Exact renormalization group for quantum spin systems and strongly correlated electrons

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In this renewal proposal we will extend the applicability of the functional renormalization group (FRG) method to general quantum systems with projected Hilbert spaces. An important example is the quantum spin systems which we have studied during the first funding period of this proposal; in this case the charge degrees of freedom of an underlying electronic lattice model are projected out, so that the effective spin Hamiltonian acts only on the spin sector of the electronic Hilbert space. Another example is the so-called $t-J$-model which retains not only the spin sector of the Hilbert space, but also the part of the charge sector involving states without doubly occupied lattice sites. Up until now a direct application of the FRG to the $t-J$-model is not possible because the projected Hilbert space does not allow a representation of its correlation functions in terms of unconstrained functional integrals over bosonic or fermionic coherent states. However, the crucial insight in our initial work on the application of the FRG to quantum spin systems is that the FRG can also be used to study quantum systems which do not have a functional integral representation. Apparently, a large part of the FRG community is not aware of this fact. The purpose of this proposal is to further develop this strategy and apply it to fermionic lattice models with projected Hilbert spaces, thus opening a new way to study strongly correlated fermionic lattice models non-perturbatively.