  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 $\chi$ 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 $S \sim {1/6}\log
\chi$ with high precision. This result can be understood as the emergence of an
effective finite correlation length $\xi_\chi$ ruling of all the scaling
properties in the system. We produce five extra pieces of evidence for this
finite-$\chi$ scaling, namely, the scaling of the correlation length, the
scaling of magnetization, the shift of the critical point, and the scaling of
the entanglement entropy for a finite block of spins. All our computations are
consistent with a scaling relation of the form $\xi_\chi\sim \chi^{\kappa}$,
with $\kappa=2$ for the Ising model. In the case of the Heisenberg model, we
find similar results with the value $\kappa\sim 1.37$. We also show how
finite-$\chi$ scaling allow 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.
