Exact renormalization group for quantum spin systems

We show that the diagrammatic approach to quantum spin systems developed in a seminal work by Vaks, Larkin, and Pikin [Sov. Phys. JETP 26, 188 (1968)] can be embedded in the framework of the functional renormalization group. The crucial insight is that the generating functional of the time-ordered connected spin correlation functions of an arbitrary quantum spin system satisfies an exact renormalization group flow equation which resembles the corresponding flow equation of interacting bosons. The $SU\left(2\right)$ spin algebra is implemented via a nontrivial initial condition for the renormalization group flow. Our method is rather general and offers a different nonperturbative approach to quantum spin systems.

Figure 1

Graphical representation of the exact flow equation (5) for the connected time-ordered $n$-spin correlation functions, which are represented by circles with $n$ external legs. Here, the red, slashed lines denote the derivative ${\partial }_{\mathrm{\Lambda }}{J}_{ij}^{\mathrm{\Lambda }}$ of the deformed exchange coupling.