Study of Spin Lattice and Lattice Gauge Theory


Principal Investigator

Chris Hamer

School of Physics,

University of New South Wales

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.



Jaan Oitmaa

Robert Bursill

Zheng Weihong

School of Physics,

University of New South Wales



g57 - VPP



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

Two papers were published in 1996 concerning the application of Green's Function Monte Carlo (GFMC) methods to SU(2) Yang-Mills theory in (2+1) dimensions. Using improved Runge smoothing techniques, accurate results were obtained for the ground-state energy and mean plaquette value, allowing a detailed investigation of finite-size scaling in the model. The results were more accurate in the weak-coupling region than any previous estimates in the Hamiltonian formulation. A secondary amplitude method was also used to estimate values for the Wilson loops and hence the string tension, but this proved to be less than satisfactory. It was recognized that better methods were needed to estimate correlation functions and excited state properties.

Accordingly, efforts in 1997 have been devoted to developing forward walking codes to estimate correlation functions, as applied to the Heisenberg spin model by Runge. As a test case we have calculated correlation functions for the transverse Ising model in (1+1) dimensions, which are found to agree well with exact results.

The technique has been applied to the U(1) Yang-Mills theory in (2+1) dimensions using a vectorised code on the VPP facility at ANU. The code achieves very high levels of vectorisation (around 95%) and production calculations were commenced in late 1997 for spacelike correlation functions which permit the calculation of the mass gap and string tension on large lattices (up

Appendix B -



to 32*32 sites) deep into the weak-coupling regime (where the original continuum field theory is recovered). QMC configurational data has been stored on the MDS system and will be recalled for further analysis and used as the basis for simulations of timelike correlations. The preliminary results are very encouraging. We hope to finish this work in the first half of 1998 and then move onto SU(3) Yang-Mills theory in (3+1) dimensions (pure gauge QCD), where results can be compared with experiment.

Also, exact diagonalisation calculations have been performed on the VPP for the Schwinger model ((1+1)-dimensional quantum electrodynamics: dynamical fermions (electrons and positrons) interacting with an electromagnetic field). This model is presently of interest because effects such as confinement and the dynamical breaking of Chiral symmetry, believed to exist in full QCD, can be studied under controlled conditions. The code achieves very high levels of vectorisation and the performance ratio obtained when comparing the VPP to superscalar (SGI Power Challenge) architectures was as high as 60, due to the large memory bandwidth of the vector machine. The results obtained so far are orders of magnitude more accurate than previous studies.

What computational techniques are used?

The basic QMC method used is a Green's function Monte Carlo (GFMC) method, and applied to lattice gauge theory. The correlation functions and other expectation values are estimated by a forward walking technique, and applied to a lattice spin model. The method is basically an iterative, stochastic process, well suited to vector computation, with branching added. The configurational data generated needs to be stored on a mass storage facility such as MDS so that analysis can be performed outside of run time and the configurational data can be reused in further simulations.

The exact diagonalisation code is a conjugate gradient diagonalisation for a sparse operator acting on a large Hilbert space (tens of millions of basis states). This is an iterative process which calculates a few eigenvalues and eigenvectors of the operator. Vectorisation occurs naturally, the vector length being very large. However, the main loops involve randomly scattered access to very large arrays, and so the flat memory structure and high memory bandwidth of the vector architecture lead to a huge performance increase over scalar machines.

- Appendix B