510 Computational methods for lattice field theories
Table 15.3.Comparison between different methods for simulating the scalar
quantum field theory on a lattice.Method Described
in sectionLattice
sizeTime
constanthCorrelation
timeCPU
timeOverall
efficiency
Metropolis 15.4.1 8 28 110 0.32
Metropolis 15.4.1 16 97 416 0.10
Heat-bath 15.4.1 8 6.5 103 1.49
Heat-bath 15.4.1 16 24 392 0.43
Andersen 15.4.3 8 0.05 170 32 0.063
Andersen 15.4.3 8 0.1 85 31 0.38
Andersen 15.4.3 16 0.1 110 124 0.29
Hybrid 15.4.3 8 0.365 15 45 1.48
Hybrid 15.4.3 16 0.22 60 175 0.38
Langevin 15.4.3 8 0.1 560 82 0.022
Langevin 15.4.3 16 0.1 2200 322 0.0056
Multigrid 15.5.4 8 1.5 1440 0.46
Multigrid 15.5.4 16 1.5 5750 0.46
Four/Lang 15.5.5 8 0.2 10 118 0.85
Four/Lang 15.5.5 16 10 523 0.76The time units are only relative: no absolute run times should be deduced from them. The
correlation time is measured in simulation steps (MD steps or MCS). For the methods
with momentum refreshment, the correlation time is measured in MD steps. The overall
efficiency in the last column is the inverse of (CPU time×correlation time). For 16× 16
lattices this number has been multiplied by four.the proportionality constant betweenhkand the inverse propagator:
hk=0.2
√
Kk,−k. (15.106)
From the table it is seen that for small lattices the heat-bath and the hybrid meth-
ods are most efficient. For larger lattices, the multigrid and Fourier-accelerated
Langevin methods take over, where the latter seems to be more efficient. However,
its efficiency decreases logarithmically with size (that of the multigrid remains con-
stant) and comparisons of the values obtained with this method and MC simulations
are always a bit hazardous, although these results may be very useful in themselves.
15.7 Gauge field theories
15.7.1 The electromagnetic LagrangianThe scalar field theory is useful for some applications in particle physics and statist-
ical mechanics. However, the fundamental theories describing elementary particles