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

TL;DR

This paper investigates the anisotropic spin-stiffness in Heisenberg models on a square lattice, revealing deviations from naive estimates and proposing a mechanism for electronic liquid crystal pinning in YBCO.

Contribution

It provides a detailed analysis of spin-stiffness anisotropy in anisotropic Heisenberg models, linking the findings to the pinning of electronic liquid crystal directions in YBCO.

Findings

Deviations from naive estimates of spin-stiffness ratios in weakly anisotropic models.

Different behaviors of spin-stiffness in spin-half and spin-one models as anisotropy varies.

Spin-wave theory's accuracy varies between spin-half and spin-one models.

Abstract

Using series expansions and spin-wave theory we calculate the spin-stiffness anisotropy $\rho_{sx}/\rho_{sy}$ in Heisenberg models on the square lattice with anisotropic couplings $J_x,J_y$. We find that for the weakly anisotropic spin-half model ($J_x\approx J_y$), $\rho_{sx}/\rho_{sy}$ deviates substantially from the naive estimate $\rho_{sx}/\rho_{sy} \approx J_x/J_y$. 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 $J_x/J_y$ 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 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 $0.18 J_x$ for $J_y/J_x=1$ towards $0.25 J_x$ for $J_y/J_x\to 0$. Spin-wave theory is extremely accurate for spin-one but breaks down for spin-half presumably due to the onset of topological terms.