In this section, we discuss two further important developments based on the previous section (Section 6.3) on the isotropic Heisenberg quantum spins. These extensions are important in treating the observed phase transitions in the two-dimensional magnetic systems. Theoretically, two-dimensional isotropic Heisenberg quantum spins remain in paramagnetic state at all temperatures [Mermin:66a]. However, all crystals found in nature with strong two-dimensional magnetic characters go through phase transitions into ordered states [Birgeneau:71a], [DeJongh:74a]. These include the recently discovered high- materials, and , despite the presence of large quantum fluctuations in the spin- antiferromagnets.
We consider the cases where the magnetic spins interact through
In the case , the system goes through an Ising-like antiferromagnetic transition, very similar to those that occur in the high- materials. In the case h = -J, that is, the XY model, the system exhibits a Kosterlitz-Thouless type of transition. In both cases, our simulation provides convincing and complete results for the first time.
Through the Matsubara-Matsuda transformation between spin-1/2 operator and bosonic creation/destruction operations and , a general quantum system can be mapped into quantum spin system. Therefore, the phase transitions described here apply to general two-dimensional quantum systems. These results have broad implications in two-dimensional physical systems in particular, and the statistical systems in general.
HPFA Applications and Paradigms