The Density Matrix Renormalization Group

The Density Matrix Renormalization Group (DMRG) method is used to obtain solutions of the many-body Schroedinger equation for a lattice Hamiltonian. It is mostly of use in strongly correlated electron systems where, due to the nature of the interactions, traditional techniques give very poor results. We have applied the technique to determine the ground state wavefunction for a variety of models. This is then used to determine various physical quantities, for example the ground-state phases, magnetization, susceptibility etc. The models we have studied have applications in several areas of condensed matter physics, including the theory of high-temperature superconductivity, heavy-fermion metals and the colossal-magnetoresistance materials.

Principal Investigator

Miklos Gulacsi
Theoretical Physics



Facilities Used



Ian McCulloch
Theoretical Physics

RFCD Codes


Significant Achievements, Anticipated Outcomes and Future Work

The major technical achievement of the project has been the extension of the DMRG algorithm to conserve non-Abelian symmetries. This has a dramatic effect on the efficiency of the algorithm; improvements of more than two orders of magnitude have been seen in some cases. This arises because the conserved quantities associated with a non-Abelian symmetry allow the Hamiltonian matrix to be block-diagonalized while also removing what would otherwise appear as degenerate eigenvalues. In addition these conserved quantities have a direct physical interpretation, which allows us to calculate physical properties which were previously prohibitively difficult to obtain. This algorithm is the sucessor to and a major improvement on the earlier version described in the 1998 Annual Report for project x13.

The most significant achievement of the project to date is the discovery of a previously unrecognized ferromagnetic phase in the one-dimensional Kondo lattice model, and the subsequent discovery of the corresponding phase in the local-moment regime of the Anderson lattice model. This is a surprising result as prior to this work it was generally believed that the phase diagram of the 1D Kondo lattice had been completed. This new phase resolves some of the confusion regarding the ground-state phases of the Kondo lattice model, while also raising many new questions which are still to be resolved.

Some of the most interesting open questions in condensed matter physics revolve around the nature of strongly correlated electrons in two dimensions. This encompasses such effects as high-temperature superconductivity and the colossal-magnetoresistance materials. With the APAC National Facility machine, the application of the DMRG method to models of these phenomena is becoming practical, so the major emphasis of the project for the future is to fully exploit the resources of the SC by parallelizing the code using MPI and/or multithreading.


Computational Techniques Used

The DMRG software has been written by I. McCulloch. The implementation makes heavy use of BLAS, LAPACK and GNUmp and ARPACK libraries. The SC has been invaluable by providing parametric parallelization, allowing the calculation of the properties of a model over a significant region of the parameter space of each model. Even using a single processor per job, the SC allowed us to make significant improvements in the accuracy of the calculations which were essential in obtaining the results described previously.

DMRG works by diagonalizing the Hamiltonian matrix of a quantum lattice model in a truncated basis set. The truncation is necessary because the dimension of the Hamiltonian matrix grows exponentially with the system size. This truncation is effected by dividing the lattice into two parts, and calculating which basis states in each part have the largest weight in the overall wavefunction. The basis is then truncated to keep only these states. The boundary of this partition is then shifted by one lattice site and the procedure is iterated until the energy and wavefunction has converged. The accuracy of the truncation depends strongly on the length of the interface between the two parts of the system, thus the algorithm performs substantially better for a one-dimensional chain than for higher dimensions.

Currently the bulk of the computational effort is used in obtaining the ground-state wavefunction and energy of the Hamiltonian. This is represented by the eigenvector and eigenvalue of largest absolute value of the Hamiltonian matrix. This matrix typically has a dimension of ~ 1 million by 1 million, and is stored in a block-sparse form. The diagonalization is done using the Davidson algorithm, which is near-ideal for this purpose as the matrix is usually strongly diagonally dominant. Also an extremely good initial guess-vector for the diagonalization is determined by a basis transformation from the previous DMRG step, thus only a small number of matrix-vector multiplies are required each step (usually less than 10).

Much of the complexity of the algorithm arises from the need to keep track of the basis at each iteration and perform the necessary basis truncations and transformations. While this represents a large proportion of the size of the code, the computational effort required here is relatively small.


Publications, Awards and External Funding

The algorithm developed by I. P. McCulloch and M. Gulacsi for use in this project is described in the following papers: Results obtained using the above algorithm on the PC and SC are published in the following papers: Results obtained using the PC and SC formed the major component of this PhD thesis: