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Concept
Density matrix renormalization group
Résumé
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 first application of the DMRG, by Steven R. 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 quantum Heisenberg model.
The main problem of quantum many-body physics is the fact that the Hilbert space grows exponentially with size. In other words if one considers a lattice, with some Hilbert space of dimension on each site of the lattice, then the total Hilbert space would have dimension , where is the number of sites on the lattice. 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 state one is most often interested in is 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 blocks + two sites + right blocks is known as the superblock.
Source officielle
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