Symmetry breaking and criticality in tensor-product states

We discuss variationally optimized matrix-product states for the transverse-field Ising chain using D x D matrices with small D is an element of {2-10}. For finite system size N there are energy minimums for symmetric as well as symmetry-broken states, which cross each other at a field value h(c)(N,D); thus the transition is first order. A continuous transition develops as N ->infinity. The asymptotic critical behavior is then always of mean-field type (the magnetization exponent beta=1/2) but a window of field strengths where true Ising scaling holds (beta=1/8) emerges with increasing D. We also demonstrate asymptotic mean-field behavior for infinite-size two-dimensional tensor-product (iPEPS) states with small tensors. The behaviors should be generic at symmetry-breaking transitions.