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Exact blocking formulas for spin and gauge models

Using the example of the two-dimensional (2D) Ising model, we show that in contrast to what can be done in configuration space, the tensor renormalization group (TRG) formulation allows one to write exact, compact, and manifestly local blocking formulas and exact coarse grained expressions for the partition function. We argue that similar results should hold for most models studied by lattice gauge theorists. We provide exact blocking formulas for several 2D spin models (the O(2) and O(3) sigma models and the SU(2) principal chiral model) and for the 3D gauge theories with groups Z_2, U(1) and SU(2). We briefly discuss generalizations to other groups, higher dimensions and practical implementations.