Exact renormalization group for quantum spin systems

Abstract

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 (2) spin algebra is implemented via a non-trivial initial condition for the renormalization group flow. Our method is rather general and offers a new non-perturbative approach to quantum spin systems.where the subscripts i, j label the N sites r i of a Ddimensional lattice, h 0 is an external magnetic field in arXiv:1807.02524v2 [cond-mat.str-el]

A rich man's derivation of scaling laws for the Kondo model

“…As already emphasized in Ref. [10], the spin degrees of freedom appear in Eq. (31) in the same way as a bosonic field, which is a consequence of the fact that time-ordered spin correlation functions satisfy bosonic Kubo-Martin-Schwinger boundary conditions.…”

“…In this work we have shown how to derive the oneloop scaling equations for the anisotropic Kondo model within the functional renormalization group approach for quantum spin systems recently proposed in Ref. [10]. Of course, the scaling laws for the Kondo model are well known and have been derived a long time ago using less sophisticated methods.…”

“…Following Ref. [10], we can derive an exact flow equation for the generating functional G Λ [η, η, h] by simply taking the derivative of Eq. (11) with respect to the cutoff parameter Λ.…”

“…We use a specific implementation of the FRG for quantum spin systems recently developed in Ref. [10], which does not require that the relevant degrees of freedom can be represented in terms of canonical fermions or bosons. Apart from its formal elegance, this spin FRG approach has the advantage that it can also be used to study the strong coupling regime of the Kondo model where Anderson's "poor man's scaling" approach is not valid.…”

Heat capacity of anisotropic Heisenberg antiferromagnet within the spin Hartree-Fock approach in quasi-1D regime

“…The accuracy in this region can be improved by taking into account systematically the correlation function of order higher than two, e.g. using the recently proposed spin-FRG approach [31].…”