Monte Carlo study of the spin-1 antiferromagnetic chain with both bilinear and biquadratic exchange interactions

Abstract

A Monte Carlo simulation study based on the heat-bath algorithm on a checkerboard arrangement is applied to the spin-1 antiferromagnetic chain model with both bilinear and biquadratic exchange interactions. Changing the parameter \ensuremath{\lambda} which denotes strength of the biquadratic interaction against the bilinear one from -1.0 to 2.0, we obtain the following results according to the measurements of energy and two-point correlation function. At \ensuremath{\lambda}=1 and -(1/3 where some physical quantities are derived rigorously, our Monte Carlo simulation results agree well with the rigorous ones. For ${\ensuremath{\lambda}}_{c}$\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}${\ensuremath{\lambda}}_{1}$ with taking 0.25\ensuremath{\le}${\ensuremath{\lambda}}_{c}$\ensuremath{\le}0.5 and ${\ensuremath{\lambda}}_{1}$\ensuremath{\ge}2.0, its model has the gapless ground state accompanied by a power-law decaying correlation function. This implies the existence of a critical line. For -1.0${\ensuremath{\lambda}}_{c}$, on the other hand, its model has the ground state with energy gap accompanied with exponentially decaying correlation function. This indicates that the valence-bond-solid (VBS) state asserted by Affleck et al. is stabilized there. As the point of \ensuremath{\lambda}=0 comes to the standard Heisenberg antiferromagnetic model and is in the region of -1.0${\ensuremath{\lambda}}_{c}$, the Haldane's conjecture is supported that this ground state is described by the VBS state for the case of integer spin, e.g., S=1.