Quantum fluctuations are capable of pushing the transition point from in the classical model, down to in the quantum spin-1/2 case, although they are not strong enough to push it down to 0. They also reduced the constant from 1.67 in the classical case to 1.18 in the spin-1/2 case.
The critical behavior in the quantum case is of the KT-type, as in the classical case. This is a little surprising, considering the differences regarding the spin space. In the classical case, the spins are confined to the X-Y plane (thus the model is conventionally called a ``planar rotator'' model). This is important for the topological order in KT theory. The quantum spins are not restricted to the X-Y plane, due to the presence of for the commutator relation. The KT behavior found in the quantum case indicates that the extra dimension in the spin space (which does not appear in the Hamiltonian) is actually unimportant. These correlations are very weak and short-ranged. The out-of-plane susceptibility remains a small quantity in the whole temperature range.
These results for the XY model, together with those on the quantum Heisenberg model, strongly suggest that although quantum fluctuations at finite T can change the quantitative behavior of these nonfrustrated spin systems with continuous symmetries, the qualitative picture of the classical system persists. This could be understood following universality arguments that, near the critical point, the dominant behavior of the system is determined by long wavelength fluctuations which are characterized by symmetries and dimensionality. The quantum effects only change the short-range fluctuations which, after integrated out, only enter as renormalization of the physical parameters, such as .
Our data also show that, for the XY model, the critical exponents are spin-S independent, in agreement with universality. More specifically, in Equation 6.18 could, in principle, differ from its classical value 1/2. Our data are sufficient to detect any systematic deviation from this value. For this purpose, we plotted in Figure 6.18(b), using versus . As expected, data points all fall well on a straight line (except the point at where the critical region presumably ends). A systematic deviation from would lead to a slightly curved line instead of a straight line. In addition, the exponent, at , seems to be consistent with the value for the classical system.
Our simulations reveal a rich structure, as shown in the phase diagram (Figure 6.21) for these quantum spins. The antiferromagnetic ordered region and the topological ordered region are especially relevant to the high- materials.
Figure: Phase Diagram for the Spin- Quantum System Shown in Equation 6.12. The solid points are from quantum Monte Carlo simulations. For large , the system is practically an Ising system. Near h=0 or h=-2, the logarithmic relation, Equation 6.16 holds.
Finally, we point out the connection between the quantum XY system and the general two-dimensional quantum system with continuous symmetry. Through the above-mentioned Matsubara-Matsuda transformations, our result implies the existence of the Kosterlitz-Thouless condensation in two-dimensional quantum systems. The symmetry in the XY model now becomes , a continuous phase symmetry. This quantum KT condensation may have important implications on the mechanism of the recently discovered high-temperature superconducting transitions.