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.

[1] 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.[when?]

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.[when?]

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.

The main problem of quantum many-body physics is the fact that the Hilbert space grows exponentially with size.

The DMRG is an iterative, variational method that reduces effective degrees of freedom to those most important for a target state.

Now a candidate for the ground state of the superblock, which is a reduced version of the full system, may be found.

The candidate ground state that has been found is projected into the Hilbert subspace for each block using a density matrix, hence the name.

[further explanation needed] Now one of the blocks grows at the expense of the other and the procedure is repeated.

Each time we return to the original (equal sizes) situation, we say that a sweep has been completed.

A practical implementation of the DMRG algorithm is a lengthy work[opinion].

A few of the main computational tricks are these: The DMRG has been successfully applied to get the low energy properties of spin chains: Ising model in a transverse field, Heisenberg model, etc., fermionic systems, such as the Hubbard model, problems with impurities such as the Kondo effect, boson systems, and the physics of quantum dots joined with quantum wires.

It has been also extended to work on tree graphs, and has found applications in the study of dendrimers.

For 2D systems with one of the dimensions much larger than the other DMRG is also accurate, and has proved useful in the study of ladders.

The method has been extended to study equilibrium statistical physics in 2D, and to analyze non-equilibrium phenomena in 1D.

The DMRG has also been applied to the field of quantum chemistry to study strongly correlated systems.

DMRG is a renormalization-group technique because it offers an efficient truncation of the Hilbert space of one-dimensional quantum systems.

The Hamiltonian matrix of the superblock (the chain), which at the first iteration has only four sites, is formed by these operators.

At this point you must choose the eigenstate of the Hamiltonian for which some observables is calculated, this is the target state .

is the target state, expectation value of various operators can be measured at this point using

The success of the DMRG for 1D systems is related to the fact that it is a variational method within the space of matrix product states (MPS).

stands for the four possibilities of the projection of the spin quantum number of the two electrons that can occupy a single orbital, thus

Such gauge freedom is employed to transform the matrices into a canonical form.

In the one-site algorithm, only one matrix (one site) whose elements are solved for at a time.

Two-site just means that two matrices are first contracted (multiplied) into a single matrix, and then its elements are solved.

Having the MPS in one of the above canonical forms has the advantage of making the computation more favorable - it leads to the ordinary eigenvalue problem.

In 2004 the time-evolving block decimation method was developed to implement real-time evolution of matrix product states.

Subsequently, a new method was devised to compute real-time evolution within the DMRG formalism - See the paper by A. Feiguin and S.R.