  A projected entangled pair state (PEPS) with ancillas can be evolved in
imaginary time to obtain thermal states of a strongly correlated quantum system
on a 2D lattice. Every application of a Suzuki-Trotter gate multiplies the PEPS
bond dimension $D$ by a factor $k$. It has to be renormalized back to the
original $D$. In order to preserve the accuracy of the Suzuki-Trotter (S-T)
decomposition, the renormalization has in principle to take into account full
environment made of the new tensors with the bond dimension $k\times D$. Here
we propose a self-consistent renormalization procedure operating with the
original bond dimension $D$, but without compromising the accuracy of the S-T
decomposition. The iterative procedure renormalizes the bond using full
environment made of renormalized tensors with the bond dimension $D$. After
every renormalization, the new renormalized tensors are used to update the
environment, and then the renormalization is repeated again and again until
convergence. As a benchmark application, we obtain thermal states of the
transverse field quantum Ising model on a square lattice - both infinite and
finite - evolving the system across a second-order phase transition at finite
temperature.
