The Density Matrix Renormalization Group

               

Principal Investigator

Miklos Gulacsi

Theoretical Physics

Research School of Physical Science and Engineering

Co-Investigators

Ian McCulloch

Theoretical Physics

Research School of Physical Science and Engineering

Projects

x13 - PC

The Density Matrix Renormalization Group (DMRG) is a
numerical technique for solving one dimensional many
particle Schroedinger equations, with advantages over other numerical techniques, e.g. Monte Carlo, etc., in that it handles with ease very long chains, strongly interacting limits and calculates any given number of excited states. The program developed and used by us is better than any currently exisiting DMRG program, because it can preserve the full symmetries of the original Hamiltonian, thus increasing the accuracy and reducing computational effort.

Essentially, DMRG does exact diagonalization of the Hamiltonian matrix of a 1 dimensional quantum system in a truncated Hilbert space. The Hilbert space needs to be truncated periodically because the number of basis states (and therefore the dimension of the matrix that needs diagonalizing) grows exponentially in the size of the system. DMRG is very versatile because it provides a good approximation to the true wavefunction in any given sector of the Hilbert space, from which practically any physical quantity can be calculated.

         
               

 
               

What are the results to date and the future of this work?

Up to date the program was used to establish the phase diagram of the Kondo lattice model. This particular model describes the interaction between an array of localised moments (generally f-electrons) and conduction, generally p- or d-electrons and as such it is used to understand the properties of heavy fermion compunds, binary alloys, impurity effects. In the future we will be using this technique for the general case, where the f-electrons are not localised (Anderson model) and we intend to use if for purely magnetic systems, such as the Heisenberg model.

Preliminary results were presented at the 1999 March Meeting of the American Physical Society, Atlanta: "Phase diagram of Kondo lattice models in one-dimension", Program and Abstract Booklet, VC29.04 and "Ordering of localized moments in the Kondo lattice model with phonons", Program and Abstract Booklet, VC29.07.

What computational techniques are used?

The software was developed by Ian McCulloch, and consists of two main components. Finding the ground state wavefunction entails finding the lowest energy eigenvalue and eigenvector of

               
Appendix A -

               

       
a large, sparse matrix. This is done using an off-the-shelf eigensolver package, and is traditionally the most CPU intensive part of DMRG. The second component (which as far as we know, is unique to our program) is to do a basis transform at each step, which maintains the conserverved quantum numbers (Casimir operators of the Hamiltonian operator), which is a significant advance because the matrix is block diagonal in the quantum numbers, therefore the dimension of the matrix to solve is greatly reduced. In fact, currently the basis transforms now take most of the CPU time, and the matrix diagonalization is relatively fast.
       
- Appendix A