Density matrix renormalization group

The Density Matrix Renormalization Group (DMRG) is a numerical technique originally intended to obtain the ground state of a quantum manybody system with high accuracy. It is a variational method, and its efficience does not decrease when the system is strongly correlated. The method has been extended to equilibrium statistical mechanics and non-equilibrium systems. Its main disadvantage is that only 1D and tree-like systems are suitable to obtain the maximum power of the method.

DMRG was introduced by Steve White and Reinhard Noack in 1993, and the first application was a toy model: to find the spectrum of a spin 0 particle in a 1D box. It had been proposed by Kenneth G. Wilson as a test for all renormalization group methods, since they all happened to fail with this simple problem. After that, the Heisenberg model was tried, with the same accuracy.

The main problem of quantum many-body physics is the fact that the Hilbert space grows exponentially with size. For a spin chain of length <math>L<math>, the Hilbert space dimension is <math>2^L<math>. Therefore, the method proceeds by a smart reduction of the number of effective degrees of freedom and a variational search within this reduced space.

After a warmup cycle, the method splits the system into two 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. Now a candidate for the ground state of the full system is found, which may have a rather poor accuracy. The method is iterative and it will be improved with the forthcoming steps.

The candidate ground state which has been found is projected into the subspace for each block using a density matrix, whence the name. 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 <math>10^{10}<math>.

A nice introduction to the subject may be found at the review of Karen Hallberg [1] (, the review of Uli Schollwoeck [2] ( and the Ph.D. thesis of Javier Rodríguez-Laguna [3] (


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