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

Using series expansions and spin-wave theory we calculate the spin-stiffness anisotropy ${\rho }_{sx}\mathrm{/}{\rho }_{sy}$ in Heisenberg models on the square lattice with anisotropic couplings $J_x,J_y$. We find that for the weakly anisotropic spin-half model ($J_x\approx J_y$), ${\rho }_{sx}\mathrm{/}{\rho }_{sy}$ deviates substantially from the naive estimate ${\rho }_{sx}\mathrm{/}{\rho }_{sy}\approx J_x\mathrm{/}{J}_y$. We argue that this deviation can be responsible for pinning the electronic liquid crystal direction, a novel effect recently discovered in YBCO. For completeness, we also study the spin-stiffness for arbitrary anisotropy $J_x\mathrm{/}{J}_y$ for spin-half and spin-one models. In the limit of $J_y\mathrm{/}{J}_x\to 0$, when the model reduces to weakly coupled chains, the two show dramatically different behavior. In the spin-one model, the stiffness along the chains goes to zero, implying the onset of Haldane-gap phase, whereas for spin-half the stiffness along the chains increases monotonically from a value of $0.18J_x$ for $J_y\mathrm{/}{J}_x=1$ towards $0.25J_x$ for $J_y\mathrm{/}{J}_x\to 0$. Spin-wave theory is extremely accurate for spin-one but breaks down for spin-half presumably due to the onset of topological terms.