79沼津工業高等専門学校研究報告
第 54 号，2020 年 1 月

On the eﬀective use of the multilevel algorithm
in SU(2) lattice gauge theory
KOMA Yoshiaki∗
We investigate the static quark-antiquark potential in SU(2) gauge theory by the lattice simulation
with the multilevel algorithm, and examine how the numerical errors of the potential are controlled
by the algorithm parameters. We ﬁnd that one of the key parameters, the physical time extent ofsublattices, is optimal when it is chosen to be around 0.37 fm, which is quite similar to that alreadyobserved in SU(3) lattice gauge theory.
Key Words: quark-antiquark potential, lattice gauge theory
INTRODUCTION

structed with a sequential chain of independent vac-

For the understanding of dynamics of quarks anduum conﬁgurations, they are accompanied by numer-gluons inside hadrons a quantum gauge ﬁeld theory

ical errors. That is to say, one always faces problem

with non-Abelian SU(2) group symmetry, which weof accuracy of the numerical result. Therefore, it usu-

just call the SU(2) gauge theory, has often been inves-

ally takes longer computer time to obtain solid results.

tigated instead of realistic quantum chromodynamics

Even the quark-antiquark potential, which is a rather(QCD) with SU(3) group symmetry [1]. This is be-well-known quantity, has still room for improvement

cause that the SU(2) group corresponds to a subgroupon the numerical accuracy.

I.

of SU(3), and is expected to describe some of essen-

In this report, we present some of preliminary re-tial features of QCD. Numerical investigation is usu-sults on the quark-antiquark potential in SU(2) gauge

ally needed for the understanding of nonperturbative

theory obtained by the lattice simulation with theproperties of the theory, and a powerful method for

multilevel algorithm [3, 4]. We compute the Polyakov

this purpose is to deﬁne the theory on a hypercubic

loop correlation function (PLCF), which is deﬁned

discrete lattice, and then compute its partition func-

by a pair of spatially separated Polyakov and anti-tion by a Monte-Carlo method using supercomputers.

Polyakov loops, and extract the ground state poten-We call this method the lattice simulation.

tial straightforwardly. We then examine how the nu-The lattice simulations generally provide us withmerical errors of the potential are controlled by theexpectation values of physical quantities. The staticalgorithm parameters, aiming to ﬁnd an optimal set

quark-antiquark potential (a variation of the poten-of parameters to achieve eﬃcient error reduction.

tial energy as a function of the distance between aquark and an antiquark) is one of the typical quanti-II.

NUMERICAL PROCEDURES

ties computed nonperturbatively by the lattice simulation. Once the potential is provided, it is possible

We consider SU(2) gauge theory in four dimen-

to investigate the bound state of a quark and an anti-

sions with lattice volume L3 ×T and lattice spacing a,quark by solving the Schrödinger equation, which are

and impose periodic boundary conditions in all space-then compared to meson spectra observed by high-time directions. We use the standard Wilson gauge

energy accelerator experiments [2]. The functional

action composed of link variables Uμ (x) ∈ SU(2) as1S = β {1− ReTr[Uμ(x)Uν(x+ μ̂)Uμ†(x+ ν̂)Uν†(x)]}.(1)2x,μ<ν

form of the potential is also used to understand thenontrivial vacuum structure.
However, since the expectation values are generally given by statistical averages of operators con-Basic building blocks of the PLCF are two-linkcorrelators deﬁned byT(x0 , �x1 , �x2 )αβγδ ≡ U0 (x0 , �x1 )αβ U0† (x0 , �x2 )γδ , (2)∗ Division of Liberal Arts / Physics groupwhere the Greek indices of U0 (x) label the component of a 2 × 2 complex matrix for the SU(2) group.

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沼津工業高等専門学校研究報告

第 54 号

A two-link correlator is nothing but the direct prod-

multilevel algorithm, which allows us to overcome theuct of two temporal link variables separated by a dis-problem on the smallness of the PLCF.

tance r = |�x1 −�x2 |. The two-link correlator acts on a

The idea of the algorithm is to compute a corre-color state in the 2 ⊗ 2̄ representation of the SU(2)lation function from the product of sublattice averages

ber n, which satisﬁes T(x0 , �x1 , �x2 )αλγ� |n; �x1 , �x2 �αγ =viding the lattice volume into several layers along theto the energy. The multiplication law of two-link cor-averages, spatial links at the sublattice boundaries aregroup |n; �x1 , �x2 �αβ with the principal quantum num-of its components, where a sublattice is deﬁned by di-−En (r)a|n; �x1 , �x2 �λ� , where En (r) (> 0) corresponds

time direction. During the computation of sublattice

relators between two neighboring time slices x0 andﬁxed. The computation of correlation functions in thisx0 + a isway is supported by the transfer matrix formalism. Ineorder to use the multilevel algorithm eﬃciently, how-{T(x0 , �x1 , �x2 )T(x0 + a, �x1 , �x2 )}αβγδ

ever, the number of time slices in a sublattice Ntsl and= T(x0 , �x1 , �x2 )αλγ� T(x0 + a, �x1 , �x2 )λβ�δ , (3)the number of internal updates Niupd must be chosen
appropriately depending on the gauge coupling β inwhere the repeated Greek indices λ and � are to beEq. (1).

summed over from 1 to 2. The PLCF is ﬁnally constructed asIII.

NUMERICAL RESULTS

TrP (�x1 )TrP (�x2 )∗Our previous study of SU(3) lattice gauge theory= {T(0, �x1,�x2 )T(a, �x1,�x2 )· · ·T(T −a, �x1,�x2 )}ααγγ . (4)showed that an optimal choice of aNtsl was around

Based on the transfer matrix formalism, the ex-

0.37 fm (1 [fm] = 10−15 [m]) [5, 6]. By taking into ac-pectation value of the PLCF is evaluated formally bycount this observation, we have performed an exten-inserting the complete set of eigenstates at all timesive test of the algorithm parameters in SU(2) latticeslices x0 = 0, a, ..., T − a, which leads to a relation

gauge theory. In this report, we present a part of the�TrP (�x1 )TrP (�x2 )∗ � =∞
e−En (r)T .

results at β = 2.45 and 2.55 for Ntsl = 4 (see, Table I),(5)and demonstrate how the choice of parameters aﬀects

n=0

the eﬀectiveness of the algorithm.

The ground state potential, V (r) ≡ E0 (r), is thenIn Figs. 1 – 4, we show numerical results of theextracted byhistory of the PLCF, the potential, the force, and the1ln�TrP (�x1 )TrP (�x2 )∗ �+O(e−(ΔE)T ). (6)Trelative error of the potential at β = 2.45 (left) andThe systematic error term of O(e−(ΔE)T ) is associatedthe on-axis data for the PLCF such that the relativewith the excited states, which is always negligible atposition vector between a quark and an antiquark iszero temperature as the size of T is usually taken large

�r/a = (n, 0, 0) with n = 1, 2, 3..., we plot both theenough. Once the potential is computed accurately,

on-axis and the oﬀ-axis data such as �r/a = (n, n, 0),

it is also possible to investigate the force from the�r/a = (n, n, n), �r/a = (2n, n, 0) for other results (per-derivative (diﬀerence) as

mutation of the x, y, and z axes is also taken into ac-V (r) = −F (r) =dV (r)V (r + a) − V (r − a)=.
dr2aβ = 2.55 (right), respectively. While we only plot

(7)

TABLE I: The list of simulation parameters. The Sommerscale r0 is determined so as to satisfy r02 F (r0 ) = 1.65,
where r0 is often regarded as 0.50 fm.

A numerical problem is that it is impossible tocompute the PLCF in Eq. (5) accurately within theβ L3 T Ntsl Niupd Ncnf r0 /a a [fm] aNtsl [fm]2.45 324 4 50000 20 5.37 0.0931 0.3724
2.55 324 4 50000 20 7.476 0.0669 0.2676

ordinary lattice simulations as the expectation values
are extremely small at long distances, which are easily
obscured by the statistical noise. We then employ the281

On the effective use of the multilevel algorithm in SU(2) lattice gauge theory

101010-4-410

L32T32b245-6

L32T32b255-6

10-8-810<PP>

PLCF-1010-1210-10

10-14

10-12

10-16

10-1410-1810-20-1610

323456 7 8 9

1042

34

1053

10

23456 7 8 9

1042

34510

Niupd

Niupd

FIG. 1: The history of the PLCF as a function of Niupd at β = 2.45 (left) and β = 2.55 (right) for all Ncnf = 20
conﬁgurations. The bunch of plateau from upper to lower corresponds to the quark-antiquark distance r/a = 1, 2, ...,
respectively, where only the on-axis data are plotted.
2.0

2.0
L32T32b245

L32T32b255
1.5
a V (r)a V (r)1.5

1.0(n,0,0)(n,n,0)(n,n,n)(2n,n,0)0.5

1.0(n,0,0)(n,n,0)(n,n,n)(2n,n,0)

0.5

0.0

0.0
05101520

0510

r/a1520

r/aFIG. 2: The potential as a function of the quark-antiquark distance r/a at β = 2.45 (left) and β = 2.55 (right), where not
only the on-axis data but also the oﬀ-axis data are plotted.

to r/a ∼ 14 at β = 2.55. For the force the result atcount to increase the statistics). The numerical errors

β = 2.45 is clearly better than that at β = 2.55 up

in Figs. 2 and 3 are evaluated by the single-elimination

to long distances as shown in Fig. 3. This tendency isjackknife method with Ncnf = 20 conﬁgurations.
Fig. 1 exhibits that it is possible to computeof course reasonable since the force is computed fromextremely small expectation values by increasing thethe diﬀerence of the potential. The scatter plot of

Niupd only when the physical temporal length aNtsl is

the relative error of the potential in Fig. 4 explicitly

optimally chosen. For the result at β = 2.45 the orderdemonstrates the eﬀectiveness of the multilevel algo-−18).

rithm on the choice of algorithm parameters. The

For the result at β = 2.55, on the other hand, theeﬀect of the error reduction is better at β = 2.45 thansmallest plateau of the PLCF is of O(10−14 ) and itthat at β = 2.55, especially at long distances.

of magnitude of the smallest plateau is of O(10seems diﬃcult to obtain further small expectation values even if the Niupd is increased. Note that the value
IV.

of aNtsl at β = 2.55 is about 30% smaller than thatSUMMARY

at β = 2.45. The potential in Fig. 2 is computed byusing the values of PLCF at Niupd = 50000. The re-We have investigated the static quark-antiquarksult is clean up to r/a ∼ 18 at β = 2.45, while up

potential in SU(2) gauge theory by using the latticesimulation with the multilevel algorithm [3, 4], and3

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沼津工業高等専門学校研究報告

0.20

第 54 号

0.20
L32T32b245

L32T32b255(n,0,0)(n,n,0)(n,n,n)(2n,n,0)a V' (r)0.15

a V' (r)0.15(n,0,0)(n,n,0)(n,n,n)(2n,n,0)

20.10

20.10

0.05

0.05

0.00

0.00
0510

15

20

0510

r/a15

20

r/a

FIG. 3: The force as a function of the quark-antiquark distance r/a at β = 2.45 (left) and β = 2.55 (right), where not only
the on-axis data but also the oﬀ-axis data are plotted.
33
L32T32b255

2( V(r)-Vave)/Vave * 100( V(r)-Vave)/Vave * 100

L32T32b245

10-1-2-3

2
10-1-2-30

51015

200r/a5101520r/aFIG. 4: The relative error of the potential (the relative diﬀerences of the potentials between each of Ncnf = 20 conﬁgurations
and their average) as a function of the quark-antiquark distance r/a at β = 2.45 (left) and β = 2.55 (right), where not only
the on-axis data but also the oﬀ-axis data are plotted.

the optimal condition aNtsl  0.37 fm [5, 6], which ishave examined how the numerical errors of the potential are controlled by the algorithm parameters, whereobserved in SU(3) lattice gauge theory, seems to holdthe key parameters are the number of time slices inalso in SU(2) lattice gauge theory.

a sublattice Ntsl and the number of internal updates

The author is grateful to M. Koma for fruitful

Niupd . We have compared the results of the PLCF,collaboration. The lattice simulations have been per-the static potential, the force, and the relative error

formed on the NEC SX-ACE at Research Center forof the potential at β = 2.45 and 2.55 for Ntsl = 4, andNuclear Physics (RCNP) and Cybermedia center, Osaka University.

have found that Ntsl = 4 is suitable for β = 2.45 andunsuitable for β = 2.55. These results suggest that[1] W. Greiner and A. Schäfer, Quantum chromodynamics,
Springer-Verelag, Berlin and Heidelberg, 1994.
[2] M. Tanabashi et al. (Particle Data Group), Phys.
Rev. D 98 (2018) 030001.
[3] M. Lüscher and P. Weisz, JHEP 0109 (2001) 010.

[4] M. Lüscher and P. Weisz, JHEP 0207 (2002) 049.
[5] Y. Koma, M. Koma, and H. Wittig, Phys. Rev. Lett.
97 (2006) 122003.
[6] Y. Koma and M. Koma, Nucl. Phys. B769 (2007)
79–107.

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