  In this paper we apply the formalism of translation invariant (continuous)
matrix product states in the thermodynamic limit to $(1+1)$ dimensional
critical models. Finite bond dimension bounds the entanglement entropy and
introduces an effective finite correlation length, so that the state is
perturbed away from criticality. The assumption that the scaling hypothesis
holds for this kind of perturbation is known in the literature as finite
entanglement scaling. We provide further evidence for the validity of finite
entanglement scaling and based on this formulate a scaling algorithm to
estimate the central charge and critical exponents of the conformally invariant
field theories describing the critical models under investigation. The
algorithm is applied to three exemplary models; the cMPS version to the
non-relativistic Lieb-Liniger model and the relativistic massless boson, and
MPS version to the one-dimensional quantum Ising model at the critical point.
Another new aspect to our approach is that we directly use the (c)MPS induced
correlation length rather than the bond dimension as scaling parameter. This
choice is motivated by several theoretical arguments as well as by the
remarkable accuracy of our results.

Quantum Physics;Statistical Mechanics;Strongly Correlated Electrons;High Energy Physics - Theory