488 Computational methods for lattice field theories
trajectory as exp(−SD[φ]), where the actionSDdiffers by some power ofhfrom
the continuum action[17]:
SD[φ]=S[φ]+O(hk) (15.69)for some positivek. The discrete action may renormalise to a continuum limit
with slightly different parameters, but as the behaviour of the model is calibrated
in the end by matching calculated physical quantities to the experimental values,
our model with discrete time step might still describe the correct continuum limit.
Indeed, Batrouniet al.[17]show that the discrete time action in the Langevin limit
(i.e. the case in which the momenta are refreshed at every time step, see below) is
a viable one at least to first order inh. A problem is that the difference between the
discrete and the continuum actions makes it difficult to compare the results of the
MD simulation with an MC simulation of the same system with the same values of
the parameters.
The discretisation error can be corrected for in exactly the same way as in the
variational and diffusion quantum Monte Carlo method (see, for example, the dis-
cussion near the end ofSection 12.2.5). The idea is to consider the leap-frog MD
trajectories as a trial step in a Monte Carlo simulation. The energy before and
after this trial step is calculated, and the trial step is accepted with probability
exp(−Hnewclass+Holdclass)(note thatHclassis a classical ‘energy’ which includes kin-
etic and potential energy). If it is rejected, the momenta are refreshed once more and
the MD sequence starts again. This method combines the Andersen refreshment
steps with microcanonical trajectory acceptance/rejection steps. In the previous
subsection we saw that the refreshment step satisfies a modified detailed balance
condition which ensures the correct (canonical) distribution. Now we show that
the microcanonical trajectories plus the acceptance/rejection step, satisfy a similar
detailed balance with a canonical invariant distribution.
We write the transition probability in the form of a trial step probabilityω,P;′,P′
and a Metropolis acceptance/rejection probabilityA,P;′,P′. The trial step probabil-
ity is determined by the numerical leap-frog trajectory and hence is nonzero only for
initial and final values compatible with the leap-frog trajectory. Time-reversibility
of the leap-frog algorithm implies that
ω,P;′,P′=ω′,−P′;,−P. (15.70)The acceptance probability is given as usual by
A,P;′,P′=min{1, exp[Hclass(,P)−Hclass(′,P′)]}. (15.71)The acceptance step is invariant underP↔−Pas the momenta occur only with
an even power in the Hamiltonian.