Following the discovery of high-temperature superconductivity, two-dimensional quantum antiferromagnetic spin systems have received enormous attention from physicists worldwide. It is generally believed that high-temperature superconductivity occurs in the planes, which is shown in Figure 6.4. Many features can be explained [Anderson:87a] in the Hubbard theory of the strongly coupled electron, which at half-filling is reduced to spin-1/2 antiferromagnetic Heisenberg model:

where are quantum spin operators. Furthermore, the neutron scattering experiments on the parent compound, , reveal a rich magnetic structure which is also modelled by this theory.

Physics in two dimensions (as compared to three dimensions) is characterized by the large fluctuations. Many analytical methods work well in three dimensions, but fail in two dimensions. For the quantum systems, this means additional difficulties in finding solutions to the problem.

**Figure 6.4:** The Copper-Oxygen Plane, Where the Superconductivity Is
Generally Believed to Occur. The arrows denote the quantum spins.
, , denote the wave functions which
lead to the interactions among them.

**Figure:** Inverse Correlation Length of Measured in Neutron
Scattering Experiment, Denoted by Cross; and Those Measured in our
Simulation, Denoted by Squares (Units in .
. At ,
undergoes a structural transition. The curve is the fit shown in
Figure 6.11.

New analytical methods have been developed to understand the low-T behavior of these two-dimensional systems, and progress had been made. These methods are essentially based on a expansion. Unfortunately, the extreme quantum case lies in the least reliable region of these methods. On the other hand, given sufficient computer power, Quantum Monte Carlo simulation [Ding:90g] can provide accurate numerical solutions of the model theory and quantitative comparison with the experiment (see Figure 6.5). Thus, simulations become a crucial tool in studying these problems. The work described here has made a significant contribution to the understanding of high- materials, and has been well received by the science community [Maddox:90a].

Wed Mar 1 10:19:35 EST 1995