Local correlation among the chiral condensate, monopoles, and color magnetic fields in Abelian projected QCD

In this subsection, we define monopoles in Abelian projected QCD in lattice formalism [40]. Like the ordinary SU(3) plaquette, the Abelian plaquette variable is defined as

where $\hat{\mu}$ is the $\mu$-directed unit vector in the lattice unit. The Abelian field strength $\theta_{\mu\nu}^{i}(s)$ ($i=1,2,3$) is the principal value of the exponent in $u_{\mu\nu}(s)$, and is defined as

with the forward derivative $\partial_{\mu}$. Note that $\theta_{\mu\nu}^{i}(s)$ is ${\rm U(1)}^{2}$ gauge invariant and corresponds to the regular Abelian field strength in the continuum limit of $a\rightarrow 0$, while $n_{\mu\nu}^{i}(s)$ corresponds to the singular gauge-variant Dirac string [40].

The electric current $j_{\mu}^{i}$ and the monopole current $k_{\mu}^{i}$ are defined from the Abelian field strength $\theta_{\mu\nu}^{i}$ as

where $\partial_{\mu}^{\prime}$ is the backward derivative and $\tilde{\theta}_{\mu\nu}$ is the dual tensor of $\tilde{\theta}_{\mu\nu}\equiv\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}{\theta}_{\alpha\beta}$. Both electric and monopole currents are $\mathrm{U}(1)^{2}$ gauge invariant, according to $\mathrm{U}(1)^{2}$ gauge invariance of ${\theta}_{\mu\nu}^{i}(s)$. In the lattice formalism, $k_{\mu}^{i}(s)$ is located at the dual lattice $L_{\rm dual}^{4}$ of $s^{\alpha}+\frac{1}{2}$ with flowing in $\mu$ direction [41].

In this way, Abelian projected QCD includes both electric current $j_{\mu}^{i}$ and monopole current $k_{\mu}^{i}$. Remarkably, lattice QCD shows monopole dominance, i.e., dominant role of monopoles for quark confinement in the MA gauge [42]. Also, lattice QCD shows monopole dominance for chiral symmetry breaking, that is, monopoles in the MA gauge crucially contribute to spontaneous chiral-symmetry breaking in both SU(2) [29, 31] and SU(3) lattice QCD [33].

In the lattice formalism, the monopole current $k_{\mu}^{i}$ appears on the dual lattice $L_{\rm dual}^{4}$ of $s^{\alpha}+\frac{1}{2}$, and therefore we define the local monopole density

where $P(s)$ denotes the dual lattices in the vicinity of $s$, i.e., $P(s)=\left\{s^{\prime}\in L_{\rm dual}^{4}\middle||s-s^{\prime}|=1\right\}$. Note here that the distance between the site $s$ and its closest dual site $s^{\prime}$ is $|s-s^{\prime}|=\sqrt{\sum_{1}^{4}(\frac{1}{2})^{2}}=1$ in the four-dimensional Euclidean space-time.

In QCD in the MA gauge, color magnetic monopoles generally appear, and play an important role in nonperturbative properties, which migth looks curious, since the original QCD action does not have monopoles.

However, some active roles of magnetic objects would be natural in QCD, because QCD itself has color magnetic instability, and spontaneous generation of color magnetic fields generally takes place, as Savvidy first pointed out in 1977 [43, 44].

In fact, in the QCD vacuum in the Minkowski space-time, the gluon condensate $\langle G_{\mu\nu}^{a}G_{\mu\nu}^{a}\rangle$ takes a large positive value, which physically means that the QCD vacuum is filled with color magnetic fields. Since the gluon condensate is expressed with color magnetic fields $\vec{H}_{a}$ and color electric fields $\vec{E}_{a}$ as

its large positivity means inevitable significant generation of color magnetic fields. Thus, some superior role of magnetic objects is expected instead of electric objects in the Minkowski QCD vacuum.

In the Euclidean space-time, because of the space-time $SO(4)$ symmetry, the roles of magnetic and electric fields become similar. Actually, the gluon condensate is written as $G_{\mu\nu}^{a}G_{\mu\nu}^{a}=2(\vec{H}_{a}^{2}+\vec{E}_{a}^{2})$, where the electromagnetic duality is manifest. Then, in Euclidean QCD, the electric field often behaves as a magnetic field, and therefore we regard the Euclidean electric field as a sort of the magnetic field in this paper.

In Abelian projected QCD, there are two Lorentz invariant quantities ${\cal F}$ and ${\cal G}$ in the Euclidean space-time:

with the color magnetic field $(\vec{H}_{i})_{j}\equiv\frac{1}{2}\epsilon_{jkl}F_{i}^{kl}$ and the color electric field $(\vec{E}_{i})_{j}\equiv F_{i}^{j4}$. These quantities are also invariant under the residual U(1)² gauge transformation and global Weyl transformation [34], i.e., permutation of the color index, in the MA gauge.

Here, ${\cal F}$ is parity-even and expresses total magnitude of magnetic fields in the Euclidean space-time, since the electric field behaves as a magnetic field there. In this paper, we simply call ${\cal F}$ “magnetic quantity” in Euclidean gauge theories. Note that ${\cal G}$ is parity-odd and is just the Abelian projected quantity of the topological charge density on instantons in QCD, which might relate to chiral symmetry breaking.

In the lattice formalism, the field strength tensor is a plaquette variable spanning at $s$, $s+\hat{\mu}$, $s+\hat{\nu}$, and $s+\hat{\mu}+\hat{\nu}$, so that we define the Abelian field strength $F_{\mu\nu}^{i}(s)$ as the local average of clover-type four plaquettes,

and consider ${\cal F}$ and ${\cal G}$ as local quantities in each Abelian gauge configuration.

In this subsection, we briefly review the local chiral condensate in lattice formalism. The local chiral condensate can be calculated with the quark propagator for each gauge configuration $U=\{U_{\mu}(s)\}$ generated with the Monte Carlo method.

As the lattice fermion, we here adopt the Kogut-Susskind (KS) fermion [3]. For the KS fermion, the Dirac operator $\gamma_{\mu}D_{\mu}$ is expressed by $\eta_{\mu}D_{\mu}$ with the staggered phase $\eta_{\mu}(s)\equiv(-1)^{s_{1}+\cdots+s_{\mu-1}}$ $(\mu\geq 2)$ with $\eta_{1}(s)\equiv 1$. The KS Dirac operator is expressed as

with $U_{-\mu}(s)\equiv U^{\dagger}_{\mu}(s-\hat{\mu})$, and the KS Dirac eigenvalue equation takes the form of

Here, the quark field $q^{\alpha}(s)$ is described by a spinless Grassmann variable $\chi(s)$ [3], and the chiral condensate per flavor is evaluated as $\langle\bar{q}q\rangle=\langle\bar{\chi}\chi\rangle$/4 in the continuum limit.

The local chiral condensate can be calculated using the quark propagator of the KS fermion with a small quark mass $m$. The chiral-limit value is estimated by the chiral extrapolation of $m\rightarrow 0$. As a technical caution, the chiral and continuum limits do not commute for the KS fermion at the quenched level, although this problem would be absent in full QCD [45].

For the gauge configuration $U=\{U_{\mu}(s)\}$, the Euclidean KS fermion propagator is given by

with the color index $i$ and $j$. This propagator is numerically obtained by solving the large-scale linear equation with a point source. The local chiral condensate for the gauge configuration $\{U_{\mu}(s)\}$ is expressed with the propagator as

Here, we consider the net chiral condensate by subtracting the contribution from the trivial vacuum $U=1$ as

where the subtraction term is exactly zero in the chiral limit $m=0$. The global chiral condensate is obtained by taking its average over the space-time $x$ and the gauge ensembles $U_{1},U_{2},...,U_{N}$,

In this subsection, we analytically investigate relation between chiral symmetry breaking and the field strength in Euclidean Abelian gauge systems. For the simple argument, we consider Euclidean U(1) gauge systems with quasi-massless Dirac fermions coupled to the U(1) gauge field, although it is straightforward to generalize this argument to Abelian projected QCD with ${\rm U}(1)^{2}$ gauge symmetry.

In the U(1) gauge system, the chiral condensate is proportional to the functional trace of the fermion propagator,

with the covariant derivative $D_{\mu}\equiv\partial_{\mu}+igA_{\mu}$, the field strength $F_{\mu\nu}$, $\sigma\cdot F\equiv\sigma_{\mu\nu}F_{\mu\nu}$ and $\sigma_{\mu\nu}\equiv\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$. Note that $D^{2}-m^{2}$ is a negative-definite operator, and all of its eigenvalues are negative. Since the trace of any odd-number product of $\gamma$-matrices is zero, we find

with

Because of the overall factor $m$ in $I$, $I$ goes to zero in the chiral limit $m\rightarrow 0$, unless the denominator becomes zero in this limit.

Since the operator $(D^{2}-m^{2})^{2}$ in the denominator is positive definite, to realize the zero denominator in $I$ in Eq. (21), we need a significant negative contribution from the other three terms including ${\cal F}$, ${\cal G}$, or $[D^{2},\sigma\cdot F]$. For instance, in the absence of the field strength, i.e., $F_{\mu\nu}\equiv 0$, one finds near $m\simeq 0$

with the UV cutoff $\Lambda$ and the degeneracy $\gamma$. According to the positive denominator in the integrand, $I_{F_{\mu\nu}\equiv 0}$ has no IR singularity to cancel $m$ of the numerator, and therefore $I_{F_{\mu\nu}\equiv 0}$ goes to zero in the chiral limit of $m\rightarrow 0$.

To cancel $m$ in the numerator of $I$, we need a significantly large amount of the field strength so as to present zero mode in the denominator of $I$ and to keep $I$ non-zero in the chiral limit. Note here that ${\cal F}(\geq 0)$ always gives a negative (non-positive) contribution in the denominator of $I$, while the contribution from ${\cal G}$ or $[D^{2},\sigma\cdot F]$ can be positive and negative. In fact, the magnetic quantity ${\cal F}$ can give the zero mode in the denominator of $I$, even without the contribution from ${\cal G}$ and $[D^{2},\sigma\cdot F]$. In contrast, in Euclidean Abelian gauge systems, ${\cal F}\equiv\frac{1}{4}F_{\mu\nu}^{2}=0$ means $F_{\mu\nu}=0$, and then ${\cal G}=[D^{2},\sigma\cdot F]=0$.

To conclude, the magnetic quantity ${\cal F}$ is expected to be significantly important to realize chiral symmetry breaking in Euclidean Abelian gauge theories, although, in some cases, the contribution from ${\cal G}$ and $[D^{2},\sigma\cdot F]$ can assist the realization of chiral symmetry breaking.

In a special case of constant $F_{\mu\nu}$, one finds $[D^{2},\sigma\cdot F]=0$ for the Abelian system, and obtains

because of ${\rm tr}\gamma_{5}={\rm tr}\sigma_{\mu\nu}={\rm tr}\gamma_{5}\sigma_{\mu\nu}=0$. For more special case of a constant magnetic field, there occurs the Landau-level quantization, and the spatial degrees of freedom perpendicular to the magnetic field is frozen in the lowest Landau level. This infrared effective low-dimensionalization of the charged spinor dynamics induces chiral symmetry breaking in the chiral limit  [46, 47, 48], which is known as magnetic catalysis [49].

To begin with, we deal with an idealized Abelian gauge system of a static monopole-antimonopole pair on a periodic lattice of the four-dimensional Euclidean space-time.

In the three-dimensional space ${\bf R}^{3}$, let us consider a static monopole-antimonopole pair with the distance of $l$ in $z$-direction. To realize such a lattice gauge system, we set the Abelian link-variable $u_{\mu}(s)$ to be

otherwise $u_{\mu}(s)=1$.

Figure 1 shows the building-block plaquette to realize a static monopole-antimonopole pair on the lattice. Here, only the red link-variables take a nontrivial value of $i$.

As for the phase variable $\theta_{\mu}(s)$, which corresponds to the Abelian gluon, one finds

otherwise $\theta_{\mu}(s)=0$. For the all-red plaqutte with $s_{x}=s_{y}=0$ and $1\leq s_{z}\leq l$, one gets

which leads to the Dirac string of $n_{xy}(s)=-1$ and zero field strength $\theta_{xy}(s)=0$, because of the definition of the field strength $\theta_{\mu\nu}(s)$ and the Dirac string $n_{\mu\nu}(s)$,

Thus, the all-red plaquette induces the singular Dirac string at its center on the dual lattice. In fact, for the idealized system in Fig. 1(b), a Dirac string appears inside the all-red plaquette.

At the terminal of the Dirac string, a monopole or an anti-monopole appears on the dual lattice, as shown in Fig. 2.

Actually, the three-dimensional spatial cube including only one all-red plauette has a static (anti)monopole at its center (on the dual lattice), because only one $n_{kl}(s)$ has nonzero value of $\pm 1$ among the six independent plaquettes composing the cube,

Thus, this idealized system includes a static monopole at $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ and a static anti-monopole at $(\frac{1}{2},\frac{1}{2},l+\frac{1}{2})$ in spatial ${\bf R}^{3}$.

This monopole and anti-monopole system has also physical magnetic flux around the line segment connecting the monopole pair. In fact, a physical magnetic field is created in the neighboring plaquette $u_{xy}(s)$ of the all-red plaquette in Fig. 1. In this idealized system, only the plaquette $u_{xy}(s)$ including one red link takes a nontrivial value as

otherwise $u_{\mu\nu}(s)=1$. Note here that, by gauge transformation, the location of the Dirac string is generally changed, but the physical field strength is never changed.

Figure 3 shows the local chiral condensate $\langle\bar{\chi}\chi(s)\rangle_{u}$ and the magnetic quantity ${\cal F}\equiv\frac{1}{4}F_{\mu\nu}^{2}=\frac{1}{2}\vec{H}^{2}$ for $l=4$ in the three dimensional space ${\bf R}^{3}$. In this demonstration, the quark mass is taken to be $m=0.01$ in the lattice unit.

For this idealized static system, there actually appears a magnetic field $\vec{H}$, i.e., nonzero flux of ${\cal F}=\frac{1}{2}\vec{H}^{2}>0$, in space between the monopole and the anti-monopole, and the local chiral condensate takes a significant value in the vicinity of the magnetic field. In contrast, one finds ${\cal G}=\vec{H}\cdot\vec{E}=0$ everywhere, since only spatial plaquettes take a nontrivial value and $\vec{E}=\vec{0}$. Thus, in this system, it is likely that the magnetic field stemming from monopoles has the primary correlation with the local chiral condensate.

Next, let us investigate a static magnetic flux system without monopoles. Owing to the spatial periodicity, the special case of $l=L_{z}$ in the static monopole-antimonopole system has no (anti)monopoles, because of the magnetic-charge cancellation. In this special case of $l=L_{z}$, there only exists a physical static magnetic flux along $z$-direction.

Figure 4 shows the local chiral condensate $\langle\bar{\chi}\chi(s)\rangle_{u}$ and the magnetic quantity ${\cal F}\equiv\frac{1}{4}F_{\mu\nu}^{2}=\frac{1}{2}\vec{H}^{2}$ for $l=L_{z}$ in spatial ${\bf R}^{3}$, taking the quark mass of $m=0.01$ in the lattice unit.

Again, the local chiral condensate takes a significant value in the vicinity of the magnetic field $\vec{H}$, i.e., nonzero flux of ${\cal F}=\frac{1}{2}\vec{H}^{2}>0$, even without (anti)monopoles. Note also that this system has ${\cal G}=\vec{H}\cdot\vec{E}=0$ everywhere, because of $\vec{E}=\vec{0}$. Therefore, in this idealized system, we conclude that the magnetic field or ${\cal F}$ has the primary correlation with the local chiral condensate.

To begin with, we pick up a gauge configuration generated in lattice QCD on $V=24^{4}$ at $\beta=6.0$, and investigate correlation between the local chiral condensate and magnetic variables.

Figure 5 shows the local chiral condensate $\langle\bar{\chi}\chi(s)\rangle_{u}$ with the quark mass of $m=0.02$, the local monopole density $\rho_{\rm L}(s)$, and the Lorentz invariants ${\cal F}(s)$ and $|{\cal G}(s)|$, respectively, as well as the monopole location in the space ${\bf R}^{3}$ at a time slice, for a typical gauge configuration of Abelian projected QCD.

From Fig. 5(a), one finds that the local chiral condensate tends to take a large value near the monopole location [33]. Since monopoles appear on the dual lattice, we show the local monopole density $\rho_{\rm L}(s)$, as the average on closest dual sites. Of course, $\rho_{\rm L}(s)$ takes a large value near the monopole. The distribution of the the local monopole density resembles that of the local chiral condensate, as was pointed out in Ref. [33]. Figures 5(c) and (d) show the Lorentz invariants ${\cal F}$ and ${\cal G}$, respectively. As a new result in this paper, we find that the distributions of ${\cal F}$ and ${\cal G}$ also resemble that of the local chiral condensate.

The close relation of monopoles with ${\cal F}$ and ${\cal G}$ might be understood, since the field strength tensor relates to monopoles as $\partial_{\mu}\tilde{F}_{\mu\nu}^{i}=k_{\nu}^{i}$. Roughly speaking, the monopole can be a kind of source of ${\cal F}$ and ${\cal G}$. In contrast, their similarity with the local chiral condensate is fairly nontrivial.

In any case, we find clear correlation of distribution similarity among the local chiral condensate, the local monopole density, and the Lorentz invariants ${\cal F}$ and ${\cal G}$ in Abelian projected QCD in the MA gauge.

In this subsection, we quantify the similarity between the local chiral condensate $\langle\bar{\chi}\chi(s)\rangle_{u}$ and magnetic variables, i.e., $\rho_{\rm L}(s)$, ${\cal F}(s)$ and ${\cal G}(s)$, defined in Section III. To this end, we use all the generated 100 gauge configurations in lattice QCD on $V=24^{4}$ at $\beta=6.0$, and calculate the local chiral condensate at $2^{4}$ distant space-time points for each gauge configuration, resulting 1600 data points at each quark mass.

Figure 6 shows the scatter plot between the local chiral condensate $\langle\bar{\chi}\chi(s)\rangle_{u}$ and magnetic variables, i.e., the local monopole density $\rho_{\rm L}(s)$, Lorentz invariants ${\cal F}(s)$ and $|{\cal G}(s)|$, respectively, using 100 gauge configurations of Abelian projected QCD in the MA gauge, with the quark mass of $m=0.01,0.015,0.02$ in the lattice unit. In Fig. 6, positive correlation is qualitatively found between the local chiral condensate and the magnetic variables, $\rho_{\rm L}$, ${\cal F}$ and $|{\cal G}|$, respectively.

Next, we consider a quantitative analysis using correlation coefficients between the local chiral condensate and the magnetic variables, as a statistical indicator of correlation. In general, for arbitrary two statistical ensembles $\{A_{i}\}$ and $\{B_{i}\}$, their correlation coefficient $r$ is defined as

using the average notation $\langle~{}~{}\rangle$ and the standard deviation $\sigma_{A}\equiv\sqrt{\langle(A-\langle A\rangle)^{2}\rangle}$ and $\sigma_{B}\equiv\sqrt{\langle(B-\langle B\rangle)^{2}\rangle}$. Here, $r=1$ means perfect positive linear correlation, and $r\gtrsim 0.7$ indicates strong positive linear correlation.

We measure correlation coefficients between the local chiral condensate $\left|\langle{\bar{\chi}\chi}(s)\rangle_{u}\right|$ and three magnetic variables, $\rho_{\mathrm{L}}(s)^{\alpha}$, $\mathcal{F}(s)^{\alpha}$ and $\left|\mathcal{G}(s)\right|^{\alpha}$, at various exponent $\alpha$, using 100 gauge configurations of Abelian projected QCD in SU(3) lattice QCD at $\beta=6.0$ on $V=24^{4}$, for the quark mass $m=0.01$ in the lattice unit. Table 1 shows the result for the correlation coefficients.

Quantitatively, the magnetic quantity ${\cal F}$ has the strongest correlation with the chiral condensate rather than $\rho_{\rm L}$ and ${\cal G}$. As a conclusion of this paper, we find a strong positive correlation of $r\simeq 0.8$ between the local chiral condensate $\left|\langle{\bar{\chi}\chi}(s)\rangle_{u}\right|$ and the magnetic quantity ${\cal F}(s)$ in the confined vacuum of Abelian projected QCD.

Finally, we investigate also a high-temperature deconfined phase in lattice QCD on $V=24^{3}\times 6$ at $\beta=6.0$, where the temperature is $T\simeq$ 330 MeV above the critical temperature. We generate 200 gauge configurations, and calculate the local chiral condensate at $2^{3}$ distant space points at a time slice for each gauge configuration, resulting 1600 data points at each quark mass.

Figure 7 shows the scatter plot between the local chiral condensate $\left|\langle{\bar{\chi}\chi}(s)\rangle_{u}\right|$ and magnetic variables, i.e., the local monopole density $\rho_{\rm L}(s)$, and Lorentz invariants ${\cal F}(s)$ and $|{\cal G}(s)|$, respectively, using 200 gauge configurations of Abelian projected QCD in the MA gauge, with the quark mass of $m=0.01,0.015,0.02$ in the lattice unit.

We show in Table 2 correlation coefficients between the local chiral condensate $\left|\langle{\bar{\chi}\chi}(s)\rangle_{u}\right|$ and three magnetic variables, $\rho_{\mathrm{L}}(s)^{\alpha}$, $\mathcal{F}(s)^{\alpha}$ and $\left|\mathcal{G}(s)\right|^{\alpha}$, at various exponent $\alpha$, using 200 gauge configurations of Abelian projected QCD in SU(3) lattice QCD at $\beta=6.0$ on $V=24^{3}\times 6$, for the quark mass $m=0.01$ in the lattice unit.

From Fig. 7 and Table  2, all the correlations between the local chiral condensate and the three magnetic variables, $\rho_{\rm L}$, ${\cal F}$ and ${\cal G}$, become weaker in the deconfined phase, where the chiral condensate itself goes to zero in the chiral limit.