Applications of Density Matrix Renormalization Group to Problems in Magnetism
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The application of real space renormalization group methods to quantum lattice models has become a topic of great interest following the development of the density matrix renormalization group by White. This method has been used to find the ground and low-lying excited state energies and wave functions of quantum spin models in which the form of the ground state is not clear, for instance because the interactions are frustrated. It has also been applied to fermion problems where the tendency for localization due to the strong Coulomb repulsion is opposed by the lowering of the kinetic energy which occurs as a result of electron transfer. The approach is particularly suitable for one-, or quasi-one-dimensional problems. The method involves truncating the Hilbert space in a systematic and optimised manner. Results for the ground state energy are thus variational bounds. The results for low-lying energies and correlation functions for one-dimensional systems have unprecedented accuracy and the method has become the method of choice for solving one-dimensional quantum spin problems. We review the method and results obtained for the spin-1 chain with biquadratic exchange as well as the spin-1/2 model with competing nearest and next nearest neighbour exchange will be described. More recently, the density matrix renormalization group has been applied to reformulate the coupling constant renormalization group approach which is appropriate for the study of critical properties. This approach has been applied to the anisotropic spin-1/2 Heisenberg chain. Finally, we discuss recent work which has borne promising applications in two dimensions - the Ising model and the two-dimensional Hubbard model.
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