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Using series expansions and spin-wave theory we calculate the spin-stiffnessanisotropy $\rho_{sx}/\rho_{sy}$ in Heisenberg models on the square latticewith anisotropic couplings $J_x,J_y$. We find that for the weakly anisotropicspin-half model ($J_x\approx J_y$), $\rho_{sx}/\rho_{sy}$ deviatessubstantially from the naive estimate $\rho_{sx}/\rho_{sy} \approx J_x/J_y$. Weargue that this deviation can be responsible for pinning the electronic liquidcrystal direction, a novel effect recently discovered in YBCO. Forcompleteness, we also study the spin-stiffness for arbitrary anisotropy$J_x/J_y$ for spin-half and spin-one models. In the limit of $J_y/J_x\to 0$,when the model reduces to weakly coupled chains, the two show dramaticallydifferent behavior. In the spin-one model, the stiffness along the chains goesto zero, implying the onset of Haldane-gap phase, whereas for spin-half thestiffness along the chains increases monotonically from a value of $0.18 J_x$for $J_y/J_x=1$ towards $0.25 J_x$ for $J_y/J_x\to 0$. Spin-wave theory isextremely accurate for spin-one but breaks down for spin-half presumably due tothe onset of topological terms.

physical review. b, condensed matter and materials physics

Spin-stiffness of anisotropic Heisenberg model on square lattice and  possible mechanism for pinning of the electronic liquid crystal direction in  YBCO