Quantum Monte Carlo study of weakly coupled spin ladders

Abstract

We report a quantum Monte Carlo study of the thermodynamic properties of arrays of spin ladders with various widths $(n)$, coupled via a weak interladder exchange coupling $\ensuremath{\alpha}J$, 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 $n=3$ with weak interladder coupling $\ensuremath{\alpha}=0.05$, estimated from the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}J$ 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 ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}0.07$ 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 $\ensuremath{\sigma}$ 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 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 ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ are discussed.