59

沼津工業高等専門学校研究報告
第 56 号，2022 年 2 月

Numerical Evaluation of the Lattice Green Functions
KOMA Yoshiaki ∗Abstract: The Green functions take the roles of propagators in quantum ﬁeld theories, which
provide information on the interaction between particles. We evaluate the Green functions for a
massless boson deﬁned on a three-dimensional Euclidean lattice numerically, and investigate thelattice eﬀects by comparing the results with the continuum ones. We ﬁnd an inevitable diﬀerence
at very short distances quantiﬁed approximately by an exponential function.
Key Words: Green function, Propagator, Lattice formulation
1.

In this report, we present part of our numerical

Introduction

Revealing the properties of quantum ﬁeld theories often requires numerical methods, and a lattice
formulation of spacetime oﬀers a powerful method forthis purpose. In this formulation, ﬁeld variables of atheory are treated as the site or link variables, whichallows us not only to solve the ﬁeld equations by using
numerical algorithms but also to compute the parti-

results on the behavior of the lattice Green functions
for a massless boson in three-dimensional Euclidean
space both in ﬁnite and inﬁnite volumes. We thencompare these results with those in the continuum
theory. We partially resort to the coordinate spacemethod to evaluate the lattice Green functions in inﬁnite volume as demonstrated in [1–3].

tion function that contains the quantum eﬀects by using the Monte-Carlo method. Only the caveat may be

2.

The lattice Green functions

that the numerical results obtained in this way suﬀer

The lattice Green function for a massless bo-from systematic errors due to ﬁnite cutoﬀ and ﬁnite

son in inﬁnite volume in the coordinate representation

volume. This is because one cannot perform computa-

G∞ (x) is generally deﬁned by the equation

tion practically in inﬁnite volume with inﬁnitesimallysmall lattice spacing. Therefore, it is always impor-

ΔG∞ (x) = −δx0 ,(1)

tant to control the lattice artifacts in the numerical

where Δ denotes the lattice Laplacian and δx0 is theresults.

Kronecker delta, δx0 = 1 for x = 0 and δx0 = 0 for

The evaluation of the Green functions deﬁned
on the lattice, which we call the lattice Green functions, may give some hints. The Green functions takethe roles of propagators in quantum ﬁeld theories,x �= 0. In the N -dimensional Euclidean space, theleft-hand side of Eq. (1) is written as
ΔG∞ (x) =which provide information on the interaction between=

particles. For a massless boson, such as the photon
in quantum electrodynamics, carrying momentum pμ ,the propagator is given by 1/p2 in the momentum representation, and the Fourier transform corresponds tothe well-known Coulombic potential ∼ 1/r. If one
dares to compute the same propagator and potential

on the lattice, however, the results could be diﬀerent
from those in the continuum and inﬁnite volume limits
in some respects. Thus, the comparison of the lattice
results with the continuum ones that can be obtained
analytically will be useful to expose the lattice artifacts.

N


μ=1
N


μ=1

∇∗μ ∇μ G∞ (x)[G∞ (x+ μ̂)+G∞ (x− μ̂)−2G∞ (x)] ,(2)where ∇μ and ∇∗μ are the forward and backward diﬀer-

ences to a direction μ, respectively. The lattice spacing is set to one throughout this report.
The solution of Eq. (1) is formally obtained by
performing the Fourier transformation of the Greenfunction in the momentum representation. On thelattice, momentum carried by a particle is deﬁned bypμp̂μ = 2 sin,(3)2
where pμ ∈ [−π, π]. With this lattice momentum theGreen function in the momentum representation is
given by∗ Division of Liberal Arts

G̃∞ (p) =

1
1
1

= 
, (4)
pμ =
2
2
p̂
2N − 2 μ cos pμ
4 μ sin 2

60

沼津工業高等専門学校研究報告

the idea is based on Vohwinkel’s observation that theand the Fourier transform is
 π N
d p ipx
G∞ (x) =e G̃∞ (p) ,N−π (2π)where p x =第 56 号

lattice Green function satisﬁes a relation
 π N
d p ipx(∇∗μ + ∇μ )G∞ (x) = xμe ln(p̂2 ) . (11)N−π (2π)(5)N

μ=1 pμ xμ .

Summing over all directions μ in Eq. (11), one obtains

In ﬁnite volume with periodic boundary condi-

sort of a recursion relation

tions in all directions, pμ in Eq. (3) is discretized aspμ = 2πnμ /Lμ with integers nμ = 0, 1, 2, ..., Lμ − 1.

N

The Green function in the momentum representation

G∞ (x+μ̂) = G∞ (x−μ̂)+

is then given by

(12)

1
1
1
G̃L (p) = 2 = 
, (6)

p =
p̂
4 μ sin2 2μ 2N −2 μ cos 2πnμwith ρ =L

and the Fourier transform is
μ1  i 2πnx
GL (x) =e Lμ μ G̃L (p) ,V

Eq. (1) at the origin leads toG∞ (1, 0, 0) = G∞ (0) −μ=1 Lμ . In order to avoid divide-by-zeromode nμ = 0 for all μ reluctantly, which yields a ﬁnite

symmetry. Moreover, the combination of the Greenfunctions,1
ΔGL (x) = −δx0 +
.
V

(8)

k(n) = (n−1)G∞ (n, 0, 0) + 2nG∞ (n, 1, 0)+(n+1)G∞ (n, 1, 1) − nG∞ (n−1, 0, 0)

The Fourier transformation of G̃L (p) is easily
achieved by using the FFT algorithm, while that−2(n−1)G∞ (n−1, 1, 0)−(n − 2)G∞ (n−1, 1, 1) , (14)of G̃∞ (p) needs subtle approach. One may rewrite

is found to be a constant of motion, satisfying k(n +Eq. (5) by using the modiﬁed Bessel functions In as
G∞ (x) =

dα e

0

(13)

applies to G∞ (0, 1, 0) and G∞ (0, 0, 1) due to cubic

volume correction in Eq. (1) as

N


1
,
6

where G∞ (0) = G∞ (0, 0, 0). The same relation also

computation, one may exclude the zero momentum−2N α

μ=1 xμ for ρ �= 0.

(7)

N

 ∞

N

In three dimension [2], the direct evaluation ofnμ �=0where V =2xμ 
[G∞ (x)−G∞ (x−ν̂)]ρ ν=1Ixμ (2α) ,1) = k(n) for n ≥ 1, and the behavior of the Greenfunctions at n → ∞ leads to k(n) = 0. By setting(9)n = 1 in Eq. (14), one further obtains a relation

μ=1

and evaluate this integral numerically. However, theG∞ (1, 1, 1) =convergence property of the integration depends on

1(G∞ (0) − 3G∞ (1, 1, 0)) .
2(15)

the spatial dimension N , since the modiﬁed Bessel

Likewise the Green functions at any distances can befunction behaves as1 2α
In (2α) ∼e
4παexpressed as(10)

G∞ (x) = r1 (x)G∞ (0)+r2 (x)G∞ (1, 1, 0)+r3 (x) , (16)for large α, where the integrand will be of O(α−2 ) for

where r1 (x), r2 (x), and r3 (x) are rational numbers

N = 4, O(α−3/2) for N = 3, and O(α−1determined recursively by using Eq. (12).

) for N = 2.

The two inputs G∞ (0) and G∞ (1, 1, 0) in

These asymptotic behaviors indicate that it may be
diﬃcult to obtain precise numerical values for lower

Eq. (16) are also computed from the recursion re-dimensions N ≤ 3 as the convergence is expected tolation itself.

be slow.

In Table 1, we show the values of

G∞ (0) and G∞ (1,1,0) obtained by solving simulta-

An interesting way to avoid the convergenceneous equations for G∞ (n, 0, 0) and G∞ (n, 1, 0) at

long distances n → ∞, which are expected to be-

problem is to use the coordinate space method ashave as G∞ (n, 0, 0) ∼ 1/(4πn) and G∞ (n, 1, 0) ∼demonstrated in Refs. [1–3]. According to Ref. [1],261

Numerical Evaluation of the Lattice Green Functions

Table 1: The G∞ (0) and G∞ (1, 1, 0) determined by setting the maximum number n.
n G∞ (0)20 0.2527310098586630026512899780573065761804420557026865017
40 0.2527310098586630030260020266135700840185454806081835215
60 0.2527310098586630030260020266135701299925645074087004469
n G∞ (1, 1, 0)
20 0.0551914336877373167861103588492548550092182265478188015
40 0.0551914336877373170165449460300639096093757534683259826
60 0.0551914336877373170165449460300639378812455101391984498√
1/(4π n2 + 1), respectively. Clearly, the values show

0.30

convergence for large n. The presented values seem tobe unnecessarily lengthy, but they should be as pre0.20

cise as possible in order to reduce accumulation ofrounding errors during the numerical recurrence. If
it is possible to perform computation with quadruple

0.10

precision, the values determined at n = 40 can be

GL (0)
G∞ (0)

used, where the diﬀerence of the lattice Green function from the continuum one at n = 40 is expected to

0.00

be of O(1/n2 ) ∼ 10−3 . It is interesting to note thatthe value of G∞ (0) determined in this way agrees with
√

50 100 150 200 250 300 350 400 450 500
L

Fig. 1: The behavior of GL (0) as a function of L, which iscompared with G∞ (0) (dotted line).

the half of the Watson integral I3 [4, 5],
I3=
G∞ (0) =20  1 2  11 2
3 − 1 Γ 24
Γ 24
. (17)
3192π

a constant gap even after taking the limit L → ∞.

As shown in Fig. 3, however, we ﬁnd that oncethe values at the origin are subtracted from the lat3.

Numerical results

tice Green functions, they seem to fall into one curve
corresponding to G(r)−G∞ (0). We look at the diﬀer-

We shall present numerical results in three dimensions. In the continuum case, the Green function

ence between them carefully in Fig. 4, which clearlyat the origin is divergent due to the 1/r behavior. In

shows that the diﬀerence tends to disappear as thecontrast, the lattice Green function at the origin issize L is increased. Only at very short distances foralways ﬁnite both in ﬁnite and inﬁnite volumes. Inr ≤ 5 there is an inevitable diﬀerence. A chi-square

Fig. 1, we plot the values of GL (0) as a function of L,

ﬁtting analysis for the inﬁnite volume data indicateswhere the dotted line corresponds to G∞ (0). We ﬁndthat the diﬀerence approximately obeys an exponential functionthat GL (0) depends on the size L, which approaches
G∞ (0) from below as the size L is increased.

f (r) = 0.0151(3)e−0.84(2)r .

tions both in ﬁnite and inﬁnite volumes, GL (r) ≡In ﬁnite volume, Eq. (13) is modiﬁed to

In Fig. 2, we plot the on-axis lattice Green func-

GL (r, 0, 0) and G∞ (r) ≡ G∞ (r, 0, 0), which are com-

11
GL (1, 0, 0) = GL (0) − (1 − ) ,6Vpared with the continuum Green function G(r) =

(18)

(19)

1/(4πr). The G∞ (r) seems to be identical to G(r)where the ﬁnite volume correction originates from theexcept for the distances near the origin, while GL (r)removal of the zero momentum mode as already ex-

are clearly diﬀerent from G(r) and are dependent on

plained. Thus, the ratio of GL (1) − GL (0) to G∞ (1) −G∞ (0) is just given by 1 − 1/V . For L = 16 it gives

the size L. Although GL (r) approaches G(r) gradu-

0.9998, so that the relative error is only of 0.02 %.

ally as the size L is increased, there still seems to be3

62

沼津工業高等専門学校研究報告

0.10

Although analytical relations between GL (r) − GL (0)G16 (r)G32 (r)G64 (r)G∞ (r)
G(r)0.05

第 56 号

and G∞ (r) − G∞ (0) for r ≥ 2 are not known yet, theresults at small r in Fig. 4 indicate that the ﬁnite volume eﬀect at short distances can easily be controlled.
Whether or not one can control the ﬁnite volume effect at long distances depends on available computerresources. The results suggest that the size L should

0.00

be taken at least four times larger than the distance
0

5

10

15
r

20

25

of interest, since the eﬀect of periodic boundary con-30

ditions appears around the distances of L/4.
On the other hand, the control of the ﬁnite cutoﬀ

Fig. 2: The behaviors of the lattice Green functions
GL (r) and G∞ (r), which are compared with the continuum Green function G(r) = 1/(4πr).

eﬀect as characterized by Eq. (18) may cause problem when one aims to clarify short-distance properties with massive particles. In a quantum ﬁeld the-0.00
G16 (r) − G16 (0)G32 (r) − G32 (0)G64 (r) − G64 (0)G∞ (r) − G∞ (0)G(r) − G∞ (0)−0.10

ory that the propagator is given by 1/(p2 + m2 ) withmass m, the Fourier transform corresponds to the
Yukawa-type potential ∼ e−mr /r. In this case, theexponential function e−mr certainly reﬂects a physicaleﬀect. What is complicated is that, in some of quan-−0.20

tum ﬁeld theories, masses are dynamically generated
by the quantum eﬀects. If this is the case, the discrim-−0.30

ination of the ﬁnite cutoﬀ eﬀect is crucial, which will
0

5

10

15
r

20

25

30

require systematic investigation including the scaling
analysis.

Fig. 3: The behaviors of the normalized lattice Green functions, where the values at the origin are subtracted.

4.

·10−2

1

G16 (r)− G16 (0) −(G(r)−G∞ (0))
G32 (r)− G32 (0) −(G(r)−G∞ (0))
G64 (r)− G64 (0) −(G(r)−G∞ (0))
G∞ (r)− G∞ (0) −(G(r)−G∞ (0))

0.5

f (r) = 0.0151(3)eSummary

We have investigated the lattice Green functions
for a massless boson numerically both in ﬁnite and inﬁnite volumes in three-dimensional Euclidean space.

−0.84(2)r

The presented results exhibit an inevitable diﬀerence
from the continuum Green function at very short distances for r ≤ 5, which is quantiﬁed by using an ex-0ponential function. Similar analyses can be performed
also on the oﬀ-axis lattice Green functions. The con0

5

10

15
r

20

25

30

trol of the short-distance eﬀect in our actual latticecomputation [6] is in progress.

Fig. 4: The diﬀerence between the normalized lattice
Green functions and the normalized continuum Greenfunction.

The author is grateful to Miho Koma for fruitfulcollaboration.

[1] M. Lüscher, P. Weisz, Nucl. Phys. B445 (1995) 429-450.
[2] S. Necco, R. Sommer, Nucl. Phys. B622 (2002) 328-346.
[3] D.S. Shin, Nucl. Phys. B525 (1998) 457-482.
[4] G.N. Watson, Quart.J.Math., Oxford 10 (1939) 266-276.

[5] G.S. Joyce, I.J. Zucker, Proc. Amer. Math. Soc. 133(2005) 71-81.
[6] Y. Koma, E.-M. Ilgenfritz, T. Suzuki, and H. Toki,
Phys. Rev. D64 (2001) 014015.

4