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Currently, I am self-learning Renormalization and its application to PDEs, nonequilibrium statistical mechanics and also condensed matter. One particularly problem I face is on the conservation of symmetry of hamiltonian during renormalization.

Normally renormalization of hamiltonian, say Ising model in 2D square lattice, must preserve its form, i.e.

H = S_i S_j + h S_i

a) H' = S'_i S'_j + h' S'_i (S' is block spin)

b) H' = S'_i S'_j + h' S'_i + α S'_i S'_j S'_k

Only a) is permissible while b) is not allowed for non-zero α.

1. I want to understand why, for this transformation of S to S', the symmetry is preserved and which symmetry is preserved under this transformation.

2. Why the hamiltonian in this form is guaranteed to work under transformation? i.e., why such transformation guarantees no extra term coming into the hamiltonian? Is there any mathematical theorem behind it?

3. Is there any other example?

4. In reality, the renormalization operation is discrete rather than continuous which discretizes the RG flow into steps of mapping in parameter space. So, we cannot evaluate the transformation of parameters "smoothly". However, this may only be true in Ising model. Is there any possible system which permits a continuous rescaling for continuous renormalization?

Thank you very much.

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# A Symmetry of hamiltonian under renormalization

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