Fractional instanton of the SU(3) gauge theory in weak coupling regime

Fractional instanton of the SU() gauge theory in weak coupling regime

Etsuko Itou
Department of Physics, and Research and Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan
Department of Mathematics and Physics, Kochi University, Kochi 780-8520, Japan
Research Center for Nuclear Physics (RCNP), Osaka University, Osaka 567-0047, Japan
July 20, 2019

Motivated by recent studies on the resurgence structure of quantum field theories, we numerically study the nonperturbative phenomena of the SU() gauge theory in a weak coupling regime. We found that topological objects with a fractional charge emerge, if the theory is regulated by an infrared (IR) cutoff via the twisted boundary condition. Some integer-instantons are generated as a semi-classical configuration in the Montecarlo simulation even in the weak coupling regime. Furthermore, some of them consist with several fractional-instantons. We also measure the Polyakov loop to investigate the center symmetry and confinement. The fractional-instanton corresponds to the solution linking two of degenerate -broken vacua in the deconfined phase.

1 Introduction

Instanton is one of the classical solutions of the quantum field theory, and labels a vacuum state. In pure Yang-Mills theory in the four-dimensional Euclidean spacetime (), the instanton has a topological charge (instanton number) given by


where denotes a totally anti-symmetric tensor. It always takes an integer value [1]. In fact, the topological charge has been measured by the lattice ab initio calculations [2]. The distribution of is broad in the hadronic (confined) phase in the low temperature, while it is narrow in the quark-gluon-plasma (deconfined) phase in the high temperature [3, 4]. According to the lattice numerical study [5, 6], it is also known that the first-order phase transition occurs between these phases in the pure SU() gauge theory.

Figure 1: A roadmap to see a resurgence structure for the Yang-Mills theory. In this work, we focus on two boxes in the center.

Actually, most lattice calculations have been performed on the hypertorus () with the standard periodic boundary condition, though the Yang-Mills theory on the periodic hypertorus does not have a self-dual configuration as a classical solution. To obtain a stable configuration on the finite lattice, we have to impose the twisted boundary conditions [7]. The reason, why the ordinary lattice calculation with the periodic boundary condition can observe the nontrivial- solutions and the distribution of , is that the boundary effect is negligible in the strong coupling regime, where we have mainly focused on in the nonperturbative studies of the Yang-Mills theory.

The question arises as to the property of the topological objects in the weak coupling regime of the Yang-Mills theory. To see the quantities in the perturbative regime on the lattice, for instance to calculate of the running coupling constant, we need to set the renormalization scale to be higher than the Lambda scale (). The renormalization scale is inversely proportional to the spacial lattice extent (), so that we have to use the lattice extent satisfying . We expect that the choice of the boundary condition in such a small box must have an influence on the property of classical solution. Then, we have to consider which is a proper boundary condition to investigate the weak coupling regime. In fact, to realize the matching with the perturbative calculation on , we need to utilize the nontrivial boundary conditions on hypertorus. Otherwise, the classical solution does not connect to the standard perturbative vacuum [7, 8, 9] because of a gauge inequivalent configuration, which is called toron, of the degenerate minimal action [10, 11]. For instance, the Schrödinger functional boundary [12, 13] or the twisted boundary conditions [14, 15] have been introduced in the calculation of the running coupling constant.

On the other hand, it is well-known that the perturbative expansion of the SU() gauge theory on spacetime does not converge in the higher order approximation. To solve this, for the quantum mechanical models and low-dimensional quantum field theories, which suffer from a similar problem, the resurgence scenario has been proposed [16, 17]. In the scenario, as a proper choice of the spacetime structure, a compact dimension and/or a boundary condition with -holonomy are introduced. Then, the perturbative series and the nonperturbative effects of the theory on the modified spacetime are related to each other in the weak coupling regime, and the physical quantities are determined without any imaginary ambiguities. The characteristic phenomenon in the nonperturbative side of this scenario is the appearance of the topological objects with a local fractional charge, which contribute to the perturbative vacua, in (semi)-classical solutions of the theory. The fractionality of the topological charge is related to the cancelation of the renormalon pole [18, 19] whose action is of order in comparison with the action for the integer-instantons [20]. Recently, the resurgence structure has been revealed in several quantum-mechanical models and low-dimensional quantum field theories [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Using the lattice numerical simulation, a signal of the fractional behavior for the energy density has observed in the Principle Chiral Model [30].

For the gauge theory, a recent paper [31] has proposed a promising regularization formula on . The authors pointed out that the IR cutoff is necessary, which should be a higher energy scale than the dynamical IR scale, namely scale, otherwise the trans-series expansion of physical observables breaks down in the Yang-Mills theory. Therefore, they introduce the twisted boundary conditions for the two compactified dimensions using one-form center symmetry. The twisted boundary condition induces the IR cutoff to the theory, and in the nonperturbative side the fractional instanton is allowed in the weak coupling regime. Furthermore, the center symmetry is dynamically restored because of the tunneling behavior between -degenerated vacua. It seems to be a promising to discuss the adiabatic continuity, where no phase transition occurs toward the decompactified limit in contrast with the first order phase transition at the finite-temperature. Although the resurgence structure of the Yang-Mills theory on the modified spacetime has not yet been proven, these phenomena are very similar with the ones in low-dimensional models, which are successfully resurgent.

According these situations, in this work, we formulate of the lattice gauge theory on and study its nonperturbative properties by the numerical simulation. We summarize the relationship among several theories and lattice setups, that we mention above, in Fig. 1. This work corresponds to the second box in the center column in Fig. 1. Here, we maintain a large aspect ratio between two radius for the three-dimensional torus () and the temporal circle () as . We utilize the same twisted boundary condition as the one in Ref. [31]. Actually, we have already studied the running coupling constant of the Yang-Mills theory by using the same boundary condition in Ref. [15], so that we can tune the lattice parameter to be a weak coupling regime. The spacetime structure will be equivalent with the one in Ref. [31] in the continuum and limits. As a comparison, we also perform the ordinary periodic-lattice simulation with the same lattice parameter (third box in the center column in Fig. 1) and show the different nonperturbative-phenomena from the ones on the twisted lattice. The search for the fractional topology and the research of the basic properties of the Yang-Mills theory on the twisted lattice are interesting in itself. Furthermore, it is a challenge to define a regularized Yang-Mills theory from the perturbative to the nonperturbative regimes, although two parts with the question-mark in Fig. 1, namely the resurgent structure of the modified theory proposed in Ref. [31] and its adiabatic continuity, remain to be proven.

To show whether the configurations with fractional charge are generated, we directly measure the topological charge of configurations on the twisted lattice. We find that the multi-fractional-instanton solutions exist in the nontrivial- configurations even in the weak coupling regime. The total topological charge takes an integer-value as in the strong coupling regime in the ordinary periodic lattice, but some of them consist of several fractional-instantons. Furthermore, we also study the relationship between the fractional-instantons and the other nonperturbative phenomena, namely the tunneling behavior, the center symmetry, and the confinement. The fractional-instanton connects two of the degenerated -broken vacua. By investigating the scaling law of Polyakov loop, we conclude that although the center symmetry seems to be partially restored, the free-energy of single probe quark is still finite and it is consistent with the deconfinement behavior.

The structure of the paper is as follows: In §. 2, we give a review of the twisted boundary conditions for the two compactified directions on the lattice. The toron configurations get a higher energy than the perturbative one and the gluon propagator does not have the zero-mode in the lattice setup. We give a comment on the absence of the zero-mode, which is related with the existence of the fractional-instanton as shown in Ref. [31]. In §. 3, the strategy of the Montecarlo simulation is explained. We tune the simulation parameters to be in the perturbative regime () in the large extent enough to generate multi-instanton configurations. We also describe our sampling method for a tiny lattice spacing with a long autocorrelation. Section 4 presents the simulation results. In §. 4.1, the nontrivial- configurations show up on the twisted lattice in the weak coupling regime, while there is no such configuration on the ordinary periodic lattice with the same lattice parameters. We find the local topological object with a fractional charge in the nontrivial solutions. The Polyakov loop distribution on the twisted lattice has the different behavior from the periodic case as shown in §. 4.2. We describe the tunneling structure between the -degenerated vacua by studying the local topological charge and the Polyakov loop on each site in §. 4.3. In §. 4.4, the deconfinement property of these configurations is discussed. The last section contains the summary and several future directions.

2 Twisted boundary condition on hypertorus lattice

We review the properties of the lattice gauge theory with the two-dimensional twisted boundary condition. On the hypertorus with the ordinary periodic boundary condition, the saddle points, which we would like to study their fluctuations in the perturbation expansion, are not only degenerated through gauge transformations, but an extra degeneracy exists due to the global toroidal structure [10, 11]. The configuration related with a part of the zero-mode is called toron. The toron configuration, however, has the higher energy than the vacuum one by the two-dimensional twisted boundary condition. Simultaneously, the gluon propagator is regulated by an IR momentum cutoff [9, 14, 15]. We also review the typical property of the fractional instanton related with the IR momentum cutoff given in Ref. [31].

Let us start the Wilson-Plaquette lattice gauge action for the Yang-Mills gauge theory;


Here, and denote the lattice bare coupling constant and the plaquette,


respectively. The represents the link-variable from a site to its neighbor in the -direction, and takes the value with the SU() group elements.

Now, we introduce the twisted boundary conditions in the and directions;


while the ordinary periodic boundary conditions are imposed in the and directions;


Here, denotes the lattice extent for each direction in lattice unit. For simplicity, only in this subsection, we consider the lattice extents for all directions have the same length. The SU() matrices, (), are the twist matrices, which have the following properties in the case of ,


for a given and ().

At the corner on the lattice in the - plane, a translation of the link variable is given by


The interchanging and in the equation should lead to the same result. The gauge transformation for the original links;




Then, the Wilson-Plaquette gauge action with the twisted boundary conditions at the boundary, for instance , is given by


The toron configurations, which are related to the closed winding around the whole torus, are not transformed into themselves by the twisted conditions [10]. In fact, they do not have a degenerate energy with the standard vacuum, since their boundary plaquette gives a different contribution from the standard one.

Next, we consider the gluon propagator in this lattice setup. The link variable can be parameterized by the gauge fields () as


The plane-wave expansion of the gauge field is given by


where is a complex matrix. The corresponding twisted condition for the gauge field for implies


The non-zero solution is realized only if the momentum components satisfy




Here, denotes an ordinary physical momentum for and , while there is an additional unphysical degree of freedom for the twisted directions. We can find an one-to-one correspondence between the unphysical momenta and the color degrees of freedom of  [14]. In fact, the number of the combination of is due to the traceless condition of the gauge field.

If we take the Feynman gauge, then the gluon propagator in the momentum space is

where and . Because of the factor , the zero-modes including the torons are excluded in the propagator. Therefore, the IR cutoff proportional to in momentum space is induced in this lattice setup.

The IR momentum cutoff is also related with the existence of the fractional instanton on (). According to Ref. [31], all size-moduli of the integer-instanton on are associated with the translation-moduli of the fractional-instanton on the compactified spaces (). Then, the fractional-instanton with the smallest instanton number () has no size modulus, and hence the size of fractional-instantons is unique. The size is related with the compactification radius (). The intuitive understanding of the absence of the size-modulus is the following111We appreciate K. Yonekura for useful discussions.: If the instanton-size is smaller than the compactification radius, then the instanton becomes the ordinary integer-instanton, since the situation is the same with spacetime. On the other hand, the large instanton-size is also forbidden, since the gluon has a non-zero “mass” coming from the shifted momentum () of the twisted directions. Then, the size of the fractional instanton with the smallest charge is fixed. The fractional-instanton with the larger charge can be constructed as the composite states of the smallest ones.

In the end of this section, let us show the explicit form of the twist matrices in our numerical calculations [15, 32],


3 Simulation strategy

3.1 Lattice parameters

The simulation strategy to investigate the nonperturbative properties in the perturbative regime is as follows. We utilize the Wilson-Plaquette gauge action given by Eq. (2) as a lattice gauge action. We put the lattice parameter . The other lattice parameters, which we can control by hand, are the lattice extents in spacial () and temporal directions (). The lattice spacing “” is put unity during the numerical simulation. Once we introduce the physical quantity as a reference scale, for instance the Sommer scale [33] and scale in the gradient flow method [34], then we obtain one-to-one correspondence between and “”.

To investigate a nontrivial semi-classical solution, the lattice parameters () satisfy the following three conditions:
(1) the twisted boundary condition on the two compactified dimensions to introduce the IR momentum cutoff and to kill the toron modes
(2) sufficiently large lattice-extent to generate a multi-number of the topological objects
(3) tuned lattice gauge coupling to realize the perturbative regime

For the condition (1), we use the lattice, where the twisted boundary condition is imposed to the and directions in the three spacial directions. For simplicity, the -direction has the same lattice extent with the -directions, but its boundary condition is periodic.

The size of the fractional-instanton is predicted as the same with the compactification radius (). Then, to generate the multi-fractional-instanton, for the condition (2), at least one-direction (here ) is -times larger than . We choose for this work.

For the condition (3), we take . According to Fig.  in Ref. [15], the running coupling constant at the scale () indicates . It is well consistent with the result of the –loop approximation, which is independent of the renormalization scheme. If we fix the scale where the running coupling constant in the Twisted-Polyakov-Loop scheme diverges as shown in Ref. [15], the lattice setup with () () corresponds to . It satisfies the Dunne-Ünsal condition, , where the theory is in the weakly coupled regime but still there are some nonperturbative features. The lattice spacing is [fm], if we use [MeV]. Then, the box size of this lattice is extremely small, but it would be suitable to study the semi-classical behavior of the gauge theory in the weakly coupled regime. In fact, the action density () is roughly in ()=(), which is close to the classical limit, where the action takes a minimum value.

3.2 Sampling method of the configurations in high

To collect the gauge-configuration samples in a weak-coupling regime, we have to care about the autocorrelation problem: a new generated-configuration, which is updated using the random-number series and the old configuration, is very similar with the old one in Montecarlo simulation. The autocorrelation length depends on observables, and generally the quantities related to the low-mode physics have a long autocorrelation. In a typical calculation for the Yang-Mills theory (quenched QCD) at the zero-temperature with [fm], the autocorrelation length of the topological charge is a few ten- or hundred-sweeps (see e.g. [35]). The length grows in proportion to  [35, 36, 37], so that our lattice calculations with must suffer from a severe autocorrelation problem.

To avoid this, we prepare the seeds of random-number generation, here we label them as . We independently update sweeps using each random-number series. Here, we call one sweep as a combination of one Pseudo-Heat-Bath (PHB) update and over-relaxation steps. We collect configurations as samples in a fixed -th sweep, and we name them “conf.” using its seed of the random-number series. For the comparison, we also generate the other configurations using the same method and the same lattice parameters, while the boundary conditions are periodic for four directions. From here, we denote “TBC lattice” for the lattice with () (twisted, twisted, periodic, periodic) boundaries, while “PBC lattice” for the one with the periodic boundaries for all directions.

4 Results

4.1 Topological charge

The lattice operator of the topological charge defined by Eq. (1) can be written by the clover-leaf operator on the lattice (see Eq.(2.18) in Ref. [38]).

Figure 2: Cooling step dependence of the instanton charge (TBC).

The measured topological charge of the quantum gauge configuration, which is generated by Montecarlo method, do not take an integer-value, because of UV fluctuations. We utilize the cooling method [39, 40, 41], that is the evolving step to minimize the gauge action by smoothing the configurations. Figure 2 presents an example plot for the value of for several configurations (conf.# ) in the TBC lattice as a function of the cooling step. The value of rapidly converges to an (almost) integer-value with a few cooling steps. We perform the cooling process until steps and confirm that the plateau continues. Here, the small discrepancy from the exact integer-value, at most , comes from the lattice artifacts. In this paper, we neglect the small discrepancy, and identify the value of in the plateau of the cooling step as an integer-value.

Now, we fix the number of cooling steps as (-cool ) and the number of sweeps as . The left panel of Fig. 3 shows the result of the PBC lattice. The horizontal axis denotes the configuration-number labeled by the seed of random-series.

Figure 3: Total instanton charge on the PBC (left) and the TBC (right) lattices for each configuration.

We see that all configurations have . On the other hand, there are several configurations with non-zero values on the TBC lattice as shown in the right panel of Fig. 3. In our calculation, it distributes . The number of configurations with non-zero topological charges is , and the remain configurations live in the sector.

Figure 4: Typical distributions of the local instanton charges. The integer-instanton (and anti-instanton) exists in the right-top panel. On the other hand, one can clearly see that the topological charges is “fractionalized” in the left-bottom panel (see Eq. (19)).

Let us focus on the results on the TBC lattice. We found that the charges on each lattice site have a strong -dependence in the four-dimensional spacetime. Taking the sum only for the three-dimensional spaces, we define a local charge,


as a function of the imaginary-time coordinate. We plot the local charge for several typical configurations in Fig. 4. The left-top panel of Fig. 4 presents the local charge of the configuration with (conf.#2). All configurations in the sector indicate that its local-charge in every -coordinate is also zero. Here, we call them “Type-I”. On the other hand, the local charge of the configurations with can be roughly classified by three types as follows:
Type-II(a): it has a single peak (the right-top panel in Fig. 4)
Type-II(b): it has several peaks (the left-bottom panel)
Type-II(c): it takes a continuous non-zero value (the right-bottom panel)
In the case of Type-II(a), if we take the sum of around the single peak, then the result agrees with (almost) integer and is consistent with the value of . For instance, the confs. (red-circle) and (blue-square) have and , respectively. These peaks can be interpreted as the integer instanton and anti-instanton, respectively. In the case of Type-II(c), after summed up to for all coordinates, we can find that it gives an integer-value. The sums of in the confs. (read-circle) and (blue-square) shown in Fig. 4 become and , respectively. We find that an uniform behavior for -direction of the local charge on site-by-site as similar with the one for the action density in Principal Chiral Model given in Ref. [30]. On the other hand, the values of the local charge on all sites display the same order in contrast to the appearance of more than hierarchy in the data of Type II(a) and Type II(b).

The configurations in Type-II(b) are the most interesting. We can take the sum of for each peak by dividing regime into several domains, whose boundaries are the coordinates taking the local minimum of . Then, each charge takes -value within error, where is the integer-value except for a multiple number of . The confs. (red-circle) and (blue-square) plotted in Fig. 4 have the total instanton number and , respectively. The local charge for each peak can be calculated by


Thus, some instantons with an integer charge contain several fractional-instantons in the weak coupling regime.

Figure 5: Cooling step dependence of the local charges, (filled symbol) and (open symbol), for the conf. (red-circle) and (blue-square).

To see the fractionality of the charge is not a just quantum fluctuation, let us show the stability of these local fractional-instantons during the cooling processes. Figure 5 displays the local charges () for the confs. (red-circle) and (blue-square) as a function of cooling step. Although if it is a just quantum fluctuation then the charge would disappear by the averaging process, we found that the position of the local minimum of and each charge are very stable.

Next, we investigate the topology changing during the PHB updates. In ordinary lattice calculations for the Yang-Mills theory focusing on the strong coupling regime, the total instanton charge can be changed within a few ten or hundred Montecarlo sweeps, since the potential barrier is finite on the lattice. As we explained, the lattice spacing in our calculation is very tiny. The topology changing does not so frequently occurs, but it is happened in sweeps. Furthermore, we found that the local topological charge is rather frequently changed during the PHB updates.

Figure 6: Example plots of the sweep dependence of the local topological charge . The magenta-triangle, blue-square, and red-circle symbols denote the configurations, which are generated by the initial random-number , and , respectively.

The typical results are shown in Fig. 6. In all panels, the number of cooling processes is fixed as -cool. Let us focus on each color symbol in three panels from the left to the right. The magenta-triangle, blue-square, and red-circle symbols denote the local charge of confs., , and , respectively. During the PHB updates from -th to -th sweep, the total-charge changes as follows:

Meanwhile, the combination of the local charges indicates a rich variation:

Thus, the multi-fractional-instanton merges into the integer-instanton and vice versa, and the fractional instanton with a large charge deforms into the multi-fractional one with a small charge.

In fact, in model, it is known that the fractional-instantons can transform into the integer-instanton if the moduli parameter is changed by hand (see Fig.  in Ref. [42]). If two fractional-instantons approach to each other and merge into one, then the translation moduli of the fractional-instanton is back to the size-moduli of the integer-instanton. On the other hand, if the size of integer-instanton becomes larger than the radius of the compactified direction, then the integer-instanton divides into two fractional-instantons. We consider that the similar phenomenon dynamically occurs during the Montecarlo simulation for the Yang-Mills theory.

We put three remarks: The first one is related to the bion configuration. We investigated configurations in total, which are independently generated by using seeds of the random number from -th to -th sweep with the -sweep interval. Among them, there is no configuration contained by a pair of the fractional-instanton and the fractional-anti-instanton. Such a configuration is called bion, and it plays an important role for the resurgence structure in the model [27, 28]. The absence of the bion will be discussed in the summary section.

The second one is a decrease of the topological charge in the long PHB sweeps. At -th sweep, the number of configurations in the and sectors are and , respectively. As increasing the number of sweep, in fact at -th sweeps, the corresponding numbers become and . Thus, the number of configurations in the sector increases. It might suggest that only the sector remains after the infinitely updations. Actually, the self-dual configuration on this lattice setup is not an exact solution, and we need the additional twisted boundary conditions also in the and directions to make it [7]. This situation is the same with the ordinary QCD calculations. If the instanton size is small compared to the extension of the lattice, then we can see the various instanton configurations, since it is not sensitive of the choice of boundary conditions. To see an actual distribution of the topological charges on the two-dimensional twists must be found through the accumulation of the data in the large-volume and the low- simulations.

The third one is for the size-modulus of the fractional-instanton. According to Ref. [31], there is no size-modulus in the fractional-instantons. Several (exactly in total) fractional-instantons with are plotted in Figs. 4 and 6. It seems that they all has an unique shape, namely a similar height of the peak and a similar curve around it. The actual data of the peak-height takes between and . In our analyses, where we neglect at most error, it is hard to give a solid conclusion beyond the error. On the other hand, some fractional-instantons with the other charges have a broader width. We consider that the lost of the uniqueness of the shape for the fractional-instantons comes from the introduction of an additional one-compact (periodic) direction in comparison with the spacetime structure in Ref. [31]. The periodicity forces the total charge to take an integer-value. This constraint deforms the shape of configuration. We expect that a further large-volume calculation for the temporal direction and the continuum extrapolation will improve the situation, and will reveal the uniqueness of the shape for the fractional-instantons.

4.2 Polyakov loop and center symmetry

The Polyakov loop () is the order parameter of the center symmetry breaking, and it changes with and , under the center transformation. In the PBC lattice, the Polyakov loop in the -direction is given by the product of the link variables of the direction;


Here, we take the spacetime average for each configuration, and for and for the others. On the other hand, because of the twisted boundary condition, the definition of the Polyakov loop in the twisted directions are modified as,


in order to satisfy the gauge invariance and the translational invariance.

Figure 7: Scatter plot of Polyakov loop for each direction. Left panels denote the spacial (bottom) and the temporal direction (top). All directions have the periodic boundary condition. Right-top panel shows the results for the spacial direction with periodic boundary, while the right-bottom panel gives the ones for the spacial direction with twisted boundary (blue-triangle) and the temporal direction with periodic boundary conditions (red-square).

The scatter plots of the Polyakov loop for each direction are given in Fig. 7. Each data point denotes the data for one configuration. Here, all configurations are at -th sweep from the random start. The results for the PBC lattice are shown in the left panels of the figure. The , and directions are equivalent, so that here we present the Polyakov loop only in the and directions in the left-top and left-bottom panels, respectively. At with , it is clearly in the deconfined phase because of its scale (). Then, the Polyakov loop in the -direction locates in one of three degenerated vacua, whose complex phases are and . In the continuum limit keeping the physical lattice-size, one of them is chosen, therefore the center symmetry is spontaneously broken. This behavior is the same with the situation of the Yang-Mills theory in the high-temperature phase. The Polyakov loop in the direction seems to be invariant under the center symmetry, since they locate around the origin.

On the other hand, the right panels in Fig. 7 present the scatter plots of the Polyakov loop in the TBC lattice. The Polyakov loop for the direction, where the twisted boundary conditions is imposed, is shown in the right-bottom panel. Because of the twist matrix, the Polyakov loop locates around the origin in the complex plane. The result for the -direction is identical with the one for the -direction. For the -direction, the behavior is the same with the one in the PBC lattice. The -direction as shown in the right-top panel indicates a curious behavior, even though the boundary condition for the -direction is not changed from the PBC lattice. The distribution of the Polyakov loop are spread over the whole (almost) triangle, where the matrix of the Polyakov loop before taking the trace satisfies the unitarity condition. The location of each data is changed under the center symmetric transformation if , so that the center symmetry of the configurations is generally broken in the same meaning with results of the finite-temperature simulation. However, its breaking structure becomes milder than the one in the PBC lattice.

Note that in the right-top panel of Fig. 7, the red-circle symbols located in one of the -degenerated vacua denote the configurations with , while the blue-diamond symbols, which are inside of the triangle, corresponds to the configurations. The figure clearly suggests that there is a relationship between the value of and the Polyakov loop in the -direction.

4.3 Tunneling behavior and fractional instanton

Now, let us investigate the relationship shown in the end of the previous subsection. We introduce the Polyakov loop in the -direction on each lattice site;

Figure 8: Histogram of for each type of local charge distributions. The corresponding data of the local charge are shown in Fig 4.

The histogram of the for a typical configuration is shown in Fig. 8. Here, the corresponding data of the local charge are displayed in Fig. 4. We see that in the case of Type-I and Type-II(a), the site-by-site distribution of the Polyakov loop locates at one of the -degenerated vacua. In the case of Type-II(a), where the integer instanton exists, the tunnel phenomenon is not apparent, since it occurs within a tiny slice of -coordinate. The situation is similar with that for an ordinary integer-instanton in the strong coupling regime on , where the instanton configuration wind around the whole angle.

On the other hand, in the case of Type-II(b) configurations, two of the -degenerated vacua have been chosen. To see the manifest relationship between the fractional-instanton and the distribution of the Polyakov loop, we plot the averaged complex phase of conf. as a function of (the blue-circle symbols) in Fig. 9. Here, the averaged complex phase expresses as . Simultaneously, we also present the local topological charge as the red-square symbols, where it is multiplied by so as to be easily seen.

Figure 9: -coordinate dependence of the complex phase of the Polyakov loop in -direction (blue-circle) and the local topological charge (red-square).

We can find that if the value of the local charge is almost zero, then stays in or  222The small discrepancy of from at comes from the mix with a small excess around in the left-bottom panel of Fig. 8., while around the peak of the local charge (), where the fractional-instanton exists, the complex phase of the Polyakov loop start changing its value. It describes that the fractional-instanton is related to the tunneling between the -degenerated vacua.

The phenomena occur in the two-dimensional model that can be obtained by the dimensional reduction of the four-dimensional Yang-Mills theory with the twisted boundary conditions [31, 42, 43, 44, 45]. In fact, in the limit where the () directions shrink, the reduced theory of four-dimensional Yang-Mills becomes the two-dimensional nonlinear sigma model, whose boundary condition in the compactified direction () has the -holonomy. The non-zero expectation value of the Polyakov loop for the shrinking direction () corresponds to the vacuum expectation value (v.e.v.) of the complex scalar field in the reduced theory, where the v.e.v. depends on the location of . The fractional instanton can be interpreted as a classical solution connecting two different vacua. Our results in Fig. 9 show that the similar interpretation is valid even for the fractional-instantons of the four-dimensional Yang-Mills theory.

In the case of Type-II(c), the histogram of the Polyakov loop on site-by-site has three peaks at three degenerated vacua equally. No clear -dependence in the distribution of the Polyakov loop exists, so that we expect that the tunneling behavior among three vacua occurs also through and directions. The magnitude of the Polyakov loop given in Eq. (20) is very small (), so that the Polyakov loops in all directions locate near the origin in the complex plane. That means the center symmetry in such configurations is dynamically restored. In fact, such a dynamical restoration of the center symmetry is predicted in Ref. [31] on spacetime. In our numerical calculation on lattice, the appearance rate of Type-II(c) configuration is rare, in fact three per one-hundred. Although we naively expect that Type-II(c) might be dominant in the continuum and the limits, it must be an important future-work to find which type of configurations remains in the double limits.

4.4 Polyakov loop and confinement

In this section, we focus on the other nonperturbative phenomenon; the confinement. Although the center symmetric distribution described as can be interpreted as a confinement property in the ordinary lattice calculations, we have tuned the value of the lattice parameters to realize the weak coupling regime, where the colored particles can be described by asymptotically free particles. Here, we make a definition of the confinement and the deconfinement in the theory clear, whose center symmetry is restored even in the perturbative regime.

We have seen that the center symmetry seem to be restored by observing the Polyakov loops in the and directions. The center symmetric property in the and directions is forced by hand, since it comes from the twist matrices in Eqs. (21). On the other hand, the Polyakov loop in the -direction, where a periodic boundary condition is imposed, also indicates the center symmetric property. In our calculation, both and directions have the same periodic boundary condition. Thus, if we take (or ), then we could find the Polyakov loops in all directions exhibit the center symmetric distribution.

Generally, the Polyakov loop is related to the free-energy of a single (probe) quark, . The large appears in the confined phase, and it diverges in the infinite-volume limit. In our calculation, we take a large lattice extent () with an extremely small lattice spacing, so that we have to make the origin of the smallness of clear; whether it comes from the sizable or the large with a finite value of .

Figure 10: Magnitude of Polyakov loop in -direction as a function of . We take from right to left.

Figure 10 shows the absolute value of the Polyakov loop in the -direction for two lattice parameters; on the TBC lattice and on the PBC lattice. It is known that the latter lattice exhibits the confinement behavior, since the critical temperature in the finite temperature simulation is determined as in  [46]. The blue-circle symbols denote the results for the PBC lattices as a function of . We see that all data in are consistent with each other within - statistical error bar, so that there is no dependence. It is natural in the confined phase, since the Polyakov loop does not feel the lattice extent because of the confinement. On the other hand, the -dependence of the TBC lattice (red-circle symbols) clearly decreases as increasing  333We cannot directly compare the absolute values of Polyakov loop between and , since the lattice raw data have to be multiplicatively renormalized, where the renormalization factor depends on the value of  [47]. . The data at , which we mainly focus on in this work, is still in the middle of its decreasing beyond the statistical error bar. We conclude that the configurations with the fractional-instanton have the deconfinement property as expected.

5 Summary and future works

We have numerically studied the nonperturbative phenomena of the SU() gauge theory in the weak coupling regime on with the large aspect ratio between two radius. Furthermore, we introduce the twisted boundary condition in two directions to realize the perturbative standard vacuum on the hypertorus. The twisted boundary condition induces the IR momentum cutoff for the gluon propagator, and is also related with the existence of the fractional instanton. According to the analogy of the quantum-mechanical models and the low-dimensional quantum field theories, such a deformed spacetime structure might be necessary to see the the resurgence structure. This work is the first step to find a fractional-instanton and its nonperturbative properties within the lattice gauge theory. It looks a promising regularization method to give a well-defined construction of the Yang-Mills theory from the perturbative to the nonperturbative regimes.

The numerical results on the TBC lattice exhibit that the total topological charge () can take nonzero values even in the weak coupling regime, while the PBC lattice calculations with the same lattice parameters generate only the trivial solutions. We have shown that some multi-fractional-instantons locally exist in the configurations with . The fractional-instantons can merge into the integer-instanton and vise versa during the PHB update processes. The fractional-instanton will be a standard integer-instanton in the decompactified limit, since if the lattice extent becomes larger then the confining scale (), then the situation back to the ordinary zero-temperature calculation in the strong coupling regime.

We have also investigated the center symmetry by observing the Polyakov loop for each spacetime direction. The Polyakov loop in the -direction on the TBC lattice shows the different behavior from the one on the PBC lattice, though the boundary condition of the direction does not change. The distributions of the Polyakov loop are scattered over the possible unitary triangle in the complex plane. The configuration lives in one of the -degenerated vacua if it has the zero total charge, while the others, whose Polyakov loop locates inside the unitary triangle, correspond to the ones with . We have shown that the local fractional-instanton connects two of the -degenerated phases of the Polyakov loop in the -direction. On the other hand, the Polyakov loop in the -direction seems to be center-symmetric, but its scaling property indicates the deconfinement property. Furthermore, we have found that there are some configurations (Type-II(c)), whose Polyakov loops in all directions exhibit the center symmetric property.

For future works, we address following points:
Resurgence structure and renormalon effect
To see a resurgence structure, we would like to find the renormalon pole and its contribution to a physical observable in the perturbative expansion on this twisted spacetime, since it would cancel a contribution of the fractional topological objects in the sector. Until now, the perturbative expansion of the plaquette value up to has been numerically investigated on with the twists in three dimensions using the stochastic perturbation theory [48]. They have determined the normalization of the Borel singularity and estimated the effect of the renormalon to the gluon condensate with a not-so-small error bar. As shown in the present work, the spacetime structure gives a large influence on the physics both qualitatively and quantitatively in the perturbative regime. Finding the divergent structure and the corresponding effect from the renormalons on with two-dimensional twists by using a similar calculation with Ref. [48] is a next task .

Adiabatic continuity to the nonperturbative regime
It is expected that the Yang-Mills theory with the twisted boundary condition goes to the ordinary confined phase in the large volume and the strong coupling limit, since in these limits the boundary effects can be negligible. The question is whether the phase transition occurs between the perturbative and the nonperturbative regimes. According to the PBC calculation in the context of the finite-temperature Yang-Mills theory, the pure SU() gauge theory exhibits the weakly first-order phase transition. However, as shown in §. 4.2 and §. 4.4, the TBC lattice simulation on indicates a curious behavior: the center symmetry seems to be partially restored even in the weak coupling regime, while the configuration still exhibits the deconfinement property. Furthermore, in comparison with the finite-temperature phase-transition (see Fig.  in Ref. [49]), the -dependence for the Polyakov loop in the TBC lattice with the small lattice size has a gentle downward slope toward the strong coupling regime (shown in Fig.  in Ref. [15]). These numerical results on the twisted lattice might suggest the absence of the phase transition toward the decompactified limit, which is conjectured as an adiabatic continuity (Ref. [50] and references therein). In fact, the continuity has been established in the case of and sigma models in the large- limit [51].

Other lattice calculations to find a fractional instanton:
Schrödinger functional boundary, the four-dimensional twisted, and other approaches
It might be worth to mention the other formulations, which show the fractional instantons on the lattice.

The most popular lattice setup, which connect to the perturbative vacuum, is the one with the Schrödinger functional boundary condition [12]. The similar discussion with this work might be possible by using the lattice setup with the Schrödinger functional boundary with a large aspect ratio between the extents of the spacial and temporal directions. At that time, the zero mode still remains on the lattice with the ordinary Schrödinger-functional boundary condition, then to find the fractional-instanton it may need an additional gauge fixing or the other technique [52] to kill all zero modes and to stabilize the fractional-instanton.

We can also consider the twisted boundary condition for three or four directions. In fact, although it has been discussed in the strong coupling regime, the lattice numerical simulations with four-dimensional twists have been carried successfully for the SU() pure gauge theories. The fractional instanton and the action density have been observed [53, 54]. Note that the theory with four-dimensional twisted boundary conditions locally has the same gauge symmetry with SU(), but the global symmetry becomes SU(). On the other hand, the three-dimensional twisted lattice still keep the same global symmetry with the SU(). As same as our calculation, the total on the lattice with three-dimensional twists always takes an inter-value. On the other hand, the discussions in §. 4.2 and §. 4.3 of this work will be changed, since the additional twisted boundary condition for the -direction must deform the distribution of Polyakov loop in the -direction.

The other approaches to find the fractional topological charge on the lattice have been done by using the Dirac operator with the higher-dimensional representations on the periodic lattices [55, 56, 57]. The index theorem for the higher-dimensional Dirac operator and the one for fundamental representation are related with each other, where the color degree of freedom appears as a factor between them. Furthermore, the caloron solution at the finite temperature with a nontrivial homonomy has also been constructed [58, 59, 60, 61]. The energy density and the zero-mode density of the caloron for the SU() and SU() gauge theories have been numerically investigated [62].

Finding bion configurations
We cannot find a pair of instanton and anti-instanton with neither integer nor fractional charge in the sector, so that there is no evidence of the bion configuration in our simulation. In the resurgence scenario, the existence of the bion is important since the fluctuation around such configurations is one candidate of the renormalon. In the case of the integer instanton and anti-instanton, the force between them is attractive, so that it easily annihilates each other if the distance of them is short. We expect that the force between the fractional ones is also attractive, and then the similar phenomena might occur in our simulation. We consider that the bion will appear in the decompactified limit of the temporal direction.

On the other hand, the reduced theory of the four-dimensional Yang-Mills theory, namely two-dimensional model with the twisted boundary condition, suggests the sign of the potential depends on the relative phase between the fractional-instanton and fractional-anti-instanton [24, 26, 27]. The large volume simulation and the study on the dynamics between the fractional-instantons for the Yang-Mills theory are interesting future works.

Including the dynamical quarks: -QCD and adjoint QCD
One of the most important directions is to find a similar fractional topological object in the QCD(-like) theories; namely including the dynamical fermions. It is known that there are at least two promising models; the -QCD [63, 64] and the adjoint QCD models [50, 65, 66, 67]. Both theories have the exact center symmetry of the SU() gauge group in the action, so that we expect that the similar -degenerated vacua and the corresponding fractional instanton might appear in the weak coupling regime.

The formulation of the two-dimensional twists can extend to the system including the dynamical fermions [68]. Actually, in the case of the fundamental fermions, the numerical simulation for the SU() gauge theory coupled to with the staggered (Kogut-Susskind) fermions has been carried out [15]444Note that the theory has an IR fixed point, and it is not QCD(-like) theory.. If we utilize the other lattice fermions, it is possible to reduce the number of the fermions using the exact algorithm. The advantage of the usage of the twisted boundary conditions is not only for the absence of toron but also for the induced IR momentum cutoff, as we explained. Then, we can perform the simulation with the exact massless fermions. It must be helpful to investigate the adiabatic continuity near the massless limit as discussed in Ref. [50].


The author appreciates S. Aoki and K. Yonekura for deeply discussions about the lattice numerical data, and would like to thank T. Fujimori, T. Misumi, M. Nitta, N. Sakai, and M. Yamazaki for valuable comments. We are also indebted to S. Aoki, T. Misumi, K. Nagata and N. Sakai for helpful comments on this manuscript. The author is grateful to the organizers and participants of “RIMS-iTHEMS International Work-shop on Resurgence Theory” at RIKEN, Kobe for giving her a chance to deepen their ideas. In particular, the talk given by K. Yonekura at the workshop and the private conversation were invaluable to start this work. Numerical simulations were performed on SX-ACE at the Research Center for Nuclear Physics (RCNP), Osaka University. This work is supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT)-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (Grant No. S1511006), and is also supported in part by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (KAKENHI) Grant Number 18H01217.


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