Quantum Monte Carlo Study of Weakly Coupled Spin Ladders

We report a quantum Monte Carlo study of the thermodynamic properties of arrays of spin ladders with various widths ($n$), coupled via a weak inter-ladder exchange coupling $\alpha J$, where $J$ is the intra-ladder coupling both along and between the chains. This coupled ladder system serves as a simplified model for the magnetism of presumed ordered spin and charge stripes in the two-dimensional CuO$_2$ planes of hole-doped copper oxides. Our results for $n=3$ with weak inter-ladder coupling $\alpha=0.05$, estimated from the $t-t'-t''-J$ model, show good agreement with the ordering temperature of the recently observed spin density wave condensation in La$_2$CuO$_{4+y}$. We show that there exists a quantum critical point at $\alpha_c \simeq 0.07$ for $n=4$, and determine the phase diagram. Our data at this quantum critical point agree quantitatively with the universal scaling predicted by the quantum nonlinear $\sigma$ model. We also report results on random mixtures of $n=2$ and $n=3$ ladders, which correspond to the doping region near but above 1/8. Our study on the magnetic static structure factor reveals a saturation of the incommensurability of the spin correlations around 1/8, while the incommensurability of the charge stripes grows linearly with hole concentration. The implications of this result for the interpretation of neutron scattering experiments on the dynamic spin fluctuations in La$_{2-x}$Sr$_x$CuO$_4$ are discussed.