We use the spin functional renormalization group recently developed by two of us [J. Krieg and P. Kopietz, Phys. Rev. B 99, 060403(R) (2019)] to calculate the magnetization M (H, T ) and the damping of magnons due to classical longitudinal fluctuations of quantum Heisenberg ferromagnets. In order to guarantee that for vanishing magnetic field H → 0 the magnon spectrum is gapless when the spin rotational invariance is spontaneously broken, we use a Ward identity to express the magnon self-energy in terms of the magnetization. In two dimensions our approach correctly predicts the absence of long-range magnetic order for H = 0 at finite temperature T . The magnon spectrum then exhibits a gap from which we obtain the transverse correlation length. We also calculate the wave-function renormalization factor of the magnons. As a mathematical by-product, we derive a recursive form of the generalized Wick theorem for spin operators in frequency space which facilitates the calculation of arbitrary time-ordered connected correlation functions of an isolated spin in a magnetic field.
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We show how the one-loop "poor man's scaling" equations for the Kondo model with arbitrary impurity spin can be obtained within the framework of the functional renormalization group approach for quantum spin systems recently developed by Krieg and Kopietz [arXiv:1807.02524]. We argue that our method supersedes the "poor man's scaling" approach and can also be used to study the strong coupling regime of the Kondo model.
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