Magnetic Transition in Antiferromagnetic Spin-$\frac{1}{2}$ Chains with Staggered Long-Range Interactions

Antiferromagnetic spin-$\frac{1}{2}$ chains with non-frustrated long-range couplings are studied using the powerful Quantum Monte Carlo algorithm based on a Stochastic Series Expansion of the partition function [1]. The case of power-law decaying interaction $J(r)=-(-1)^r r^{-\alpha}$ is investigated for the general one-dimensionnal XXZ Hamiltonian $$ {\mathcal{H}}=\sum_{i,j}J(|i-j|)\left(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y} +\Delta S_{i}^{z}S_{j}^{z}\right).$$ Very large scale numerical results obtained on systems up to $L=8000$ spins are compared and discussed through bosonization and spin-waves predictions [2] for the onset of antiferromagnetic ordering in the ground-state in function of $\beta$.\\ $[1]$ O. F. Sylju{\aa}sen and A. W. Sandvik, Phys. Rev. E {\bf 66}, 046701 (2002).\\ $[2]$ E. Yusuf, A. Joshi, and Kun Yang, Phys. Rev. B {\bf 69}, 144412 (2004).