Scaling Hypothesis for Matrix Product States

We study 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. The critical point, exponents, and central charge are determined by optimizing them 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 parameter mu(2) and lattice spacing a, we demonstrate a double data collapse for the correlation length delta xi(mu, lambda, D) = (xi) over tilde((alpha - alpha(c))(delta/a)(-1/nu)) with D the bond dimension, delta the gap between eigenvalues of the transfer matrix, and alpha(c) = mu(2)(R)/lambda the parameter which fixes the critical quantum field theory.