Quantum Lattice Models in Condensed Matter and Particle Physics

Significant Achievements, Anticipated Outcomes and Future Work

This project provides a significant portion of the supercomputing required by the Department of Theoretical Physics at UNSW, and comprises a number of distinct subprojects.

(1) Hamiltonian lattice gauge theory
Lattice gauge theory (LGT) is a method by which theorists describe the fundamental particles of nature such as electrons and quarks, and their interactions (which are mediated by other particles such as photons (light) and gluons), formulated on a lattice. The lattice formulation allows for large scale numerical simulations that can be compared with experiment. In this project significant progress has been made in the development and use of "density matrix renormalisation group" (DMRG) approaches for a formulation of LGT known as Hamiltonian LGT. We have shown that, for a particular model system, the Schwinger model of QED (quantum electrodynamics) in (1+1)D, our DMRG techniques are orders of magnitude more accurate than any other approach. We plan to exploit this success by using the same method for more complicated models in 1 space and 1 time domension. In 2003 we commenced work on the 2-species Schwinger model with the help of Research Associate John Markham. A series of production calculations has been carried out on the SC at the APAC National Facility, and work is continuing into 2004.

We have also been carrying out "Path Integral Monte Carlo" studies of LGTs on anisotropic lattices, aiming to demonstrate the connection between the standard Euclidean (or isotropic) formulation and our Hamiltonian formulation. A very successful calculation for the Abelian U(1) theory in (2+1) dimensins was completed by a graduate student, Mushtaq Loan, and published in Physical Review. Dr Tim Byrnes applied similar methods to the SU(3) theory in (3+1) dimensions, related to real quantum chromodynamics (the theory of quarks and gluons).

(2) Series studies of models of low-dimensional antiferromagnetism, organic conductors and anomalous superconductors
The UNSW group is the clear world leader in "series expansion" approaches to quantum lattice models in condensed matter physics. We have shown that our techniques are the most powerful available for numerically solving the key models for some significant classes of problems in antiferromagnetism and superconductivity. As can be seen below, some 9 publications in quality journals were produced by this project in 2003, on a number of topics, and a similar output is expected in 2004. The highlights of 2003 were: (1) studies of ferrimagnetism on 3D decorated lattices; (2) studies of the Kondo lattice model; and (3) further studies uncovering the very complex critical behaviour of the 2D J1-J2 spin model. The current series calculations running on the APAC National Facility and studies planned for 2004 include: (1) Generalisation of two-particle bound state and spectral weight calculations to 2D systems; (2) Studies of the 1/3- filled triangular t-J model of NaxCoO₂; (3) Studies of the ionic Hubbard model in one and two dimensions; (4) Series expansions of Heisenberg models in magnetic fields; and (5) Studies of magnetic models on simple cubic, BCC and anisotropic triangular lattices.

(3) Four-block DMRG Studies of Quantum Lattice Systems in Condensed Matter; and
(4) Development of Four-block DMRG Techniques in Two Dimensions
In these subprojects we have been developing and applying DMRG techniques to condensed matter quantum lattice models similar to those studied in (2) above. In 2003 our emphasis has been on developing parallel algorithms that can harness all of the processors within a compute node, to increase throughput and make efficient use of memory using the Open MP symmetric multiprocessing standard. We have had success with this, but further tuning is necessary. We are confident of achieving acceptable scaling to 8 or more processors. We have also performed some production runs to benchmark a number of algorithms for 2D systems, and we aim to publish a study in 2004. In 2004 we aim to focus on applications of our algorithms to a number of salient systems.

Computational Techniques Used

In series expansion methods for quantum lattice models a perturbation series is derived for quantities of interest, such as excitation spectra. The series is taken about some trivially solveable point, the expansion parameter typically being a coupling constant that makes the model non-trivial. In order to sum the series at the point of interest (e.g., a coupling constant relevant to experiment), an extrapolation scheme such as Pade approximants is employed. The computationally intensive work is usually in the calculation of the series coefficients. The resource requirements (memory and CPU) increase exponentially with each successive coefficient calculated, though each coefficient added can lead to a sharp rise in accuracy for the extrapolations. The APAC National Facility has enabled Dr Weihong and his group to capitalise on the superior efficiencies of their programs, and the fact that their programs can calculate many quantities which cannot be calculated by any other group or method, by affording access to large physical memory and excellent integer performance.

DMRG techniques represent a powerful alternative to "exact diagonalisation" lattice models of quantum systems on finite lattices. In quantum lattice models, the size of the Hilbert space (on which acts the energy operator that must be diagonalised to obtain the excitation spectrum and correlation functions that are measured in experiments) grows exponentially with the size of the lattice, which must be as close as possible to the bulk limit for comparison with experiment. By employing a systematic Hilbert space truncation, the DMRG allows computational physicists to attack very large lattices with very little loss in accuracy. DMRG involves intensive linear algebra operations - diagonalisation of extremely large, sparse operators, and large, dense matrices, and the accuracy and scope of the method is ultimately limited by the available memory. We have developed Open MP versions of our codes that can exploit the large memery space offered by the 16-processor node at the APAC National Facility and the 64-processor SMP system at ac3. We are also exploring the possibility of utilising MPI to explot multiple nodes and hopefully perform some of the world's largest scale DMRG codes for classic hard problems such as the doped Hubbard model of high temperature superconductivity.

Publications, Awards and External Funding

External Funding and Awards

This project was funded in 2003 by two ARC Large grants, used to employ Dr Weihong and Dr Markham, as well as an ARC QEII Fellowship held by Dr Bursill. A new ARC Discovery grant was awarded for 2003-7. This is being used to continue Dr Zheng's contract, and to hire two outstanding new Research Associates - Dr Alexander Weisse and Dr Jesko Sirker, both from Germany.

Publications

T. M. R. Byrnes, C. J. Hamer, W. Zheng, and S. Morrison, Application of Feynman-Kleinert approximants to the massive Schwinger model on a lattice, Phys. Rev. D 68, 2003, 016002.
Loan, M., Brunner, M., Sloggett, C., and Hamer, C. J., "Path integral Monte Carlo Approach to the U(1) Lattice Gauge Theory in (2+1) Dimensions", Phys. Rev. D 68, 2003, 034504.
J. Oitmaa and W. Zheng, "Ferrimagnetism and compensation points in a decorated 3D Ising models", Physica A 328, 2003, 185.
J. Oitmaa and W. Zheng, "Finite temperature strong-coupling expansions for the Kondo lattice model", Phys. Rev. B 67, 2003, 214407.
R. R. P. Singh, W. Zheng, J. Oitmaa, O. P. Sushkov, and C. J. Hamer, "A closer look at symmetry breaking in the collinear phase of the J1-J2 Heisenberg Model", Phys. Rev. Lett. 91, 2003, 017201.
Zheng, W. H., Hamer, C. J., and Singh, R. R. P., "Spectral weight contributions of many-particle bound states and continuum", Phys. Rev. Lett. 91, 2003, 037206.
W. Zheng and J. Oitmaa, "Zero Temperature Series Expansions for the Kondo Lattice Model at Half Filling", Phys. Rev. B 67, 2003, 214406.
W. Zheng and J. Oitmaa, "The Mixed Spin S = 1/2 XXZ Ferrimagnet at Zero Temperature: Cluster series expansions", Phys. Rev. B 67, 2003, 224421.
C. J. Hamer, W. Zheng and R. R. P. Singh, "Dynamical structure factor for the alternating Heisenberg chain: A linked cluster calculation", Phys. Rev. B 68, 2003, 214408.
C. J. Hamer, W. Zheng, J. Oitmaa R. R. P. and Singh, "The doped t-J ladder via series expansions", in Condensed Matter Theories, Vol. 17, eds M. P. Das and F. Green, 2003, 167-178.