What are the necessary or sufficient conditions for a renormalization group scheme to be “valid”?

Question

Suppose I have a super operator $G$ which acts on Hamiltonians to produce a new Hamiltonian that is related somehow. For the purposes of this question, suppose that these Hamiltonians are defined on an $L\times L$ 2D lattice with spins at each point (but the question applies more generally). What are the conditions that $G$ implement a "valid" renormalization group scheme (in real space)? Is there a rigorous definition?

We are typically interested in phases when applying the RG scheme, hence presumably we want to preserve whether the Hamiltonian is gapped or gapless. I suppose this requires preservation of the low energy subspace.

Typically we see that given a family of Hamiltonian with parameters $a_1, a_2, \dots$, then a good RG scheme must map to a Hamiltonian of the same form 
$$G(H(\vec{a})) = H(\vec{a}'). $$

But what can these parameters include? Can they include the local Hilbert space dimension (i.e. can the local Hilbert space dimension diverge), or are they limited to the just coupling parameters? If so, why? 

Any references would be greatly appreciated! All I can find in the current literature are very "hand-wavey"/ intuitive notions of what a renormalization group is.

Comments

Typically, the number of coupling/parameters/observables decreases. The hamiltonians are not analog. Irrelevant parameters ( from irrelevant observables ) are ignored while possible new emerging couplings are included. Actions instead hamiltonians work too.

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@igael  Exactly -- this is my understanding too. However, it's not clear to me if schemes where, for example, the local Hilbert space dimension are considered valid. Suppose I have a scheme where after $k$ iterations the local Hilbert space dimension is $\mathbb{C}^{2^k}$ (e.g. by combining neighbouring spins but not removing any of the Hilbert space). Is there a reason why this wouldn't be considered valid? However, I haven't come across such a scheme in the literature...

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Let's hope that we will get an answer from the renormalization expert... I don't yet understand well why GR is non renormalizable ( it is a semi group, if the current equations are somehow resulting from some hidden renormalization from a smaller scale, there is no clue to reverse the transformation. It is normal and must not be labelized "GR is not renormalizable" ).

Answer 1

In principle, the dimension of the Hilbert space can change, for example this holds for the Kadanoff RG which consists in merging small clusters of sites to a new site, which reduces the number of degrees of freedoms.

Each RG scheme chooses its own set of parameters, limited only by ingenuity. The set of parameters must be large enough such that the coarsening approximation introduces no significant error on the scales of interest, and useful enough such that the transformation can be carried out in some sensible approximation.

Comments

@Arnold Neumaier I was under the impression that Kadanoff block decimation left the *local* Hilbert space dimension the same after each action; you merge the spins into one and then truncate the local Hilbert space so that the new spins only have two states: up and down.

In this sense the local Hilbert space dimension remains constant and does not grow after each iteration. Though I could be incorrect.

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@Qubissential: Yes. Indeed, if you increase the local Hilbert space dimension, you are outside the standard RG framework. For how can one parameterize the result by a parameter vector $a$ of fixed structure? 

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@ArnoldNeumaier If one were to construct a RG flow where the local Hilbert space were to diverge, even if the total Hilbert space after each step were reduced, is there a reason why this is necessarily an invalid RG scheme? Or would it just be considered unusual? 

I hope that question makes sense!

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@Qubissential: To be able answer your lat question you'd have to come up with a toy model that actually does this, to see the implications and whether it succeeds in renormalizing the original problem. This determines whether the name RG is still appropriate.