We continue our study of rigorous renormalization group (RG) maps for tensor
    networks that was begun in arXiv:2107.11464. In this paper we construct a
    rigorous RG map for 2D tensor networks whose domain includes tensors that
    represent the 2D Ising model at low temperatures with a magnetic field $h$. We
    prove that the RG map has two stable fixed points, corresponding to the two
    ground states, and one unstable fixed point which is an example of a
    discontinuity fixed point. For the Ising model at low temperatures the RG map
    flows to one of the stable fixed points if $h \neq 0$, and to the discontinuity
    fixed point if $h=0$. In addition to the nearest neighbor and magnetic field
    terms in the Hamiltonian, we can include small terms that need not be spin-flip
    invariant. In this case we prove there is a critical value $h_c$ of the field
    (which depends on these additional small interactions and the temperature) such
    that the RG map flows to the discontinuity fixed point if $h=h_c$ and to one of
    the stable fixed points otherwise. We use our RG map to give a new proof of
    previous results on the first-order transition, namely, that the free energy is
    analytic for $h \neq h_c$, and the magnetization is discontinuous at $h = h_c$.
    The construction of our low temperature RG map, in particular the disentangler,
    is surprisingly very similar to the construction of the map in arXiv:2107.11464
    for the high temperature phase. We also give a pedagogical discussion of some
    general rigorous transformations for infinite dimensional tensor networks and
    an overview of the proof of stability of the high temperature fixed point for
    the RG map in arXiv:2107.11464.



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