Computational Physics

508 Computational methods for lattice field theories

By integrating out the momenta it is clear that an MD simulation at constant temper-
ature for this Hamiltonian leads to the correct Boltzmann distribution of the field.
In the Langevin leap-frog form, the equation of motion reads

φk(t+hk)=φk(t)−

h^2 k

2

∂S[φk(t)]

∂φk

+hkRk, (15.103)

whereRkis the Fourier transform of a Gaussian random number with a variance of
1 (see below). Fourier transforms are obviously carried out using the fast Fourier
transform (seeAppendix A9).
For a free field model, the dynamical system described by the Hamiltonian
(15.102)can be solved trivially, as the Hamiltonian does not contain couplings
between the differentks. In that case the action can be written as

S[φk]=

1

2 L^2 d

φkKk,−kφ−k. (15.104)

Kk,k′is the free field propagator given in(15.49). The Hamiltonian describes a set
of uncoupled harmonic oscillators with periodsTk= 2 π/

√

Kk,−k. The algorithm
will be unstable when one of the time stepshkbecomes too large with respect to
Tk(seeAppendix A7.1). The most efficient choice for the time steps is therefore

hk=αTk=α

2 π

√

4

∑

μsin

(^2) (kμ/ 2 )+m 2

, (15.105)

whereαis some given, small fraction, e.g.α=0.2. If we take all thehksmaller than
the smallest period, then the slower modes would evolve at a much smaller rate than
the fast modes. By adopting convention (15.105), the slow modes evolve at exactly
the same rate as the fast ones. Therefore, critical slowing down will be completely
eliminated for the free field model. For the interacting field with aφ^4 term present,
the time steps are taken according to (15.105), but with the renormalised mass
occuring in the denominator [17].
A remark is in place here. As the method uses finite time steps, it is not the
continuum action which is simulated, but the discrete version which deviates to
some order of the time steps from the continuum one. Therefore, comparisons with
MC or hybrid algorithms are not straightforward. The time steps chosen here are
such that the time step error is divided homogeneously over the different modes.
The algorithm for a step in the Fourier-accelerated Langevin method is as follows:
ROUTINE LangStep(ψ)
Calculate forcesFnin real space;
FFT:Fn→Fk;
FFT:φn→φk;
Generate random forcesRk;