Error estimates for extrapolations with matrix-product states

Abstract

We introduce an error measure for matrix-product states without requiring the relatively costly two-site density-matrix renormalization group (2DMRG). This error measure is based on an approximation of the full variance $\ensuremath{\langle}\ensuremath{\psi}|{(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}E)}^{2}|\ensuremath{\psi}\ensuremath{\rangle}$. When applied to a series of matrix-product states at different bond dimensions obtained from a single-site density-matrix renormalization group (1DMRG) calculation, it allows for the extrapolation of observables towards the zero-error case representing the exact ground state of the system. The calculation of the error measure is split into a sequential part of cost equivalent to two calculations of $\ensuremath{\langle}\ensuremath{\psi}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}|\ensuremath{\psi}\ensuremath{\rangle}$ and a trivially parallelized part scaling like a single operator application in 2DMRG. The reliability of this error measure is demonstrated by four examples: the $L=30,S=1/2$ Heisenberg chain, the $L=50$ Hubbard chain, an electronic model with long-range Coulomb-like interactions, and the Hubbard model on a cylinder with a size of $10\ifmmode\times\else\texttimes\fi{}4$. Extrapolation in this error measure is shown to be on par with extrapolation in the 2DMRG truncation error or the full variance $\ensuremath{\langle}\ensuremath{\psi}|{(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}E)}^{2}|\ensuremath{\psi}\ensuremath{\rangle}$ at a fraction of the computational effort.