Loop algorithm for Heisenberg models with biquadratic interaction and phase transitions in two dimensions

We present a new algorithm for quantum Monte Carlo simulation based on glob al updating with loops. While various theoretical predictions are confirmed in one dimension; we find, for S = 1 systems on a square lattice with an a ntiferromagnetic biquadratic interaction, that the intermediate phase betwe en the antiferromagnetic and the ferromagnetic phases is disordered and tha t the two phase transitions are both of the first order in contrast to the one-dimensional case. It is strongly suggested that the transition points c oincide with those at which the algorithm changes qualitatively.