Applications of Density Matrix Renormalization Group to Problems in Magnetism

Abstract

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.

Density matrix renormalization group calculations of the low-lying excitations and non-linear optical properties of poly(para-phenylene)

“…In order to study larger systems, we turn to the DMRG method [23]. The The DMRG is discussed at length in [23] and reviewed in [24] so we restrict our discussion here to features relevant to our implementation of the method for (3). The key features are the form of the system, environment and super blocks, the number m of states retained per block, and the good quantum numbers used to diagonalise the superblock Hamiltonian and the density matrix.…”

A DMRG study of the -symmetric Heisenberg chain

“…With this approach not only the energies of large systems can be calculated very accurately, the method also yields correlation functions. [2] It was tested e.g. on Heisenberg [1] and transverse Ising [3] chains and has been applied to a variety of systems like higher-spin chains [4], ladders [5], Hubbard chains [6], impurity models [7], phonon systems [8] or transfer matrices [9,10].…”

“…The system which we have studied is the U q [SU (2)]symmetric Heisenberg chain with Hamiltonian [11]…”

Density matrix renormalisation group approach to the massive Schwinger model

“…First, we aim to explore the physics of this model when an external 'background' electric field is applied, as discussed long ago in a beautiful paper by Coleman [4]. Secondly, we wish to demonstrate the application of density matrix renormalization group (DMRG) methods [6,7] to a model of this sort, with long-range, nonlocal Coulomb interactions. The DMRG approach has been used with great success for lattice spin models and lattice electron models such as the Hubbard model; here we apply it to a lattice gauge model.…”

“…Our results are based on the density matrix renormalization group (DMRG) method [6,7]. The method employed here is the "infinite system" DMRG method, as prescribed by White [6], used both with open and periodic boundary conditions (OBC and PBC).…”