Vol. 10 (2017)

Acta Physica Polonica B Proceedings Supplement

No 3

POLYAKOV LOOP FLUCTUATIONSIN TERMS OF DIRAC EIGENMODES∗Takahiro M. Doia , Krzysztof Redlichb , Chihiro Sasakib
Hideo Suganumaaba
Department of Physics, Kyoto University, Kyoto, Japan
Institute of Theoretical Physics, University of Wrocław, Wrocław, Poland(Received February 16, 2017)
We investigate, both numerically and analytically, the Polyakov loop
and its fluctuations in terms of Dirac eigenmodes in the lattice QCD. Basedon the analytical relation between the Polyakov loop and the Dirac eigenmodes, it is found that the Polyakov loop and its fluctuations, contrary tothe chiral condensate, are insensitive to the density of the low-lying Diracmodes. This result suggests that there is no direct one-to-one correspondence between quark confinement and chiral symmetry breaking in QCD.
Our argument is also generalized for the Wilson-type Dirac operator on thelattice.
DOI:10.5506/APhysPolBSupp.10.543

1. Introduction
Quark confinement and chiral symmetry breaking are important nonperturbative phenomena to understand the rich properties of the QCD vacuum.
In spite of their importance, these phenomena and their relations are still
not sufficiently understood.
In many lattice-QCD calculations at finite temperature both quark deconfinement and chiral restoration occur in the same temperature range [1,2].
However, this result does not necessarily imply that quark confinement andchiral symmetry breaking are strongly correlated, as there are some opposite findings. For example, in Ref. [3], it was shown that for physical valuesof quark masses, the deconfinement crossover temperature is significantly
higher than the chiral crossover temperature. In addition, even after removing the low-lying Dirac modes, “hadrons” as bound state of quarks andgluons can be still observed on the lattice [4], and the Polyakov loop andthe string tension are almost unchanged [5].
∗Presented at the “Critical Point and Onset of Deconfinement” Conference, Wrocław,Poland, May 30–June 4, 2016.

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T.M. Doi et al.

In this paper, we present the analytic relation between the Polyakov loop
fluctuations and Dirac eigenmodes, and show that the low-lying Dirac modes
have negligible contribution to the Polyakov loop fluctuations. Finally, as
a recent progress, we generalize our argument for the Wilson-type Diracoperator on the lattice in order to remove the fermionic doubler modes.
2. Polyakov loop fluctuations and Dirac eigenmode
In this section, we introduce the Polyakov loop fluctuations [6] and the
Dirac eigenmodes on the lattice [5].
We consider the SU(Nc ) lattice QCD on a standard square lattice withspacing a, and the notation of sites s = (s1 , s2 , s3 , s4 ) (sµ = 1, 2, · · · , Nµ ).
The link-variables Uµ (s) = eiagAµ (s) are expressed by gauge fields Aµ (s) ∈SU(Nc ) and gauge coupling g. The lattice is taken to be spatially isotropic,i.e., N1 = N2 = N3 ≡ Nσ and N4 ≡ Nτ , with Nσ ≤ Nτ . All γ-matrices aredefined to be hermitian, thus γµ† = γµ .
We define the link-variable operator Û±µ by the matrix element as
hs|Û±µ |s0 i = U±µ (s)δs±µ̂,s0with

hs, Nt |Û4 |s, 1i = −U4 (s, Nt ) . (1)The temporal anti-periodicity of the link-variable operator is imposed forthe temporal anti-periodicity of the Dirac operator at finite temperature.
Using the link-variable operators, the Polyakov loop L and the Z3 rotated
Polyakov loop L̃ are defined ason1
L≡−
Trc Û4Nτ ,L̃ ≡ Le2πki/3 ,(k = 0, ±1) ,(2)Nc V
where V is the four dimensional volume of the lattice and k is taken so thatL̃ belongs to the real sector. Then, the three types of the Polyakov loop
susceptibilities are introduced
T 3 χY =iNσ3 h 22Y−hYi,Nτ3(Y = |L|, LL , LT ) ,(3)where LL ≡ Re(L̃), LT ≡ Im(L̃), and hxi denotes an average over all gauge
configurations.
We introduce two ratios of the Polyakov loop susceptilitities, RA ≡ χχALand RT ≡ χχTL . These ratios are superior to the Polyakov loop itself intheir properties for the renormalization, because the ambiguities from therenormalization scheme can be avoided. Also, from the lattice QCD calculations, it is found that the ratio RA is very sensitive probe for the quarkdeconfinement [6].

Polyakov Loop Fluctuations in Terms of Dirac Eigenmodes

545

Using the link-variable operators, the eigenvalue equation of the naive
Dirac operator D
6 ˆ on the lattice is expressed asD
6 ˆ |ni = iλn |ni ,1D
6ˆ =2a4X

γµ Ûµ − Û−µ .

(4)

µ=1

The Dirac eigenvalue iλn is purely imaginary because PD
6 ˆ is anti-hermitian.
These Dirac eigenstates |ni have the completeness of n |nihn| = 1. The
low-lying modes of the Dirac eigenmodes are important modes for chiralsymmetry breaking because they have the dominant contribution to thechiral condensate hψ̄ψi known as the Banks–Casher relation [7].
3. Polyakov loop fluctuations in terms of Dirac eigenmode
In the following, we introduce the analytical relations connecting thePolyakov loop fluctuations and the Dirac eigenmodes on the lattice. We
then discuss the relation between the chiral symmetry breaking and quark
confinement.
The analytical relation between the Polyakov loop and the Dirac eigenmodes in lattice QCD was derived in Refs. [8] and [9]. In particular, on thetemporally odd-number lattice, where the temporal length of the lattice isodd number, the above relation is rather transparent and reads
L=−(2ai)Nτ −1 X Nτ −1λnhn|Û4 |ni .
12Vn(5)

The Dirac matrix element of the link-variable operator hn|Û4 |ni is finite,thus |hn|Û4 |ni| < 1. From Eq. (5) and the definitions of the Polyakov loop
fluctuations, one can also express different Polyakov loop susceptibilities by
the Dirac eigenmodes [10]. In particular, the important quantity RA can be
expressed in terms of the Dirac modes as DP Nτ −1 nn 2P Nτ −1 nn E2λnλn
Û4Û4−RA = DP
E2 ,P Nτ −1  2πki/3 nn 2Nτ −1nn2πki/3λnRe eÛ4−λnRe eÛ4(6)
where we have defined Û4nn = hn|Û4 |ni, and the sum is taken over n.
τ −1 , theFrom Eq. (6), it is clear that due to the damping factors λNnP
low-lying Dirac modes have relatively small contribution to each sum nover Dirac modes. Consequently, the ratio RA should be insensitive to the546

T.M. Doi et al.

density of the low-lying Dirac modes. Indeed, this analytical observation
indicating that there is no direct one-to-one correspondence between quark
confinement and chiral symmetry breaking in QCD was confirmed within
the numerical calculations in lattice QCD [10].
4. Removal of the doubler modes
In this section, we improve the above results by removing the fermionicdoubler modes on the lattice. In fact, the naive Dirac operator (4) includes
the unphysical modes which are referred to as fermionic doubler modes [11].
An easy procedure for removing the doubler modes is to use the Wilson–
Dirac operator defined as
4i1 XhD̂W = 4r/a −(r − γµ )Ûµ + (r + γµ )Û−µ ,
2a(7)

µ=1

where r is the Wilson parameter. The eigenvalue equation of the Wilson–
Dirac operator is written as
D̂W |niW = Λn |niW ,Λn ∈ C .

(8)

We consider the temporally odd-number lattice and we set the temporal
size as Nτ = 2N + 1. The derivation of the relation between the Polyakovloop and the Wilson–Dirac modes is essentially the same as the derivationof Eq. (5) [9]. The actual form of the relation is expressed as
L=2aN X(Λn )N hn|Û4N +1 |niW ,3V n(9)where we set r = 1 and abbreviate the bra-vector W hn| to hn|. This analytical
relation connects the Polyakov loop and the Wison–Dirac modes. Thus, it
is confirmed that the low-lying Wilson–Dirac modes with |Λn | ∼ 0 also havenegligible contribution to the Polyakov loop, and then its fluctuations.
From the viewpoint of the chiral symmetry on the lattice, the overlapDirac operator is better than the Wilson–Dirac operator [12]. An attempt todiscuss the relation between the Polyakov loop and the overlap-Dirac modes
is in progress.
5. Conclusion
We have derived a set of the analytical relations connecting the Polyakov
loop fluctuations and the Dirac eigenmodes for the naive discretization andthe Wilson-type of the Dirac operator. These analytical relations show that

Polyakov Loop Fluctuations in Terms of Dirac Eigenmodes547

the low-lying Dirac modes have little contribution to the Polyakov loopfluctuations. Our result suggests that there is no one-to-one correspondence
between quark confinement and chiral symmetry breaking in QCD.
T.M.D. and H.S. are supported by the Grant-in-Aid for JSPS Fellows(grant No. 15J02108) and the Grants-in-Aid for Scientific Research (grantNo. 15K05076) from Japan Society for the Promotion of Science. The workof K.R. and C.S. has been partly supported by the National Science Centre,Poland (NCN) under Maestro grant No. DEC-2013/10/A/ST2/0010. The
lattice QCD calculations were performed on NEC-SX8R and NEC-SX9 atthe Osaka University.

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