Quantum phase transitions and finite entanglement scaling

The density matrix renormalization group provides a powerful tool in the numerical study of 1D systems by finding matrix product state (MPS) approximations to the ground states. Its efficiency is underpinned by the low area-law entanglement of the ground states of gapped 1D systems. The area law of entanglement is famously violated at quantum critical points (QCP) where the gap closes and the behavior of the entanglement entropy is logarithmic. Because of this MPSs fail to capture important features of critical states such as the diverging correlation length. Yet it was conjectured that the effect of finite bond dimension of the MPS is characterized by a single length scale that is determined by the conformal field theory description of the QCP, this conjecture is known as the finite entanglement scaling. In our work, we provide a more detailed picture of the structure of the MPS approximations to the states at and near critical points beyond the scaling of the correlation length, in particular showing how the phase transitions become discontinuous. The analysis is based on analytical calculations for integrable spin chains and confirmed numerically.