  We revisit the question of describing critical spin systems and field
theories using matrix product states, and formulate a scaling hypothesis in
terms of operators, eigenvalues of the transfer matrix, and lattice spacing in
the case of field theories. Critical exponents and central charge are
determined by optimizing the exponents such as to obtain a data collapse. We
benchmark this method by studying critical Ising and Potts models, where we
also obtain a scaling ansatz for the correlation length and entanglement
entropy. The formulation of those scaling functions turns out to be crucial for
studying critical quantum field theories on the lattice. For the case of
$\lambda\phi^4$ with mass $\mu^2$ and lattice spacing $a$, we demonstrate a
double data collapse for the correlation length $ \delta
\xi(\mu,\lambda,D)=\tilde{\xi}
\left((\alpha-\alpha_c)(\delta/a)^{-1/\nu}\right)$ with $D$ the bond dimension,
$\delta$ the gap between eigenvalues of the transfer matrix, and
$\alpha_c=\mu_R^2/\lambda$ the parameter which fixes the critical quantum field
theory.
