  We discuss variationally optimized matrix-product states for the
transverse-field Ising chain, using D*D matrices with small D=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 hc(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.
