The Density Matrix Renormalization Group
The Density Matrix Renormalization Group (DMRG) method is used to obtain solutions of the many-body Schroedinger equation, which describes quantum-mechanical phenomena. This particular method is mostly of use in strongly correlated electron systems where, due to the nature of the interactions, traditional techniques give very poor results. Applications are also emerging in other areas, such as quantum chemistry and nuclear physics. 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. Recently the focus has moved to the calculation of "hidden" topological order, which we have shown to be present in a large class of one-dimensional quantum liquids. Topological order, which falls outside the usual (Landau) theory of phase transitions, is rapidly moving to the forefront of theoretical and experimental condensed matter physics, with the promise of new states of matter with significant technological applications.
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Principal Investigator Miklos GulacsiTheoretical Physics, RSPhysSE Australian National University |
Project x18 |
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Co-Investigators Yung Kai ChanTheoretical Physics, RSPhysSE Australian National University Phillip Brydon Physics and Theoretical Physics, Faculty of Science Australian National University Ian McCulloch Theoretical Physics Universiteit Leiden, Netherlands |
RFCD Codes 240203 |
Significant Achievements, Anticipated Outcomes and Future Work
In the past year we have completed work on two calculations, for the dilute Kondo lattice model (a model of electron correlations in the presence of impurities) and also on topological order in one-dimensional quantum wires. Building on the initial work on topological order in 1D, we have used numerical calculations performed on the SC in combination with analytical calculations to demonstrate that topological order is actually the essential defining property of the Luttinger Liquid state (the Luttinger Liquid describes a universality class covering most one-dimensional metallic systems), this work will be published in 2004.
We also have calculations on topological order in the Kondo lattice, which will be presented at the 2004 American Physical Society March Meeting. This is the model that was the initial impetus to start the DMRG project. Now the discovery of topological order in Luttinger Liquids provides new tools to study the Kondo lattice problem. For example, it has long been conjectured in the past that the Kondo lattice is in the universality class of the Luttinger liquid in a substantial region of the phase diagram. Our recent calculation tested this directly and conclusively. Calculations on a related model, the periodic Anderson model, will also be presented at the March meeting.
Work is currently underway to integrate the DMRG software with the ALPS project (http://alps.comp-phys.org), which will provide an XML interface capable of setting up complex calculations very quickly, and in a consistent way across a variety of numerical simulation tools. The software will also be released under an open source license.
Computational Techniques Used
The core of the program is calculating the lowest energy eigenvector (or a few eigenvectors) of a large matrix problem, originating from the Schroedinger equation on a one-dimensional lattice. The total dimension of the problem space is exponentially large (around 2^400 for a typical problem), but an approximation that works well for one-dimensional lattices is to partition the physical lattice into two halves, and construct the basis as a tensor product of left and right basis sets, each of which is truncated to a manageable size. Lattice sites are then 'shifted' from one partition to the other, iteratively, thereby improving the eigenvector by a sequence of local updates. Typically, we keep ~ 1000 basis states in each 'block', which gives a final eigenvalue problem of ~ 1,000,000 basis states at each iteration. The matrix ends up in a variation of block-sparse format, in which the matrix-vector multiply can be implemented purely with BLAS level 3 (ie. O(N^3) matrix-matrix operations) acting on relatively small sizes (up to around 100x100). The eigensolver itself uses the Davidson algorithm, which is ideal for this application as the matrix tends to be strongly diagonally dominant.
The program also makes heavy use of LAPACK dense matrix eigensolvers to calculate the basis transformations required for 'shifting' lattice sites between the two blocks. This is the only production-quality DMRG program that we know of that allows the use of full SU(2) symmetry-adapted basis states in the calculation. This component utilizes the GNU Multiple Precision Arithmetic Library (GMP) to calculate the algebraic coupling coefficients.
Publications, Awards and External Funding
External Funding and Awards
Ian McCulloch was supported by FOM/NWO in the Netherlands.
Publications
M. Gulacsi, I.P. McCulloch, A. Jouzapavicius, A Rosengren, Magnetism in the Dilute Kondo Lattice Model, Phys. Rev. B (in press).