Study of Spin Lattice and Lattice Gauge Theory 


Principal Investigator

Chris J. Hamer

School of Physics

The University of New South Wales


Jaan Oitmaa

Robert J. Bursill

Zheng Weihong

Pradeep Sriganesh

Maria Samaras

School of Physics

The University of New South Wales


g57 - VPP and MDSS

Quantum Chromodynamics (QCD) is the quantum
field theory that describes the interactions of
quarks and gluons, particles which bind together to form nucleons (such as protons and neutrons), pions, and other strongly interacting particles. QCD is a complicated theory that is difficult to treat analytically, and in order to calculate the experimental predictions of the theory numerically, it is discretisedplaced on a latticeand then simulated using Monte Carlo methods. This is called lattice gauge theory. Our aim is to develop quantum Monte Carlo (QMC) techniques using a particular formulation of lattice gauge theorythe Hamiltonian formulationwhich exploits techniques used in quantum lattice models of systems in condensed matter and which has been infrequently used in the literature in the past compared with the
Euclidean formulation.


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

For some time now we have been investigating the applicability of Greens Function Monte Carlo (GFMC) methods to lattice gauge theories. The GFMC technique was developed by Kalos and Ceperley to solve quantum many body problems in areas such as condensed matter, atomic, molecular and nuclear physics. It was first applied to Hamiltonian lattice gauge theory in the 1980s by Chin et al., to calculate the ground state energy density of pure gauge theories (theories with gauge particles only, e.g. photons (light) or gluons). We investigated the finite-size scaling (dependence of quantities on lattice size) and critical properties of two-dimensional pure gauge theories with substantially greater accuracy than had previously been obtained in the Hamiltonian formulation. However, we have since been concerned with developing methods to calculate experimentally measurable quantities from the theory.

The two most important experimentally observable quantities that we need to calculate are the string tension and mass gaps (the masses of the particles such as pions and mesons which are observed in accelerator experiments). As with the Euclidean approach to lattice gauge theory,

Appendix B -



these quantities cannot be simulated directly but must be inferred from a calculation of correlation functions. Correlation functions can be measured within the GFMC approach using a technique called forward walking, first developed for continuum quantum many body systems (e.g. liquid Helium3) by Ceperley and Kalos and applied to quantum lattice systems (spin models) by Runge. The initial phase of this project has been to extend the forward walking method to lattice gauge theory. First, we tested the forward walking algorithm on the (1+1)D Ising model (a quantum spin chain system) for which exact analytic results are available for correlation functions. It was established that forward walking can indeed efficiently recover exact results.

The next step was to try the approach on U(1) theorypure gauge Quantum Electrodynamics (QED, the interaction of electrons and photons)in (2+1)D. The advantage of using this model is that it has been studied by a plethora of techniques and accurate results are available for the string tension and mass gaps. We commenced calculations of spacelike Wilson loops in 1997. The rate of exponential decay of these loops gives the string tension. We found good agreement between our forward walking GFMC calculations and results from other methods. In 1998 we went on to measure timelike correlation functions which lead to estimates of the mass gaps. Again, agreement with other methods was very good.

In order to make comparisons with experiment, one must study QCD in (3+1)D. We have developed a GFMC code for SU(3) theory, pure gauge QCD (pure gluonic matter in the absence of quarks), in (3+1)D. We are currently testing the code and hope to perform production calculations throughout 1999. A measurement of the string tension and mass gaps in this theory will allow comparison with experimental data on the glueball, a particle composed entirely of gluons. Looking further ahead, we plan to try simulations with quarks in order to calculate the masses of hadrons such as pions and mesons which again can be compared with experiment. We plan to incorporate fermions (quarks) into the simulations using the constrained path Quantum Monte Carlo (CPMC) method of Gubernatis et al. which has proved quite successful in circumventing the infamous fermion minus sign problem to study the 2D Hubbard model, an important model in the theory of High Temperature Superconductivity, and one of the "classic hard problems" in condensed matter physics.

Exact diagonalisation calculations have been performed on the VPP for the Schwinger model (i.e. (1+1)-dimensional QEDelectrons interacting with an electromagnetic field). This model is presently of interest because effects such as confinement, the U(1) axial anomaly and the dynamical breaking of Chiral symmetry, believed to exist in full QCD, can be studied under controlled conditions. Results are some orders of magnitude more accurate than previous studies.

Finally, in 1999 we are embarking on developing density matrix renormalisation group (DMRG) codes for lattice gauge theories with dynamical fermions (electrons and quarks). The DMRG was invented in 1992­93 and has revolutionised computational condensed matter physics. It is now the method of choice for solving 1D quantum lattice models (models of one dimensional conductors and magnets) where very large chains have been solved with unprecedented accuracy. It also has begun to challenge the preeminence of Quantum Monte Carlo techniques for 2D models like the Hubbard model of high temperature superconductivity. Although Euclidean Monte Carlo techniques are being used to solve lattice gauge theories with dynamical fermions, because of the minus sign problem, they are extremely inefficient and require vast

- Appendix B



supercomputing resources. We are currently developing DMRG code to solve complex (1+1)D lattice gauge theories such as QCD with a number of flavours of quarks. In the future we plan to move to problems such as the positronium spectrum of (2+1)D QED, about which very little is known. Further down the track one might even contemplate (3+1)D theories but the key aim is to make progress on the important problem of dynamical fermion systems.

What computational techniques are used?

The Quantum Monte Carlo (QMC) codes were written by A/Prof. Chris Hamer with the aid of Dr Bursill. They achieve very high levels of vectorisation (94­98%) and have arithmetically dense inner loops with operations on a number of large arrays. This makes the code ideal for execution on the VPP machine at ANUSF. The single processor speedup in going from the SGI Power Challenge at NSWCPC to the VPP is around 15. However, the other facility at ANUSF important to this work is the MDSS which is needed to store the large data files containing the gauge field configurations which can be recalled from the MDSS for offline analysis. A typical data file is 2­4 GB.

The exact diagonalisation codes, written by Sriganesh and Bursill, also have highly vectorisable inner loops acting on very large vectors. For the QMC codes memory access is always contiguous. For the exact diagonalisation codes memory access is random. This increases the Power Challenge to VPP single processor speedup to around 60­100. We also make use of the 2GB processors here as the code is very memory intensive.


R. J. Bursill and C. J. Hamer, Greens function Monte Carlo study of correlation functions in the (2+1)D U(1) lattice gauge theory, to appear in Nuclear Physics B: Proceedings Supplements.

P. Sriganesh, R. J. Bursill and C. J. Hamer Lattice calculations of the massive Schwinger model to appear in Nuclear Physics B: Proceedings Supplements.

Appendix B -