Relation between Confinement and Chiral Symmetry Breaking
in Temporally Oddnumber Lattice QCD
Abstract
In the lattice QCD formalism, we investigate the relation between confinement and chiral symmetry breaking. A gaugeinvariant analytical relation connecting the Polyakov loop and the Dirac modes is derived on a temporally oddnumber lattice, where the temporal lattice size is odd, with the normal (nontwisted) periodic boundary condition for linkvariables. This analytical relation indicates that lowlying Dirac modes have little contribution to the Polyakov loop, and it is numerically confirmed at the quenched level in both confinement and deconfinement phases. This fact indicates no direct onetoone correspondence between confinement and chiral symmetry breaking in QCD. Using the relation, we also investigate the contribution from each Dirac mode to the Polyakov loop. In the confinement phase, we find a new “positive/negative symmetry” of the Diracmode matrix element of the linkvariable operator, and this symmetry leads to the zero value of the Polyakov loop. In the deconfinement phase, there is no such symmetry and the Polyakov loop is nonzero. Also, we develop a new method for spindiagonalizing the Dirac operator on the temporally oddnumber lattice modifying the KogutSusskind formalism.
pacs:
12.38.Aw, 12.38.Gc, 14.70.DjI Introduction
Color confinement and chiral symmetry breaking are very important phenomena in nuclear and elementary particle physics and have been investigated as interesting nonperturbative phenomena in lowenergy QCD in many analytical and numerical studies NJL (); KS (); Rothe (); Greensite (). However, their properties are not sufficiently understood directly from QCD. The Polyakov loop is an order parameter for quark confinement Rothe (). At the quenched level, the Polyakov loop is the exact order parameter for quark confinement, and its expectation value is zero in the confinement phase and nonzero in the deconfinement phase. As for chiral symmetry, the order parameter of chiral symmetry breaking is chiral condensate, and lowlying Dirac modes are essential for chiral symmetry breaking in QCD, for example, according to the BanksCasher relation BanksCasher ().
Not only the properties of confinement and chiral symmetry breaking in QCD but also their relation is an interesting challenging subject SST95 (); Miyamura (); Woloshyn (); Hatta (); Langfeld (); Gattringer (); Karsch (); YAoki (); GIS (); BG08 (); LS11 (). From some studies, it is suggested that confinement and chiral symmetry breaking are strongly correlated. In finite temperature lattice QCD calculation, some studies tell that the transition temperatures of deconfinement phase transition and chiral restoration are almost the same Karsch (). Also, by removing QCD monopoles in the maximally Abelian gauge, both confinement and chiral symmetry breaking are simultaneously lost in lattice QCD Miyamura (); Woloshyn (). However, there is an opposite study that the transition temperatures of deconfinement phase transition and chiral restoration are not the same YAoki ().
In recent latticeQCD numerical studies, it is suggested that the properties of confinement are not changed by removing lowlying Dirac modes from the QCD vacuum GIS (). Since lowlying Dirac modes are essential for chiral symmetry breaking, this calculation indicates that there is no onetoone correspondence between confinement and chiral symmetry breaking in QCD.
To investigate the relation between confinement and chiral symmetry breaking, the analytical relation between the Polyakov loop and Dirac modes is very useful. For example, the Polyakov loop is expressed in terms of Dirac eigenvalues under the twisted boundary condition for linkvariables Gattringer (). However, the (anti)periodic boundary condition is physically important for the imaginarytime formalism at finite temperature. Recently, we derived a relation between the Polyakov loop and Dirac modes on a temporally oddnumber lattice, where the temporal lattice size is odd, with the normal nontwisted periodic boundary condition for link variables SDI (); DSI ().
In this study, we analytically and numerically investigate the relation between confinement and chiral symmetry breaking. In Sec. II, we derive an analytical relation connecting the Polyakov loop and Dirac modes on the temporally oddnumber lattice. In Sec. III, we develop a new method for spindiagonalizing the Dirac operator applicable to the temporally oddnumber lattice modifying the KogutSusskind (KS) formalism KS (). In Sec. IV, for more detailed analysis, we perform the numerical analysis based on the relation. Section V is summary and discussion.
Ii the relation between the Polyakov loop and Dirac modes on the temporally oddnumber lattice
In this section, we derive the relation between the Polyakov loop and Dirac modes on the temporally oddnumber lattice with the normal (nontwisted) periodic boundary condition for linkvariables in both temporal and spatial directions SDI (); DSI ().
ii.1 Operator formalism and Dirac mode in lattice QCD
As the preparation, we review operator formalism and Dirac modes in the SU() lattice QCD. We use a standard square lattice with spacing , and the notation of sites , and linkvariables with gauge fields and gauge coupling . In this paper, we define all the matrices to be hermite as .
We define the linkvariable operator by the matrix element,
(1) 
where is the unit vector in direction in the lattice unit. Using the linkvariable operator, the Polyakov loop is expressed as
(2) 
with the 4D lattice volume . Here, “Tr” denotes the functional trace of with the trace over color index.
Also, using the linkvariable operator, covariant derivative operator on the lattice is expressed as
(3) 
Thus, in the lattice QCD, the Dirac operator is expressed as
(4) 
and its matrix element is explicitly expressed as
(5) 
with . Since the Dirac operator is antihermite in this definition of , the Dirac eigenvalue equation is expressed as
(6) 
with the Dirac eigenvalue () and the Dirac eigenstate . These Dirac eigenstates have the completeness of . According to , the chiral partner is also an eigenstate with the eigenvalue . Using the Dirac eigenfunction , the explicit form for the Dirac eigenvalue equation is written by
(7)  
The Dirac eigenfunction can be numerically obtained in lattice QCD, besides a phase factor. By the gauge transformation of , is gaugetransformed as
(8) 
which is the same as that of the quark field, although, to be strict, there can appear an irrelevant dependent global phase factor , according to arbitrariness of the phase in the basis GIS ().
The Diracmode matrix element of the linkvariable operator can be expressed with :
(9) 
Note that the matrix element is gauge invariant, apart from an irrelevant phase factor. Actually, using the gauge transformation Eq.(8), we find the gauge transformation of the matrix element as GIS ()
(10) 
To be strict, there appears an dependent global phase factor, corresponding to the arbitrariness of the phase in the basis . However, this phase factor cancels as between and , and does not appear for physical quantities such as the Wilson loop and the Polyakov loop GIS ().
Note also that a functional trace of a product of the linkvariable operators corresponding to the nonclosed path is exactly zero because of the definition of the linkvariable operator Eq.(1):
(11) 
with for the nonclosed path and the length of the path . This is easily understood from Elitzur’s theorem Elitzur () that the vacuum expectation values of gaugevariant operators are zero.
Dirac modes are strongly related to the chiral condensate according to the BanksCasher relation BanksCasher ():
(12) 
where the Dirac eigenvalue density is defined by
(13) 
with the spacetime volume . From Eq.(13), the chiral condensate is proportional to the Dirac zeroeigenvalue density. Since the chiral condensate is the order parameter of chiral symmetry breaking, lowlying Dirac modes are essential for chiral symmetry breaking. In general, instead of , one can consider any (anti)hermitian operator, e.g., , and the expansion in terms of its eigenmodes BI05 (). To investigate chiral symmetry breaking, however, it is appropriate to consider and the expansion by its eigenmodes.
Note here that, although the Polyakov loop is defined by gauge fields alone, there can be some relation to the Dirac modes, as will be shown later. For, the Dirac modes are strongly affected by the gauge fields. A similar example is instantons. The instantons are defined by gauge fields alone; however they have a close connection to the axial U(1) anomaly, which relates to a fermionic symmetry. In fact, even though the Polyakov loop is defined by gauge fields alone, it has a physical meaning to consider the relation to some fermionic modes in QCD.
The role of the lowlying Dirac modes has been studied in the context of chiral symmetry breaking in QCD. In particular, the removal of lowlying Dirac modes has been recently investigated to realize the world of “unbreaking chiralsymmetry” LS11 (); GIS (). For example, propagators and masses of hadrons are investigated after the removal of lowlying Dirac modes, and paritydoubling “hadrons” can be actually observed as bound states in the chiral unbroken world LS11 (). Also, after the removal of lowlying Dirac modes from the QCD vacuum, the confinement properties such as the string tension are found to be almost kept, while the chiral condensate is largely decreased GIS ().
ii.2 The relation between Polyakov loop and Dirac modes on the temporally oddnumber lattice
We consider the temporally oddnumber lattice, where the temporal lattice size is odd, with the normal (nontwisted) periodic boundary condition for linkvariables in both temporal and spatial directions. The spatial lattice size is taken to be even.
First, as a key quantity, we introduce
(14) 
with the functional trace including also the trace over spinor index. From Eq.(4), is expressed as a sum of products of linkvariable operators. In Fig. 1, an example of the temporally oddnumber lattice is shown and each line corresponds to each term in in Eq.(14). Here, note that one cannot make any closed loops using products of oddnumber linkvariable operators on a square lattice. Since now is odd and we consider the square lattice, does not have any operators corresponding to closed paths except for the term proportional to which corresponds to a closed path and is gauge invariant because of the periodic boundary condition for time direction, which is proportional to the Polyakov loop. Therefore using Eqs.(II.1), (3) and (11), we obtain
(15) 
On the other hand, taking Dirac modes as the basis for the functional trace in Eq.(14), we find
(16) 
Combining Eqs.(15) and (16), we obtain a relation between the Polyakov loop and the Dirac eigenvalues :
(17) 
This is a relation directly connecting the Polyakov loop and the Dirac modes, i.e., a Dirac spectral representation of the Polyakov loop. Since the Polyakov loop is gauge invariant and Dirac modes can be obtained gaugecovariantly, this relation is gauge invariant. From the relation (17), we can investigate each Dirac mode contribution to the Polyakov loop individually.
Since the relation (17) is satisfied for each gauge configuration, of course, the relation is satisfied for the gaugeconfiguration average:
(18) 
The outermost bracket means gaugeconfiguration average.
We can discuss the relation between confinement and chiral symmetry breaking in QCD from the relation (17). Dirac matrix element is generally nonzero. Thus, the contribution from lowlying Dirac modes with is relatively small in the sum of RHS in Eq.(17), compared to the other Diracmode contribution because of the damping factor . In fact, the lowlying Dirac modes have little contribution to the Polyakov loop. This is consistent with the previous numerical lattice result that confinement properties, such as interquark potential and the Polyakov loop, are almost unchanged by removing lowlying Dirac modes from the QCD vacuum GIS (). Thus, we conclude from the relation (17) that there is no onetoone correspondence between confinement and chiral symmetry breaking in QCD.
The relation (17) is valid only on the temporally oddnumber lattice, but this constraint is not so serious because we are interested in continuum QCD and the parity of the lattice size is not important for physics. In fact, by a similar manner on Eq.(17), we can also derive a relation which connects the Polyakov loop and Dirac modes on the even lattice (see Appendix A).
In the derivation of the relation (17), we use only the following setup:

odd

square lattice

temporal periodicity for linkvariables
Therefore, the relation (17) is valid in full QCD and in finite temperature and density, and furthermore regardless of the phase of the system. In other words, the relation (17) holds in confinement and deconfinement phases, and in chiral broken and restored phases. Of course, the dynamical quark effect appears in the Polyakov loop , the Dirac eigenvalue distribution , and the matrix elements . However, the relation Eq.(17) holds even in the presence of dynamical quarks.
For quantitative discussion, we numerically calculate each term in the relation (17) and investigate each Diracmode contribution to the Polyakov loop individually. Using Dirac eigenfunction , Dirac matrix element is explicitly expressed as Eq.(9). Thus, the relation (17) is expressed as
(19) 
Dirac eigenvalues and Dirac eigenfunctions in Eq. (19) can be obtained by solving the Dirac eigenequation (7) using link variables in each gauge configuration. However, the numerical cost for solving the Dirac eigenequation is very large because of the huge dimension of the Dirac operator . The numerical cost can be partially reduced without approximation using the KogutSusskind (KS) formalism KS () discussed in the next section.
Iii Modified KogutSusskind formalism for temporally oddnumber lattice
In our study, we need all the eigenvalues and the eigenmodes of the Dirac operator defined by Eq.(4). This can be numerically performed by the diagonalization of . Here, to reduce the numerical cost, we use the technique of the KS formalism for diagonalizing the Dirac operator . Note here that this procedure is just a mathematical technique to diagonalize , and this never means to use a specific fermion like the KS fermion. In fact, the diagonalization of is mathematically equivalent to the use of the KS formalism.
The KS formalism is the method for spindiagonalizing the Dirac operator on the lattice. However, when the periodic boundary condition is imposed on the lattice, the original KS formalism is applicable only to the “even lattice” where all the lattice sizes are even number. In this section, modifying the KS formalism, we develop the “modified KS formalism” applicable to the temporally oddnumber lattice DSI ().
iii.1 Normal KogutSusskind formalism for even lattice
First, we review the original KS formalism and consider the even lattice, where all the lattice sizes are even number. Using a matrix defined as
(20) 
one can diagonalize all the matrices ,
(21) 
where staggered phase is defined as
(22) 
Since the Dirac operator is expressed as , one can spindiagonalize the Dirac operator
(23) 
where the KS Dirac operator is defined as
(24) 
Equation (23) shows fourfold degeneracy of the Dirac eigenvalue relating to the spinor structure of the Dirac operator. Thus, one can obtain all the eigenvalues of the Dirac operator by solving the KS Dirac eigenvalue equation
(25) 
with the KS Dirac eigenstate . Since the KS Dirac operator has only indices of sites and colors, the numerical cost for solving the KS Dirac eigenvalue equation (25) is smaller than that for solving the Dirac eigenvalue equation (7). Using the KS Dirac eigenfunction , the KS Dirac eigenvalue equation (25) is explicitly expressed as
(26) 
Also, KS Dirac matrix element is expressed as
(27) 
Because of fourfold degeneracy of the Dirac eigenvalue, there are four states whose eigenvalues are the same, and we label these states with quantum number , namely, Rothe (). In this notation, the Dirac eigenvalue equation (7) is expressed as
(28) 
The relation between the Dirac eigenfunction and the spinless eigenfunction is
(29) 
where is defined as
(30) 
Substituting Eq. (30) for Eq. (29), one can obtain the relation
(31) 
and quantum number is mixed with spinor indices. This is natural result because the quantum number is caused by the fourfold degeneracy of the Dirac eigenvalue and is relating the spinor structure of the Dirac operator.
When one imposes the periodic boundary condition on the lattice, the KS formalism is applicable only to the even lattice. In fact, the periodic boundary condition of the matrix is expressed as
(32) 
and this relation is valid only on the even lattice. A spatial periodic boundary condition is not necessarily needed physically, but a temporal periodic boundary condition is needed for the imaginarytime finitetemperature formalism. Therefore, the original KS formalism is not applicable to the temporally oddnumber lattice.
iii.2 Modified KogutSusskind formalism for temporally oddnumber lattice
Now, we present the modified KS formalism as the generalization applicable to the temporally oddnumber lattice, where the lattice size for temporal direction is odd number and the lattice sizes for spatial direction are even number.
Instead of the matrix , we define a matrix by
(33) 
The matrix is similar to the matrix , but independent of the time component of the site . Using the matrix , all the matrices are transformed to be proportional to :
(34) 
where is the staggered phase given by Eq. (22). In the Dirac representation, is diagonal as
(35) 
and we take the Dirac representation in this paper. Thus, one can spindiagonalize the Dirac operator in the case of the temporally oddnumber lattice:
(36) 
where is the KS Dirac operator given by Eq. (24)
As a remarkable feature, the modified KS formalism with the matrix is applicable to the temporally oddnumber lattice. In fact, the periodic boundary condition for the matrix is given by
(37) 
and the requirement is satisfied for all the on the temporally oddnumber lattice because the spatial lattice sizes are even number and the matrix is independent of the time component of the site . Moreover, the periodic boundary condition for the staggered phase is satisfied on the temporally oddnumber lattice because the staggered phase is also independent of the time component of the site .
From Eq. (36), it is found that two positive modes and two negative modes appear for each eigenvalue , relating the spinor structure of the Dirac operator on the temporally oddnumber lattice. Note also that the chiral symmetry guarantees the chiral partner to be eigenmode with the eigenvalue . Thus, like the case of the even lattices, one can obtain all the eigenvalues of the Dirac operator by solving the KS Dirac eigenvalue equation (26).
In the case of the temporally oddnumber lattice, according to the spinor structure of the Dirac operator given by Eq. (36), we label the Dirac eigenstates with quantum number , namely . For each KS Dirac mode , we construct these four Dirac eigenfunctions using the KS Dirac eigenfunction ,
(38) 
where is given by Eq. (30). The Dirac eigenstates have the eigenvalue in the case of and have the eigenvalue in the case of . (Recall that the Dirac eigenstates with and the Dirac eigenstates with appear in pairs because of chiral symmetry.) Substituting Eq. (30) for Eq. (38) one can obtain the relation
(39) 
Next, consider rewriting the relation (17) in terms of the KS Dirac modes. Taking the structure of the Dirac eigenfunction (38) into consideration, Eq. (17) should be written correctly as
(40) 
Using the relation (see Appendix B.2)
(41) 
RHS of Eq. (40) can be rewritten in terms of the KS Dirac modes:
(42) 
where is even on the temporally oddnumber lattice. Thus, one can obtain the relation
(43) 
using the modified KS formalism. Note that the (modified) KS formalism is an exact mathematical method for diagonalizing the Dirac operator and is not an approximation, so that Eqs.(40) and (43) are completely equivalent. Therefore, each Diracmode contribution to the Polyakov loop can be obtained by solving the eigenvalue equation of the KS Dirac operator whose dimension is instead of the original Dirac operator whose dimension is in the case of the temporally oddnumber lattice.
Note again that we never use a specific fermion like the KS fermion here. We only diagonalize the Dirac operator defined by Eq.(4) using the technique of the KS formalism, and obtain all the eigenvalues and the eigenfunctions of . Actually, even without use of the KS formalism, the direct diagonalization of gives the same results, although the numerical cost is larger.
Iv Lattice QCD Numerical Analysis and Discussions
In this section, we numerically perform SU(3) lattice QCD calculations and discuss the relation between confinement and chiral symmetry breaking based on the relation (43) connecting the Polyakov loop and Dirac modes on the temporallyodd number lattice.
The SU(3) lattice QCD Monte Carlo simulations are performed with the standard plaquette action at the quenched level in both cases of confinement and deconfinement phases. For the confinement phase, we use a lattice with (i.e., fm), corresponding to MeV. For the deconfinement phase, we use lattice with (i.e., fm), corresponding to MeV. For each phase, we use 20 gauge configurations, which are taken every 500 sweeps after the thermalization of 5,000 sweeps.
iv.1 Numerical analysis of the relation between Polyakov loop and Dirac modes
To confirm the relation (43) numerically, we calculate independently LHS and RHS of the relation (43) and compare these values. A part of the numerical results in confinement and deconfinement phases are shown in Table 1 and Table 2, respectively.
Configuration No.  1  2  3  4  5  6  7  8  9  10 

Re  0.00961  0.00161  0.0139  0.00324  0.000689  0.00423  0.00807  0.00918  0.00624  0.00437 
Im  0.00322  0.00125  0.00438  0.00519  0.0101  0.0168  0.00265  0.00683  0.00448  0.00700 
0.00961  0.00161  0.0139  0.00324  0.000689  0.00423  0.00807  0.00918  0.00624  0.00437  
0.00322  0.00125  0.00438  0.00519  0.0101  0.0168  0.00265  0.00683  0.00448  0.00700 
Configuration No.  1  2  3  4  5  6  7  8  9  10 

Re  0.316  0.337  0.331  0.305  0.313  0.316  0.337  0.300  0.344  0.347 
Im  0.00104  0.00597  0.00723  0.00334  0.00167  0.000120  0.000482  0.00690  0.00102  0.00255 
0.316  0.337  0.331  0.305  0.314  0.316  0.337  0.300  0.344  0.347  
0.00104  0.00597  0.00723  0.00334  0.00167  0.000120  0.000482  0.00690  0.00102  0.00255 
From Table 1 and Table 2, it is found that the mathematical relation (43) is exactly satisfied for each gauge configuration in both confinement and deconfinement phases, and this result is consistent with the analytical discussions in Sec. II. Then, one can discuss the relation between confinement and chiral symmetry breaking based on the relation (43) even with one gauge configuration. Of course, the relation is satisfied for the gaugeconfiguration average.
In the deconfinement phase, the center symmetry is spontaneously broken, and the Polyakov loop is proportional to for each gauge configuration at the quenched level Rothe (). In this paper, we name the vacuum where the Polyakov loop is almost real (=0) “real Polyakovloop vacuum” and the other vacua “rotated vacua.” At the quenched level, we have numerically confirmed that the relation (43) is exactly satisfied in the rotated vacua as well as the real Polyakovloop vacuum.
When dynamical quarks are included, the real Polyakovloop vacuum is selected as the stable vacuum, and the rotated vacua become metastable states. Then, the real Polyakovloop vacuum would be more significant than other vacua in the deconfinement phase. Even in full QCD, the mathematical relation (43) is expected to be valid, and we will confirm the relation and perform the numerical analysis in full QCD in the next study.
iv.2 Contribution from lowlying Dirac modes to Polyakov loop
Next, we numerically confirm that lowlying Dirac modes have little contribution to the Polyakov loop based on the relation (43). This is expected from the analytical relation (43) as discussed below Eq.(17), however, such a numerical analysis is also meaningful because the behavior of the matrix element is nontrivial.
Since RHS of Eq. (43) is expressed as a sum of the Diracmode contribution, we can calculate the Polyakov loop without lowlying Diracmode contribution as
(44) 
with the infrared (IR) cutoff for Dirac eigenvalue. The chiral condensate is expressed as
(45)  
where is the current quark mass and the total number of zero modes of .
We show the lattice QCD result of the Dirac eigenvalue distribution in confinement and deconfinement phases in Fig. 2. In the deconfinement phase, the number of lowlying Dirac modes is significantly reduced and , which means that the chiral condensate is almost zero and the chiral symmetry is restored. Then, in the deconfinement phase, it may be less interesting to investigate the effect of lowlying Dirac modes to the Polyakov loop, because lowlying Dirac modes are almost absent.
The chiral condensate after the removal of contribution from the lowlying Dirac modes below IR cutoff is expressed as
(46) 
In this paper, we take the IR cutoff of . In the confined phase, this IR Diracmode cut leads to
(47) 
and almost chiralsymmetry restoration in the case of physical currentquark mass, GIS ().
A part of the numerical results for and with the IR cutoff of in both confinement and deconfinement phases are shown in Table 3 and Table 4, respectively.
Configuration No.  1  2  3  4  5  6  7  8  9  10 

Re  0.00961  0.00161  0.0139  0.00324  0.000689  0.00423  0.00807  0.00918  0.00624  0.00437 
Im  0.00322  0.00125  0.00438  0.00519  0.0101  0.0168  0.00265  0.00683  0.00448  0.00700 
Re  0.00961  0.00160  0.0139  0.00325  0.000706  0.00422  0.00807  0.00918  0.00624  0.00436 
Im  0.00321  0.00125  0.00437  0.00520  0.0101  0.0168  0.00264  0.00682  0.00448  0.00698 
Configuration No.  1  2  3  4  5  6  7  8  9  10 

Re  0.316  0.337  0.331  0.305  0.314  0.316  0.337  0.300  0.344  0.347 
Im  0.00104  0.00597  0.00723  0.00334  0.00167  0.000120  0.0000482  0.00690  0.00102  0.00255 
Re  0.319  0.340  0.334  0.307  0.317  0.319  0.340  0.303  0.347  0.350 
Im  0.00103  0.00597  0.00724  0.00333  0.00167  0.000121  0.0000475  0.000691  0.00102  0.00256 
From Table 3 and Table 4, it is found that is almost satisfied for each gauge configuration in both confinement and deconfinement phases. In the deconfinement phase, we have confirmed that is satisfied for both real Polyakovloop vacuum and rotated vacua. Thus, the configuration average is of course almost satisfied. Therefore, the lowlying Dirac modes have little contribution to the Polyakov loop and are not essential for confinement. From Eq. (47), however, the lowlying Dirac modes below the IR cutoff are essential for chiral symmetry breaking. Thus, we conclude that there is no onetoone correspondence between confinement and chiral symmetry breaking. This result is consistent with the previous numerical lattice analysis that the confinement properties such as the Polyakov loop and the string tension, or confinement force, are almost unchanged by removing lowlying Dirac modes from QCD vacuum GIS ().
iv.3 New “positive/negative symmetry” on Dirac matrix element in confinement phase
Since Eq. (43) is the Dirac spectral expression of the Polyakov loop, one can investigate the contribution from each Dirac mode to the Polyakov loop. We calculate the matrix element and each Diracmode contribution in both confinement and deconfinement phases. The Polyakov loop is obtained by multiplying the sum of each Diracmode contribution by the overall factor in Eq. (43).
iv.3.1 Confinement phase case
Figure 3 shows the numerical results for the matrix elements Re and Im plotted against Dirac eigenvalues in the lattice unit for one gauge configuration in the confinement phase.
Figure 4 shows each Diracmode contribution to the Polyakov loop and plotted against Dirac eigenvalues in the lattice unit.
In the confinement phase, the real part of the matrix element is generally nonzero in the whole region and is not small in lowlying Diracmode region from Fig. 3. However, the Diracmode contribution to the Polyakov loop, , is small in lowlying Diracmode region because of the damping factor from Fig. 4. Thus, the damping factor has an essential role in Eq. (43).
On the other hand, from Fig. 3, the imaginary part of the matrix element is relatively small in lowlying Diracmode region, in comparison with . In any case, is small in lowlying Diracmode region, as shown in Fig. 4.
Remarkably, as shown in Fig. 3, there is a new symmetry of “positive/negative symmetry” in the confinement phase for the distribution of Diracmode matrix element , i.e., and . Then, the distribution of each Diracmode contribution to the Polyakov loop, , has the same symmetry. Since the Polyakov loop is proportional to the total sum of each Diracmode contribution, , this new symmetry leads to the zero value of the Polyakov loop, i.e., , in the confinement phase. Moreover, the contribution to the Polyakov loop from arbitrary Diracmode region is zero due to the symmetry in the confinement phase:
(48) 
This behavior in the confinement phase is consistent with the previous works GIS ().
Note that the distribution of the matrix elements is not statistical fluctuation on the gauge ensemble because the results shown here are for one configuration. We find the same behavior for other gauge configurations.
As for the dependence of the matrix element in the confinement phase, we find almost the same results that there is the positive/negative symmetry and lowlying Dirac modes have little contribution to the Polyakov loop.
iv.3.2 Deconfinement phase case
Since the deconfinement phase does not have confinement and chiral symmetry breaking, it may be less interesting to consider their relation there. In the deconfinement phase, the center symmetry is spontaneously broken, and there appear three types of vacua corresponding to the Polyakov loop proportional to , while the confinement phase has a unique vacuum of on the symmetry. Here, we mainly consider the real Polyakovloop vacuum, since it is selected as the stable vacuum when dynamical quarks are included.
We show in Figs. 5 and 6 the matrix elements and each Diracmode contribution in the deconfinement phase with real Polyakov loop, plotted against the Dirac eigenvalue , in quenched lattice QCD.