We obtained many good results which were previously unknown. Among them, the correlation functions are perhaps the most important. First, the results can be directly compared with experiments, thus providing new understanding of the magnetic structure of the high-temperature superconducting materials. Second, and no less important, is the behavior of the correlation function we obtained which gives a crucial test of the assessment of various approximate methods.
In the large spin-S (classical) system, the correlation length goes as
at low temperatures. This predicts a too-large correlation length, compared with experimental results. As , the quantum fluctuations in the system become significant. Several approximate methods [Chakravarty:88a], [Auerbach:88a] predict a similar low-T behavior. , , and p=0 or 1. is a quantum renormalization constant.
Our extensive quantum Monte Carlo simulations were performed [Ding:90g] on the spin- system as large as at low temperature range -1.0. The correlation length, as a function of , is plotted in Figure 6.11. The data points fall onto a straight line, surprisingly well, throughout the whole temperature range, leading naturally to the pure exponential form:
where a is the lattice constant. This provides a crucial support to the above-mentioned theories. Quantitatively,
Figure 6.11: Correlation Length Measured at Various Temperatures. The straight line is the fit.
Direct comparison with experiments will not only test the validity of the Heisenberg model, but also determine the important parameter, the exchange coupling J. The spacing between Cu atoms in plane is . Setting , the Monte Carlo data is compared with those from neutron scattering experiments [Endoh:88a] in Figure 6.5. The agreement is very good. This provides strong evidence that the essential magnetic behavior is captured by the Heisenberg model. The quantum Monte Carlo result is an accurate first principle calculation; no adjustable parameter is involved. Comparing directly with the experiment, the only adjustable parameter is J. This gives an independent determination of the effective exchange coupling:
Note that near , the experimentally measured correlation is systematically smaller than the theoretical curve, shown in Equation 6.4. This is a combined result of small effects: frustration, anisotropies, inter-layer coupling, and so on.
Various moments of the Raman spectrum are calculated using series expansions and comparing with experiments [Singh:89a]. This gives an estimate, (), which is quite close to the above value determined from correlation functions. Raman scattering probes the short wavelength region, whereas neutron scattering measures the long-range correlations. The agreement of J's obtained from these two rather different experiments is another significant indication that the magnetic interactions are dominated by the Heisenberg model.
Equation 6.4 is valid for all the quantum AFM spins. The classic two-dimensional antiferromagnetic system discovered twenty years ago [Birgeneau:71a], , is a spin-one system with . Very recently, Birgeneau [Birgeneau:90a] fitted the measured correlation lengths to
The fit is very good, as shown in Figure 6.12. The factor () comes from integration of the two-loop -function without taking the limit, and could be neglected if T is very close to 0. For the spin- AFM , Equation 6.4 also describes the data quite well [Higgins:88a].
Figure 6.12: Correlation Length of Measured in Neutron Scattering Experiment with the Fit.
A common feature from Figures 6.11 and 6.12 is that the scaling equation Equation 6.4, which is derived near , is valid for a wide range of T, up to . This differs drastically from the range of criticality in three-dimensional systems, where the width is usually about 0.2 or less. This is a consequence of the crossover temperature [Chakravarty:88a], where the Josephson length scale becomes compatible with the thermal wave length, being relatively high, . This property is a general character in the low critical dimensions. In the quantum XY model, a Kosterlitz-Thouless transition occurs [Ding:90b] at and the critical behavior remains valid up to .
As emphasized by Birgeneau, the spin-wave value
S=1, , fits the experiment quite well, whereas for , spin-wave value differs significantly from the correct value 1.25 as in Equation 6.4. This indicates that the large quantum fluctuations in the spin- system are not adequately accounted for in the spin-wave theory, whereas for the spin-one system, they are.
Figure 6.13 shows the energy density at various temperatures. At higher T, the high-temperature series expansion accurately reproduces our data. At low T, E approaches a finite ground state energy. Another useful thermodynamical quantity is uniform susceptibility, which is shown in Figure 6.14. Again, at high-T, series expansion coincides with our data. The maximum point occurs at with . This is useful in determining J and for the material.
Figure 6.13: Energy Measured as a Function of Temperature. Squares are from our work. The curve is the 10th order high-T expansion.
Figure: Uniform Susceptibility Measured as a Function of Temperature. Symbols are similar to Figure 6.13.