Density matrix renormalization group

The density matrix renormalization group (DMRG) is a numerical variational technique devised to obtain the low-energy physics of quantum many-body systems with high accuracy. As a variational method, DMRG is an efficient algorithm that attempts to find the lowest-energy matrix product state wavefunction of a Hamiltonian. It was invented in 1992 by Steven R. White and it is nowadays the most efficient method for 1-dimensional systems. The density matrix renormalization group (DMRG) is a numerical variational technique devised to obtain the low-energy physics of quantum many-body systems with high accuracy. As a variational method, DMRG is an efficient algorithm that attempts to find the lowest-energy matrix product state wavefunction of a Hamiltonian. It was invented in 1992 by Steven R. White and it is nowadays the most efficient method for 1-dimensional systems. The main problem of quantum many-body physics is the fact that the Hilbert space grows exponentially with size. For example, a spin-1/2 chain of length L has 2L degrees of freedom. The DMRG is an iterative, variational method that reduces effective degrees of freedom to those most important for a target state. The target state is often the ground state. After a warmup cycle, the method splits the system into two subsystems, or blocks, which need not have equal sizes, and two sites in between. A set of representative states has been chosen for the block during the warmup. This set of left block + two sites + right block is known as the superblock. Now a candidate for the ground state of the superblock, which is a reduced version of the full system, may be found. It may have a rather poor accuracy, but the method is iterative and improves with the steps below. The candidate ground state that has been found is projected into the Hilbert subspace for each block using a density matrix, hence the name. Thus, the relevant states for each block are updated. Now one of the blocks grows at the expense of the other and the procedure is repeated. When the growing block reaches maximum size, the other starts to grow in its place. Each time we return to the original (equal sizes) situation, we say that a sweep has been completed. Normally, a few sweeps are enough to get a precision of a part in 1010 for a 1D lattice. The first application of the DMRG, by Steven White and Reinhard Noack, was a toy model: to find the spectrum of a spin 0 particle in a 1D box. This model had been proposed by Kenneth G. Wilson as a test for any new renormalization group method, because they all happened to fail with this simple problem. The DMRG overcame the problems of previous renormalization group methods by connecting two blocks with the two sites in the middle rather than just adding a single site to a block at each step as well as by using the density matrix to identify the most important states to be kept at the end of each step. After succeeding with the toy model, the DMRG method was tried with success on the Heisenberg model (quantum).