Exact renormalization group for quantum spin systems and strongly correlated electrons

在这个更新方案中，我们将泛函重整化群(FRG)方法的适用性扩展到具有投影希尔伯特空间的一般量子系统。一个重要的例子是我们在这个提议的第一个资助期内研究的量子自旋系统；在这种情况下，基本电子晶格模型的电荷自由度被投影出来，因此有效的自旋哈密顿量只作用于电子希尔伯特空间的自旋部分。另一个例子是所谓的$t-J$-模型，它不仅保留了希尔伯特空间的自旋扇区，而且保留了没有双占据晶格格位的态的电荷扇区的一部分。到目前为止，将FRG直接应用于$t-J$-模型是不可能的，因为投影的Hilbert空间不允许用玻色态或费米子相干态上的无约束泛函积分来表示其关联函数。然而，在我们将FRG应用于量子自旋系统的初步工作中，关键的见解是FRG也可以用于研究没有泛函积分表示的量子系统。显然，FRG社区的很大一部分人没有意识到这一事实。本方案的目的是进一步发展这一策略，并将其应用于具有投影希尔伯特空间的费米子晶格模型，从而为非微扰研究强关联费米子晶格模型开辟了一条新的途径。

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.