  We explain the recent numerical successes obtained by Tao Xiang's group, who
developed and applied Tensor Renormalization Group methods for the Ising model
on square and cubic lattices, by the fact that their new truncation method
sharply singles out a surprisingly small subspace of dimension two. We show
that in the two-state approximation, their transformation can be handled
analytically yielding a value 0.964 for the critical exponent nu much closer to
the exact value 1 than 1.338 obtained in the Migdal-Kadanoff approximation. We
propose two alternative blocking procedures that preserve the isotropy and
improve the accuracy to nu=0.987 and 0.993 respectively. We discuss
applications to other classical lattice models, including models with fermions,
and suggest that it could become a competitor for Monte Carlo methods suitable
to calculate accurately critical exponents, take continuum limits and study
near-conformal systems in arbitrarily large volumes.
