We report a quantum Monte Carlo study of the thermodynamic properties of arrays of spin ladders with various widths $\mathrm{}\left(n\right)\mathrm{}$, coupled via a weak interladder exchange coupling $\alpha J$, where J is the intraladder 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 ${\mathrm{CuO}}_2$ planes of hole-doped copper oxides. Our results for $n=3\mathrm{}$ with weak interladder coupling $\alpha =0.05\mathrm{}$, estimated from the $t-t^{\prime }-t^{″}-J$ model, show good agreement with the ordering temperature of the recently observed spin-density-wave condensation in ${\mathrm{La}}_2{\mathrm{CuO}}_{4+y}.$ We show that there exists a quantum critical point at ${\alpha }_c\simeq 0.07$ for $n=4\mathrm{}$, 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 $n=2\mathrm{}$ and $n=3\mathrm{}$ ladders, which correspond to the doping region near but above 1/8. Our study of 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 ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ are discussed.

• Received 18 February 1999

DOI:https://doi.org/10.1103/PhysRevB.60.3294