I'm currently looking at a way of renormalizing a particular Hamiltonian. One of the questions I'm currently trying to answer is whether, in a renormaliztion group (RG) flow, it is valid to allow the local Hilbert space dimension to become arbitrarily large during the RG process. I'll try illustrate this with an example.

Consider a 1D chain of spins each with local Hilbert space $\mathbb{C}^d$. We choose the Hamiltonian such that the ground state is consists of entangled blocks of spins. The first block is length 2, the second length 3, and the $n^{th}$ length $n+1$, until the entire chain is filled with such blocks.

Thus the ground state of the first two spins might be

$$ \frac{1}{\sqrt{2}}(|00> + |11>) $$ and likewise for the next block of three spins the ground state might be $$\frac{1}{\sqrt{3}}(|000>+|111>) $$ etc. where the ground state of the block of size $L$ generalised in the obvious way.

Now consider a renormalization group (RG) scheme which involves grouping the spins together. We could theoretically just combine three spins together into a new spin with local Hilbert space dimension $\mathbb{C}^{d^3}$ and keep all the states. If we do this $n$ times, then we would get a new state with local dimension $\mathbb{C}^{d^{3^n}}$. Then, when one of these renormalized spins contains a fully entangled block, then we remove all the states that are not the ground state (which is known) hence reducing the Hilbert space of one of the modified spins.

This RG scheme would presumably preserve the ground state energy. However, as the length of the chain increases, the entangled blocks grow arbitrarily large, hence we would need an arbitrarily large Hilbert space dimension to implement the RG procedure. Does this property stop this from being a ''good'' RG scheme in some sense?


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