A Two-dimensional Algorithm Of The Density Matrix Renormalization Group

In this thesis we discuss the density matrix renormalization group. It was presented by White and was successfully applied to the one-dimensional system. However, some difficulties were encountered when it was applied to the two-dimensional system. We give out our algorithm to treat with this problem.From chapter 1 to chapter 4, we reviewed the density matrix and the theories of the critical phenomenon. Mean field theory can explain the critical phenomenon but not accurate, then scaling theory and renormalization group theory were presented.From chapter 5 to chapter 7, we discussed the density matrix renormalization group method, and give out our algorithm to the two-dimensional system. This numerical method is very accurate when it was used to calculate the ground state energy of the one-dimensional system, as the truncation error can be 10"7. There are many algorithms trying to apply this method to the two-dimensional system, but the results are not consistent, for the interaction is not treated properly.In our algorithm, the interaction is treated properly. We divide the two-dimensional system into many closed chains and calculate the spin operators of every site in the chain with the density matrix renormalization group method. Considering the symmetry, we only need to work out the operators of one edge of the closed chain. Then the Hamiltonian of the chain and the interaction between neighboring chains can be worked out. So the Hamiltonian of the system can be calculated out similarly to the one-dimensional system with the density matrix renormalization group method. The interactions along x-axis and y-axis are both considered in our algorithm. In every step, a closed chain was added to the system. So the results of a L X L system can be derived out directly from a (L-2)X(L-2) system. An example of spin-1/2 Heisenberg model was given in our text.