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arXiv:cond-mat/0004228

Renormalization Group, hidden symmetries and approximate Ward identities in the XYZ model, II

G. Benfatto, Vieri Mastropietro

Published Apr 13, 2000• 0 citations
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Abstract

An expansion based on renormalization group methods for the spin correlation function in the z direction of the Heisenberg-Ising XYZ chain with an external magnetic field directed as the z axis is derived. Moreover, by using the hidden symmetries of the model, we show that the running coupling constants are small, if the coupling in the z direction is small enough, that a critical index appearing in the correlation function is exactly vanishing (because of an approximate Ward identity) and other properties, so obtaining a rather detailed description of the XYZ correlation function.

DOI
10.48550/arxiv.cond-mat/0004228

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A simple modification of the standard Renormalization Group (RG) technique for the study of quantum spin systems is introduced. Our method which takes into account the effect of boundary conditions by employing the concept of superblock, may be regarded as a simple way for obtaining first estimates of many properties of spin systems. By applying this method to the xxz spin 1/2 Heisenberg chain, we obtain the ground state energy with much higher accuracy than the standard RG. We have also obtained the staggered magnetization and the z-component of spin-spin correlation function which confirms the absence of long-range order in the massless region of the 1D xxz model.
We use the spin functional renormalization group recently developed by two of us [Krieg and Kopietz, Phys. Rev. B 99, 060403(R) (2019)] to calculate the magnetization M(H,T)M(H,T)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 $H\ensuremath{\rightarrow}0$, 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=0H=0H=0 at finite temperature TTT. 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 byproduct, 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.
Correlation functions of dimer operators, the product operators of spins on two adjacent sites, are studied in the spin-12\frac{1}{2}21​ XXZ chain in the critical regime. The amplitudes of the leading oscillating terms in the dimer correlation functions are determined with high accuracy as functions of the exchange anisotropy parameter and the external magnetic field through the combined use of bosonization and density-matrix renormalization-group methods. In particular, for the antiferromagnetic Heisenberg model with SU(2) symmetry, logarithmic corrections to the dimer correlations due to the marginally irrelevant operator are studied, and the asymptotic form of the dimer correlation function is obtained. The asymptotic form of the spin-Peierls excitation gap including logarithmic corrections is also derived.