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QCD: Quantum Chromodynamics

from the Independent Sets and Generators

Source: Bjoern Medeke, Department of Mathematics, Institute of Applied Computer Science, University of Wuppertal, 42097 Wuppertal, Germany. Phone: +49 202 439-3776. Email:
Discipline: Physics
Accession: December 2000

Background. Lattice gauge theory is a discretization of quantum chromodynamics which is generally accepted to be the fundamental physical theory of strong interactions among the quarks as constituents of matter. The most time-consuming part of a numerical simulation in lattice gauge theory with Wilson fermions on the lattice is the computation of quark propagators within a chromodynamic background gauge field. These computations use up a major part of the world's high performance computing power.

Quark propagators are obtained by solving the inhomogeneous lattice Dirac equation Ax = b, where A = I - kD with 0 <= k < kc is a large but sparse complex non-Hermitian matrix representing a periodic nearest-neighbour coupling on a four-dimensional Euclidean space-time lattice.

From the physical theory it is clear that the matrix A should be positive real (all eigenvalues lie in the right half plane) for 0 <= k < kc. Here, kc represents a critical parameter which depends on the given matrix D. Denoting

\gamma_5 = \left( \begin{array}{cccc}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{array}\right)
the Wilson fermion matrix A is Gamma-5 symmetric,

\Gamma_5 A = A^H \Gamma_5,\;\;\;\;\Gamma_5 = I \otimes ( \gamma_5 \otimes I_3 )

Due to the nearest neighbour coupling, the matrix A has 'property A'. This means that with a red-black (or odd-even) ordering of the grid points the matrix becomes

A = I - kD
D = \left( \begin{array}{cc}0 & D_{\rm oe}  \\D_{\rm eo} & 0  \end{array}\right)

Set of QCD Matrices. The QCD matrices provided in the set QCD consist of realistic matrices D generated at different physical temperatures b.

matrix D b order nonzeros kc
conf5.0-00l4x4-1000.mtx 5.030721198080.20611
conf5.0-00l4x4-1400.mtx 5.030721198080.20328
conf5.0-00l4x4-1800.mtx 5.030721198080.20265
conf5.0-00l4x4-2200.mtx 5.030721198080.20235
conf5.0-00l4x4-2600.mtx 5.030721198080.21070
conf6.0-00l4x4-2000.mtx 6.030721198080.15968
conf6.0-00l4x4-3000.mtx 6.030721198080.16453
conf5.4-00l8x8-0500.mtx 5.44915219169280.17865
conf5.4-00l8x8-1000.mtx 5.44915219169280.17843
conf5.4-00l8x8-1500.mtx 5.44915219169280.17689
conf5.4-00l8x8-2000.mtx 5.44915219169280.17835
conf6.0-00l8x8-2000.mtx 6.04915219169280.15717
conf6.0-00l8x8-3000.mtx 6.04915219169280.15649
conf6.0-00l8x8-8000.mtx 6.04915219169280.15623

References. A survey of lattice gauge theory is given in

  1. M. Creutz: Quarks, Gluons, and Lattices, Cambridge University Press, (1986)
  2. I. Montvay and G. Munster: Quantum Fields on the Lattice, Cambridge University Press (1994)
More background information can be found at various locations on the Web. Search for "High Energy Physics" (HEP). A list of HEP Web sites is available at CERN (European Laboratory for Particle Physics).

A PostScript version of this information is also available.

Matrices in this set:

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