Excitation spectrum gap and spin-wave velocity ofXXZHeisenberg chains: Global renormalization-group calculation

Abstract

The anisotropic $XXZ$ spin-$\frac{1}{2}$ Heisenberg chain is studied using renormalization-group theory. The specific heats and nearest-neighbor spin-spin correlations are calculated throughout the entire temperature and anisotropy ranges in both ferromagnetic and antiferromagnetic regions, obtaining a global description and quantitative results. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Ising-like ferromagnetic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of characteristics of purely quantum nature are found: The in-plane interaction ${s}_{i}^{x}{s}_{j}^{x}+{s}_{i}^{y}{s}_{j}^{y}$ induces an antiferromagnetic correlation in the out-of-plane ${s}_{i}^{z}$ component, at higher temperatures in the antiferromagnetic $XXZ$ chain, dominantly at low temperatures in the ferromagnetic $XXZ$ chain, and, in-between, at all temperatures in the $XY$ chain. We find that the converse effect also occurs in the antiferromagnetic $XXZ$ chain: an antiferromagnetic ${s}_{i}^{z}{s}_{j}^{z}$ interaction induces a correlation in the ${s}_{i}^{xy}$ component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of either type (Ising or $XY$) of anisotropy; and (ii) in complete contrast, in the ferromagnet, the value and temperature of the specific heat peak increase with either type of anisotropy.