for the purposes of extrapolation, we show that one can use a highly biased estimator for the variance with excellent results. The key is that the bias vanishes smoothly as the state is made more accurate, and can be eliminate by the extrapolation.