# Renormalization Group technique in many-body hamiltonians when a hopping is present

I have recently started reading about the NRG and DMRG and their early applications to the Kondo and Heisenberg models. I am just beginning, but, as far as I have been able to understand, something key in these problems is that, when adding new levels, the Hilbert space can be seen to have the structure of a tensor product, and the hamiltonian itself of a system of size N can be written recurrently in terms of the hamiltonians of smaller units. Consider, however, something like the Heisenberg model, but now with a one-electron hopping term between neighboring levels, something like $$\hat{H}_N = \sum_{j=1}^N \hat{\mathbf{S}}_j \cdot \hat{\mathbf{S}}_{j+1}+ t\sum_{j=1}^{N} \hat{c}^\dagger_j \hat{c}_{j+1}+ \text{h.c.}$$ where the $$\hat{\mathbf{S}}_j$$ are the spin operators of site $$j$$, $$\hat{\mathbf{S}}_j=\sum_{\sigma \sigma^\prime} \hat{c}^\dagger_{j \sigma} \mathbf{P}_{\sigma \sigma^\prime} \hat{c}_{j \sigma^\prime},$$ with $$\mathbf{P}$$ the Pauli matrices, etc. When considering such hamiltonians, the presence of the hopping term, which allows electrons to fluctuate between levels, completely prevents me from regarding the Hilbert space of possible many-body states as a tensorial product, and the hamiltonian $$\hat{H}_N$$ is no longer (at least not easily) writeable in a recursive fashion in terms of smaller hamiltonians, which seemed to be key in the Renormalization group.

How is then this type of problems treated?

My initial motivation was to think if the RG could somehow be applied to Kanamori-like hamiltonians, but I don't seem to be able to find any recurrency there for the hamiltonian (I googled something about it and I found some papers on the topic, but nothing self-contained/introductory).

• I'm not sure what you're after exactly, but your Hamiltonian $\hat{H}_N$ has the form of a $t-J$-model. There have definitively been published RG and DMRG studies of that model. Maybe knowing that keyword will help you find something useful. – Anyon Mar 30 at 0:32