Scaling of entanglement support for matrix product states

Abstract

The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground-state properties of a system is limited by the size $\ensuremath{\chi}$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S\ensuremath{\sim}\frac{1}{6}\text{log}\text{ }\ensuremath{\chi}$ with high precision. This result can be understood as the emergence of an effective finite correlation length ${\ensuremath{\xi}}_{\ensuremath{\chi}}$ ruling all the scaling properties in the system. We produce six extra pieces of evidence for this finite-$\ensuremath{\chi}$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, the scaling of the entanglement entropy for a finite block of spins, the existence of scaling functions, and the agreement with analogous classical results. All our computations are consistent with a scaling relation of the form ${\ensuremath{\xi}}_{\ensuremath{\chi}}\ensuremath{\sim}{\ensuremath{\chi}}^{\ensuremath{\kappa}}$, with $\ensuremath{\kappa}=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $\ensuremath{\kappa}\ensuremath{\sim}1.37$. We also show how finite-$\ensuremath{\chi}$ scaling allows us to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density-matrix renormalization-group results and their classical analog obtained with the corner transfer-matrix renormalization group.