Crossovers due to anisotropy and disorder in quantum critical fluctuations

Scaling relations are used to study cross-overs, due to anisotropic spin interactions or single ion anisotropy, and due to disorder, in the thermodynamics and correlation functions near quantum-critical transitions. The principal results are simple with a wide range of applications. The region of attraction to the stable anisotropic fixed point in the quantum-critical region is exponentially enhanced by the dynamical critical exponent $z$ compared to the region of attraction to the fixed point in the quantum disordered region. The result implies that, even for small anisotropy, the region of attraction to the stable incommensurate Ising or planar metallic anti-ferromagnetic critical points, which belong to the universality class of the XY model with $z \to \infty$, covers the entire quantum-critical region. In crossovers due to disorder, the instability of the pure fixed point in the quantum disordered region is exponentially enhanced by $z$ compared to that in the quantum-critical region. This result suggests that for some classes of disorder and for large enough $z$, one may find singularities in the correlations as a function of frequency and temperature down to very low temperatures even though the correlation length in space remains short range.