Large- reduction in QCD with two adjoint Dirac fermions
We use lattice simulations to study the single-site version of lattice gauge theory with two flavors of Wilson-Dirac fermions in the adjoint representation, a theory whose large volume correspondent is expected to be conformal or nearly conformal. Working with as large as , we map out the phase diagram in the plane of bare ‘t Hooft coupling, , and of the lattice quark mass, , and look for the region where the center symmetry of the theory is intact. In this region one expects the large- equivalence of the single site and infinite volume theories to be valid. As for the case (see Phys. Rev. D 80: 065031), we find that the center-symmetric region is large and includes both light fermion masses and masses at the cutoff scale. We study the -dependence of the width of this region and find strong evidence that it remains of finite width as . Simulating with couplings as small as , we find that the width shrinks slowly with decreasing , at a rate consistent with analytic arguments. Within the center-symmetric region our results for the phase structure, when extrapolated to , apply also for the large volume theory, which is minimal walking technicolor at . We find a first-order transition as a function of for all values of , which we argue favors that the theory is confining in the infrared. Finally, we measure the eigenvalue densities of the Wilson-Dirac operator and its hermitian version, and use large Wilson loops to study the utility of reduction for extracting physical observables.
There has been a recent revival of interest in the possibility of using complete volume reduction for the infinite (number of colors) limit of QCD and QCD-like theories. If this reduction holds, then the theory, defined nonperturbatively on a lattice, gives predictions that, at infinite , are independent of the number of sites. Specifically this means that the theory defined on a single site, or a small, fixed number of sites, is large- equivalent to the corresponding infinite volume theory with the same bare parameters EK ().
Reduction to a single-site has long been known to fail for the pure gauge theory (and thus also for QCD in ’t Hooft’s large- limit, since quark contributions are suppressed by in this limit) BHN (); KM (); Okawa (). This failure is due to the breakdown of one of the conditions needed for a large- orbifold equivalence between the single-site and large volume theories (see Refs. Neuberger2002 () and KUY1 ()). This key condition is that the center symmetry of the single-site theory must be unbroken. This symmetry breaks spontaneously111 Strictly speaking, the symmetry is spontaneously broken only for . In practice, however, effective spontaneous symmetry breaking is seen in simulations at finite but large values of , and we use the terminology of phase transitions throughout this article. in the single-site pure gauge theory (the Eguchi-Kawai [EK] model EK ()). This is expected from perturbation theory (PT), where, at leading order, the effective potential for eigenvalues of the holonomy around the compact direction (the Polyakov loop) leads to attraction and thus clumping BHN (); KM (). Several years ago, it was realized that the addition of massless fermions that reside in the adjoint representation and that have periodic boundary conditions in the compact directions leads (in perturbation theory) to a repulsion between eigenvalues, which in turn leads to a uniform distribution of these eigenvalues KUY2 (). In this case the center symmetry is unbroken and reduction holds.
Two of us have previously investigated the single-site theory with a single Dirac adjoint fermion (discretized using Wilson fermions), finding that, for small to rather large values of the inverse ’t Hooft coupling, , there is a large range of values of the quark mass for which the center symmetry appears to be unbroken AEK (). This result was unanticipated because several leading-order perturbative calculations, done with a single compact direction (or a single site along only one Euclidean direction) show that the symmetry breaks if the physical mass exceeds a value of BBPT (); HollowoodMyers2009 (); AHUY (). As increases from to , the perturbative calculations indicate that the eigenvalue density of the link in the short direction will form a number of clumps, starting with clumps at very small masses, and decreasing to a single clump at infinite mass. The results of Ref. AEK () (which used up to 15) do not follow this pattern, instead finding no clumping for masses up to of for all values of . A semi-quantitative understanding of these results has recently been given in Refs. AHUY (); UY10 (). With more than two compact dimensions, the fluctuations in the eigenvalues can overwhelm the tendency to clump, and this happens up to masses of . The numerical results for the phase diagram obtained in Ref. AEK () have also been checked, and extended, in Ref. AHUY (). We also note that simulations with , and massless overlap adjoint fermions also find no center-symmetry breaking, at least at large HN1 (); HN2 ().
In the present paper we extend our investigations to . The main motivation for doing so is that the corresponding infinite volume theory is expected to be nearly conformal, and thus a candidate “walking technicolor” model.222For recent reviews of technicolor models on the lattice, see Refs. Rummukainen:2011xv () and DelDebbio:2011rc (). Indeed the theory with two colors is the theory with the smallest field content that lies close to the conformal window, and has been dubbed the “minimal walking technicolor” (MWT) model. If reduction holds, then we should be able to study a close relative of MWT, i.e. the theory with . One naively expects only a weak dependence on , because the number of both gluonic and fermionic degrees of freedom scale as .
We also note that the AEK (Adjoint Eguchi-Kawai) model is related by a combination of orbifold and orientifold equivalences to the QCD-like theory with fermions in the two index anti-symmetric (AS) irrep ASV (); KUY2 ().333This equivalence holds only in charge-conjugation even subsectors. This theory in turn is the large- limit of QCD with quarks, but with the limit taken with the quarks in the AS irrep (which is equivalent to the anti-fundamental for ). This is the Corrigan-Ramond large- limit of QCD CorriganRamond (), which differs from the ’t Hooft limit in having fermion loops.444Note that here taking the Corrigan-Ramond limit moves one from a theory which is not close to the conformal window (4 quarks in the fundamental irrep) to one that is (4 quarks in the AS irrep), suggesting that corrections are probably large, at least in this respect.
Our main effort herein is to determine the phase structure of the AEK model. To do so we have upgraded our simulation algorithm from a Metropolis algorithm, with CPU scaling as , to a Hybrid Monte-Carlo (HMC) algorithm, for which we find . This allows us to reach much larger values of , and to improve the statistics. Together, these advances allow us to study the nature of the symmetry breaking in more detail than in the study AEK (), allowing us to compare with the theoretical expectations of Refs. AHUY (); UY10 (). Our main result is that we find the phase diagram to be qualitatively similar to that for , with center symmetry remaining unbroken for masses up to . Specifically, our evidence suggests that, at fixed coupling, although the range of masses for which the symmetry is unbroken shrinks somewhat as increases, it remains of width as . Our strongest evidence for this is at , but our results suggest that this holds for , i.e. for the entire range of coupling that one could possibly be interested in. Thus our results suggest that one can use adjoint fermions of almost any mass to “stabilize” reduction. This is only expected to fail in the extreme weak coupling limit ().
This is an encouraging result, and so we have made the first steps in trying to see if reduction can be used to obtain results for physical quantities. The key question is how large a value of is needed so that the physical contributions are larger than those from effects. We have investigated this by studying the large extrapolation of the plaquette, by calculating the spectrum of the Wilson-Dirac operator and its hermitian counterpart, and by calculating large Wilson loops in order to see if we can extract the heavy-quark potential.
Work along similar lines has recently been reported in Ref. CGU (). These authors simulate the theory with two adjoint Wilson fermions on a lattice, and use . They report evidence that, for , there is a region of quark masses around the putative critical value, including quarks of masses , for which the center symmetry is unbroken.
This paper is organized as follows. In the following section we describe the AEK model and the properties of the large-volume theory to which it would be equivalent were reduction to hold. In Sec. III we describe the algorithm that we use, and show some results concerning its performance. Section IV is the core of the paper, in which we use our numerical results to determine the phase diagram of the AEK model. We then, in Sec. V, present first results for “observables”—the spectra of the Wilson-Dirac operator and its hermitian counterpart, and large Wilson loops. We close in Sec. VI with a summary and a discussion of the outlook for future work. An appendix describes models for eigenvalues of the Wilson-Dirac operator that are used in Sec. V.
Ii The AEK model and its putative large-volume equivalent
The partition function of the single-site theory is
where the four are SU(N) matrices, while and are Grassmann Dirac variables of flavor , living in the adjoint representation of . We use the Wilson gauge action
where is the product of links around the plaquette in the plane, and is the inverse ’t Hooft coupling,
We also use Wilson’s lattice Dirac operator
where is the adjoint representative of , and is the usual hopping parameter, related to the bare quark mass by
Periodic boundary conditions on both gauge and fermion fields have been built into the form of . Throughout this paper we set .
The theory has a center symmetry, under which
where and are integers. Note that is invariant under this transformation, so that the fermion action is also invariant. There is also the single-site version of the gauge symmetry
Finally, there is an flavor symmetry, most easily seen by writing the action in terms of four Majorana fields.555In the continuum this symmetry becomes an symmetry, but, with our choice of fermion discretization, only its subgroup remains an exact symmetry of Eq. (1)).
If reduction holds, this single-site theory is equivalent, when , to a theory that has any number of lattice sites in each of the periodic directions , including the case of . The action of the theories has the same form except that , , and , are now fields having a site index, contains a sum over the position of the plaquettes, and connects fermion fields at adjacent sites. We stress that an important feature of reduction is that it relates the single-site and infinite-volume theories having the same bare parameters, and .
We recall some important properties of the infinite-volume theory, since these will be inherited by the single-site theory if reduction holds. First, the theory is asymptotically free— fermions are required to change the sign of the first coefficient of the -function. Second, although the bare quark mass vanishes when , this critical value of is additively renormalized because Wilson fermions do not preserve chiral symmetry. The critical value is shifted to , and the physical quark mass becomes
Here we have introduced the lattice spacing, , which can be determined, in principle, by fixing the value for a physical scale, such as a particle mass. Since the theory is asymptotically free at short distances, one approaches the continuum limit () by sending , and in this limit .
The nature of this critical line depends on the infrared behavior of the theory. One possibility is that the theory lies below the conformal window, so that chiral symmetry is spontaneously broken, much as in QCD. Then, for near , one can study the long-distance behavior and vacuum structure of the lattice theory using chiral perturbation theory (ChPT). In particular, close to the continuum limit, one can use a modified ChPT which includes discretization effects SharpeSingleton (). For adjoint fermions, the symmetry breaking pattern differs from that in QCD, and is . The required generalization of the analysis of Ref. SharpeSingleton () has been given in Ref. DelDebbio08 (). One finds that, as in QCD, there are two possible scenarios: either there is a first-order transition line, at which the degenerate pseudo-Goldstone “pions” attain their minimal, non-zero mass, or there are two second-order lines, along which the pions are massless, and between which there is an Aoki-phase Aoki_phase (). Within the Aoki-phase, the vector symmetry is broken.666We note for completeness that Refs. nofirstorder1 (); nofirstorder2 () have recently raised concerns about the consistency of the “first-order scenario.” The width of the Aoki-phase is , and thus shrinks rapidly as one approaches the continuum limit.
A different possibility for the critical line arises if the massless theory is conformal in the infrared, i.e. if there is an infrared fixed point. There is growing numerical evidence that this is the situation in the theory. For this theory, the simulations of Refs. Catterrall:MWT08 () and Kari:MWT08 () map out parts of the phase diagram. In particular, there is single, second-order transition line emanating from , while for the line becomes a first-order transition. (A similar picture holds for an improved fermion action, but in this case the second-order line extends to stronger coupling dGSS11 ().) This is not established definitively, and also does not directly apply to the theory that we are interested in. Nevertheless, this possibility provides a quite different phase diagram than that which applies when one is outside the conformal window. One of our aims is to see which possibility holds at (assuming that reduction holds).
Iii Simulation algorithm
We simulate the single-site theory using the hybrid Monte-Carlo (HMC) algorithm HMC (). Integrating out the fermions leads to
As usual, hermiticity implies that is real, so we can write
where is the Hermitian Wilson-Dirac operator. Since has positive eigenvalues, we can represent its determinant using pseudofermions. Introducing momenta conjugate to the link variables, we end up with the HMC Hamiltonian
The are traceless hermitian matrices, while the pseudofermion is complex, lives in the adjoint representation of , and has an implicit Dirac index. It thus has complex components.
In practice, we represent in color space as a traceless bifundamental, i.e. as a traceless matrix, on which acts as
In this way we do not need to explicitly construct .777We thank Simon Catterall for stressing this point to us. In fact, one could, in principle, keep the trace of , since the singlet field that it represents has no impact on the dynamics. In particular, one can show that, in exact arithmetic, the molecular dynamics (MD) trajectories that are followed are identical with or without included, as is the change in . We find, however, that the number of CG iterations required for a given accuracy is larger if is included, presumably because one has to do some work to find the solution for the singlet part. Thus we always set .
Our implementation of the HMC algorithm is standard. We invert using the conjugate gradient (CG) algorithm, with a weaker stopping criterion during the MD evolution than for the accept-reject step. We require that the residue, , with the source, satisfies during MD evolution, finding that any further increase of the cut-off leads to a drop in the acceptance. For the accept-reject step we use , which makes the error in negligible. Our CG always starts from a vanishing guess, , which assures reversibility of the trajectory. We use trajectories of unit length, and adjust the step size to attain acceptances of .
In deriving the gluonic force, one must account for the fact that each link appears twice in each plaquette. Nevertheless, the final result has the standard large-volume form:
The fermionic force is
where and are Dirac indices, , and . Both forces maintain the tracelessness of .
We now discuss the scaling of CPU time with , which is a key factor in determining how large one can take . The core operation—multiplication of matrices—scales as . The use of the bifundamental form of , rather than the adjoint, is crucial here, reducing the scaling from to , as pointed out in Ref. CGU (). The next contribution to the overall scaling comes from the number of CG iterations, . This turns out to depend on the proximity to the critical line. An example is shown in Fig. 1, for the stopping criteria given above. Away from the critical line, is independent of , while near the line it grows roughly like . The third ingredient is the inverse step size, or equivalently the number of MD steps () per trajectory (for a given acceptance rate). We find that, to good approximation, this grows linearly with . Thus, for trajectories of unit length, CPU time scales as away from the critical line, and roughly as near to the line. Both scalings are considerable improvements over that for the Metropolis algorithm used in Ref. AEK (), which is for each SU(2) subgroup update and for an entire update. On the other hand, our scaling is not as good as the estimate of given in Ref. CGU (), which assumed and that is independent of , and explicitly excluded the possible effects of critical slowing down.
We have done both horizontal (fixed ) and vertical (fixed ) scans in the plane, studying the ranges and , although our main focus has been on the smaller ranges and . We use in these scans. Rather than quote a complete set of run parameters we give a representative example. We have, for , , and , used 27 values of ( in steps of ). At each , we start from the “configuration” output from the previous value, thermalize for 500 trajectories, and then run for 7500 (), 5000 (), 2000 () or 1000 () trajectories during which we make measurements every 5 trajectories and store the configurations every 50. Each scan is done in both directions—the UP and DOWN scans denoting increasing or decreasing parameter values (either or ). To give an example of the CPU time required, the UP scan took 33, 155, 342 and 618 CPU-hours of a single core on 3.0 GHz Intel Xeon processor, for , , and , respectively. Our simulations have been done on local workstations and using up to 32 CPU cores on a computing cluster.
We have also done longer runs at several points in the plane, in which we have gone up to . Details of these runs will be given below.
Iv Phase diagram of the AEK model
In this section we present our main results, from which we deduce the phase diagram sketched in Fig. 2. The most important conclusion is that there is a “funnel” in which the center symmetry is unbroken, on either side of the first-order transition which we identify with . The diagram is qualitatively similar to that found for AEK ().
iv.1 Measured quantities
To study the gross features of the phase diagram, we calculate the average of the plaquette, , defined by
To study center symmetry breaking, we consider general “open loops”:
where . These loops transform non-trivially under the center symmetry, unless all four are integer multiples of . They are thus order parameters for center-symmetry breaking.888We have used to keep the quantity of data manageable. This has the disadvantage that the traces are then insensitive to symmetry breaking such as . Histograms of link eigenvalues, to be discussed below, are, however, sensitive to such symmetry breaking. The simplest choices, on which we focus, are the four Polyakov loops, and the 12 corner variables, and , with .
As in the quenched Eguchi-Kawai model, the corner variables turn out to be particularly useful because they are sensitive to partial symmetry breaking QEKBS (). We illustrate this with simple examples. First, if for all , then all the Polyakov loops and corner variables are unity. This corresponds to complete breaking of the center symmetry. If instead
(where we have assumed that is divisible by 4, so that ), then the Polyakov loops vanish, while all corner variables are unity. In this case, there is a unbroken subgroup: both and are invariant under the subgroup of generated by
The themselves are invariant under the combination of Eq. (18) and a gauge transformation, the latter being the similarity transformation which interchanges the first diagonal entries with the second entries.
Such partial symmetry breaking can be discussed in a gauge invariant way by considering the eigenvalues of link matrices. For each link we can write
with containing the eigenvalues:
Gauge transformations can permute the eigenvalues, but not change their values. Center-symmetry transformations change the eigenvalues by a uniform translation: . Thus a direct way of looking for certain symmetry breaking schemes, and understanding their nature, is to look at the distributions of the . For example, unbroken center symmetry implies a distribution which is invariant under translations by . Partial symmetry breaking occurs when a subgroup of such translations is unbroken. In the first example above, the eigenvalues are all clumped, and all translation symmetries are broken. In the second example, the eigenvalues form two clumps, and translation by remains a symmetry.
We use the link eigenvalues in Sec. IV.5, plotting histograms and considering the correlations between links in different directions.
iv.2 Scans at moderate coupling ()
In this section we use scans of the plaquette, Polyakov loops and corner variables to map out the gross features of the phase diagram.
The most interesting values of are roughly ; this was the range studied in the model AEK (). For , this corresponds to , a range running from couplings similar to those used in large-volume simulations to very weak coupling. We have made detailed scans at , , and . That at shows a great deal of structure that is hard to analyze (including large hysteresis and the influence of a bulk transition), while that at interpolates between the results at and . Thus we show, in Figs. 3 and 4 respectively, scans of the plaquette at and . We have simulated with , , and , but, for the sake of clarity, show results only for and . We also show, in the central region, an approximate estimate of the result at , obtained by fitting results at the four values of (or more values, if available) to . Such fits will be discussed in Sec. IV.3.
The results at show three main features: (i) a change in slope at (and possibly another at ), (ii) a jump at , and (iii) a transition region at . These correspond on the phase diagram of Fig. 2 to (i) the transition region from center-symmetry broken phases to the unbroken central region, (ii) the transition line at , and (iii) the transition to the region of multiple broken phases for large . We focus first on the central feature, presenting our evidence concerning symmetry-breaking below. An important issue is whether the jump at finite survives as a first-order transition at . Our extrapolations suggest that it does: although the jump in the plaquette decreases with , it appears to remain finite at .
The conclusion of a first-order transition is clearer in the results at . These show the same qualitative features as for , but the jump in the plaquette is larger, and there is hysteresis for and .999We note in passing a peculiar phenomenon we have seen for smaller values of in the range we consider. At , the UP scans for show, in the hysteresis region, points having average plaquettes with non-vanishing imaginary parts. This breaks the charge conjugation symmetry of the theory, and is reminiscent of results found in the twisted EK model TEKfailure (). It is because of these points that we do not have an extrapolated result for the UP scans at in Fig. 4. We suspect that this occurs only in metastable phases, and view it as a sign of the complicated vacuum structure of the single-site theory.
To study center symmetry breaking, we first use scans of the absolute values of Polyakov loops and the corner variables. Both should vanish as if the symmetry is unbroken. The corner variables are more informative and we show an example, for , in Fig. 5. Results for are qualitatively similar.
For both small and large , and , the corner variables indicate that the center symmetry is broken. The nature of this breaking is clarified by the Polyakov loops, , whose plots we do not show for the sake of brevity. For we find to be non-vanishing as , indicating that the center symmetry is completely broken. This is the phase shown in Fig. 2. For and , however, are consistent with zero at , indicating only partial symmetry breaking. The nature of this partial breaking can be elucidated using the distributions of and in the complex plane, and using histograms of link eigenvalues. Some examples of the latter will be shown in Sec. IV.5.
A key issue for reduction is the realization of center symmetry in the central funnel, . For the values of used in the scans, we find no indication of symmetry breaking. Our evidence is as follows. Scatter plots of and in the complex plane show a single distribution centered around the origin, with averages consistent with zero. Similarly, the higher-order traces, , which we have evaluated at several positions inside the funnel, are all consistent with zero. Finally, histograms of link eigenvalues, examples of which are shown in Sec. IV.5, are also consistent with the absence of symmetry breaking.
The other key question is whether the funnel remains of finite width as . We can see from Fig. 5 that the funnel narrows with increasing . In particular, the lower edge of the funnel, which we call , increases from at to at . To study this question further requires larger values of , and we defer consideration until Sec. IV.4.
To complete the study of the phase diagram we have done several vertical scans with , and . We show an example of the results in Fig. 6, which displays the for and . For all , there is no indication of symmetry breaking at strong coupling, , just as for the EK model. There are possible transitions, however, as we increase . For example, at we see two transitions: one at and a second at . The first is from a center symmetric phase to one in which both Polyakov loops and corner variables are non-vanishing, consistent with complete symmetry breakdown. The second is to a phase with large corner variables and smaller Polyakov loops, which we interpret as a partially broken phase. This is the same “” phase apparent in Fig. 5 for .
For all other there is no hysteresis, so we show only UP scans. At , we see only a single transition, at , and this is directly to a partially broken phase with only non-zero. For higher , however, the symmetry is unbroken on both sides of the jump at , and we interpret this as a bulk transition. It is unclear, however, whether this corresponds to a phase transition or a crossover as .
Vertical runs at higher values of fill in gaps left by the horizontal scans, and are part of the input which leads to the phase diagram of Fig. 2. For the sake of brevity, however, we do not show any plots here.
iv.3 Scaling of the plaquette, ,
In order to study the key question of whether the symmetry unbroken funnel remains as , we have extended the calculations to larger values of at several several values of and .
We begin by looking at the average plaquette. If reduction holds, then, away from , the single-site theory is equivalent at large to a large-volume lattice theory with quarks having physical masses of . The long-distance physics of such a theory is that of a pure-gauge theory with action modified from the pure Wilson form by fermionic effects. If is much smaller than , these modifications should be small, since they are proportional to powers of the small quantity (as can be seen using the hopping parameter expansion). The large-volume theory is thus close to the pure-gauge theory with Wilson action. We can therefore make the semi-quantitative prediction that, near the lower boundary of the funnel, , the average plaquette should lie close to the large-volume, pure-gauge (Wilson action) value, but depart from that value as one approaches . On the other side of the transition, however, we do not expect the plaquette to tend to this same value as increases. This is because is now larger, so the action differs more significantly from the pure-gauge Wilson form.
Figures 3 and 4 show that the plaquette has considerable dependence on , with the slope of this dependence varying with . We show in Fig. 7 an example of an extrapolation in which the plaquette decreases with , and in Fig. 8 an example in which it increases. Results are for and are plotted versus . We use this variable because we have found that we cannot obtain a reasonable fit without taking the leading correction to be proportional to (rather than ). Examples of fits to and are shown in the figures, fitting either to all the data or dropping the two lowest values of . We find that fits of to the highest six values of are tolerable (probability ) for all choices and that we have considered, and use such fits to obtain the results given in Table 1.
The results for the plaquette at are in striking agreement with the semi-quantitative prediction explained above. In particular, for and they are consistent with the pure-gauge large-volume results, while for (close to ) they begin to differ. Thus, looking back at the scans of the plaquette in Figs. 3 and 4, we see that the extrapolation to leads to an almost constant value between the onset of the funnel at and , with the value being close to that of the pure-gauge theory.
We also have results for a single point above : at . We find here that the extrapolated plaquette differs, with high significance, from that below . This is consistent with our semi-quantitative prediction, and also indicates that the first-order transition at survives the large limit.
We now return to our numerical finding that the leading corrections to the plaquette scale as . This result has also been found in the numerical studies of Ref. AHUY (). It differs from the naive expectation that, with fields in the adjoint irrep, corrections should be powers of . We find, however, that fits to are only possible with very large coefficients (, ) of alternating signs. We consider these fits unreasonable since we expect coefficients of .
In fact, there are (at least) two possible sources of terms. The first can be seen from the perturbative result for the plaquette when one has a center-symmetric vacuum Okawa ()
which manifestly contains a correction.101010We thank Ari Hietanen for reminding us of this result. This correction arises from the fact that in the decomposition (19) non-trivial fluctuations in lie in . In other words, the fluctuations must be off-diagonal, leading to the factor . We expect that the one-loop form (21) should work reasonably well at (as it does for in Table 1) and that the predicted correction should be most applicable for the smallest values of (where fermionic contributions to the plaquette are minimized). Indeed the result for at , lies close to the prediction of .
A second source for corrections are the “would-be zero modes” of the Wilson-Dirac operator , which we discuss in more detail in Sec. V.1. There are of these (corresponding, as in the gauge case above, to the diagonal generators of SU(N) in perturbation theory), and they form an fraction of the total number of modes. Unless the contribution of these modes is exactly canceled by an contribution from the remaining eigenvalues, these modes can cause observables to depend on odd powers of . Our results for the spectrum of suggest that they play an important role in the dynamics for the values of at which we simulate.
We now turn to the extrapolations of and . In the large- limit, these can be written, using factorization, as and , respectively, both of which vanish if the center symmetry is unbroken. Thus an important test of our tentative phase diagram is that and extrapolate to zero within the funnel.
Ordinarily, corrections to factorization are proportional to , but, in light of our experience with the plaquette, we might also see corrections. In Fig. 9 we plot versus for and two values of . There is qualitative agreement with a fall-off for both ’s, but fits to a pure form, examples of which are shown in the figure, have low confidence-levels. Satisfactory fits (one example of which is shown) can be found by adding a term and dropping the lowest two values of .
Results from such fits, for all the values of and at which we have done runs up to , are collected in Table 2. The fits to all have reasonable confidence levels. The coefficient of the term is small in all cases, and in fact is consistent with zero (within ) except for , . We also show results of fits to a constant plus term. The fits are of very similar quality, and the constant turns out to be very small, and consistent with zero except, again, at , . We conclude, aside from this one point near to the edge of the funnel, that the behavior of Polyakov loop is consistent with the hypothesis that reduction holds in the funnel.
Turning to the corner variables, examples of which are shown in Fig. 10 with full results collected in the Table, we find a surprising result: at , , starts to increase once exceeds , and clearly does not extrapolate to zero. The simplest interpretation of this result is that the center symmetry is broken for . There is, however, no other evidence for such symmetry breaking. In particular, the distributions of the and are approximately uniform around the origin, the traces of Eq. (16) are all consistent with zero, and the link eigenvalues (to be discussed below) are distributed uniformly.
Instead, our favored interpretation is that the increase in with is due to the lower edge of the funnel, , increasing with . This increase can be seen (albeit for ) in Fig. 5 by comparing the results at and . It is quite possible that, for , as increases, approaches , possibly exceeding this value for . This would lead to the observed increase in since this quantity increases as one approaches the transition (as can be seen in Fig. 5). This could also explain why our fits to were less satisfactory at , .
For all the other values of that we have considered decreases monotonically with . This is exemplified by the results in Fig. 10. As for the Polyakov loops, a pure fit fails in most cases, but here we find (cf. Table 2) that the addition of a term usually leads to a better fit than the inclusion of a constant. We also find that, in almost all cases, the required (or constant) term has a coefficient which differs significantly from zero. We have also done fits to (an example is shown if Fig. 10) but the fits require very large coefficients having opposite signs, a fine-tuning which we consider unlikely to be the correct description. Overall, we think the most reasonable fits are those to , because they have the highest confidence levels, and because we have seen in the plaquette that terms are needed.
The results presented so far are consistent with the funnel (in which center symmetry is unbroken) remaining of finite width as , so that reduction holds for masses up to . We cannot definitively draw this conclusion, however, because of the following scenario. Imagine that the funnel width vanishes (for any fixed ) as . Then, for each point in the putative funnel, symmetry breaking would occur at a (possibly large, but) finite, value of . Nevertheless, there would be a finite range of for which the symmetry is unbroken, within which the arguments for reduction hold. Appropriate variables (such as the plaquette) would equal infinite volume values up to corrections proportional to powers of . Thus, within this range, it might appear that one can extrapolate to the symmetry-unbroken limit, but this would in fact not be the case. The results for the plaquette at , in Fig. 9 are an example of such misleading scaling, since we have strong evidence from the corner variables that at this for large enough .
iv.4 -scaling of the funnel width
In the light of the results in the previous subsections it is important to directly study the dependence of the funnel width. We have done so by focusing on , the lower edge of the funnel. If we can show that remains below as , then the funnel remains open.
To investigate this issue we have done fine scans of the small region, an example of which is shown in Fig. 11. There are two phases before one enters the funnel: a phase for and a phase from . The transition between the first and second phase shows significant hysteresis, while that between the second phase and the funnel does not.
Determining to high precision is a significant numerical challenge. We have thus focused on a single value of coupling, . For this , Fig. 3 shows that lies in the range . We have done very fine scans near the edge of the funnel (roughly ) with up to 53, and find that the corner variables are the most useful in determining the transition. We are able to pin down the transition, conservatively, to about . The transition from the funnel is to a phase for most (as in Fig. 11), but to a phase for and . The results are plotted against in Fig. 12, and show remarkable linearity (note that, as earlier, we have excluded and —including them requires adding a term to obtain a satisfactory fit). Two fits are shown. The first is to , and has a very small , with a reasonable coefficient of . It yields , a value far below . The second fit is to , with the intercept fixed to . This fit is extremely poor, and it gets even worse for . We have also done the corresponding fits to the alternative quantity , using or , and find consistent results. We conclude that, at least at this value of , the funnel has finite width when , so that reduction holds.111111Note that, if we use the linear fit, then the funnel at passes when . Thus the successful extrapolations of the plaquette, and for , , presented in Tables 1 and 2, are examples of the phenomenon described above in which reduction only holds for a window of values of .
iv.5 Distributions of link eigenvalues
We find that histograms of link eigenvalues provide very useful information on the nature of symmetry breaking on either side of the funnel. They are also sensitive to patterns of symmetry breaking in which both Polyakov loops and corner variables vanish, and thus provide a more stringent test that the symmetry is indeed unbroken in the funnel. In this section we present examples of the results that allow us to fill in the details of the phase diagram of Fig. 2.
We begin with an example of a histogram within the funnel, shown in Fig. 13(a). The eigenvalues are taken to have the range , and are collected in bins of width . Thus symmetry implies invariance under periodic translations by multiples of 3 bins. In fact the distribution is consistent with being uniform.121212As a check on our code, we have calculated the distribution in the , limit, i.e. for the Haar measure on the links, and obtain the theoretically expected form, which is -invariant, but does oscillate within each segment, although the amplitude of the oscillations falls as NN03 ().
When we move outside the funnel the attraction between eigenvalues leads to formation of clumps in the complex plane. The number of clumps identifies the approximate remnant symmetry and generally decreases as we move away from the funnel. In Fig. 13 we provide several examples of clumping patterns. Figs. 13(b) and 13(c) present and phases in the small region while Figs. 13(d), 13(e) and 13(f) show , and phases in the large region. Note that the partial symmetry breaking can also be seen in the corner variables giving complex patterns (compare Fig. 5). Polyakov loops are much less sensitive to this partial symmetry breaking, since they almost vanish due to the approximate symmetry.
We find that the remnant symmetry is not always exact. For example, in Fig. 13(c) we have two clumps for and in Fig. 13(d) we have three clumps for . Therefore the eigenvalues cannot be equally distributed between the clumps and the symmetry is only approximate. Even in Fig. 13(f), which shows five clumps for , we see that the clumps are not even and correspond to 7,6,7,4,6 eigenvalues, respectively. We also find that different runs can have different patterns of eigenvalue clumping, e.g. 7,7,6,6,4 versus 7,6,6,6,5, but that it is rare for the clumping to change during a run. Thus it appears that there are competing “vacua” which are not exactly related by center symmetry transformations.
To fully understand the symmetry breaking, we need to know whether there are correlations between the eigenvalues of different links. What we find is that, whenever the center symmetry is broken, the eigenvalues for all four links are highly correlated. To illustrate this, we use a case with three clumps which makes the results easy to visualize. Figure 14 shows the resultant clumping and correlations. Here we apply a gauge transformation which diagonalizes and orders the phases , and then look at the phases of the diagonal elements of . These matrices are close to diagonal, so these phases are presumably close to those of their eigenvalues. Recall that there is no ambiguity in the ordering of the diagonal elements once we specify the order for . The result shows that the clumps (of 6, 4 and 6), while being positioned at different angles, are almost completely correlated between all four links, and do not change during the Monte-Carlo evolution. Because of these correlations, the approximate remnant of the center symmetry for the parameters of Figs. 13(d) and 14 is , and not .
Once outside the funnel, the number of clumps decreases as we move to higher . We have extended some runs to and find that the UP scans end up in a two clump state, while the DOWN runs, which begin from an ordered start, begin with a single clump, and then have a transition, as is decreased, to two (well separated) clumps. In fact, the transition appears to occur in stages where more and more eigenvalues peel off from the initial clump. As is further decreased, the number of clumps increases until we enter the funnel and there is no longer any clumping. The largest number of clumps depends on , and the largest we have observed is five, as shown in Fig. 13(f).
The changes in clumping for the large values, surveyed above, are qualitatively consistent with the arguments presented in Ref. AHUY () based on the one-loop free-energy for the link eigenvalues. Decreasing from a large value corresponds to reducing the quark mass . For large , gluonic interactions dominate the free energy, and lead to attraction, and thus a single clump. As is reduced, the fermionic contributions lead to repulsion at large eigenvalue separations (corresponding to large momenta, so that the mass term is unimportant), while there remains attraction for small separations. This allows the possibility of two clumps. Reducing further the repulsion becomes important for smaller eigenvalue separations, and clumps are pulled apart into a greater number of stable clumps. At the same time, quantum fluctuations within each clump are always present, so that the clumps have a finite width (which is proportional to for weak coupling). Eventually, as the number of clumps increases, the distance between the clumps is smaller than the widths, and the clumping is washed out.
We would expect a similar sequence of clumpings to occur as we increase from zero, since this also corresponds to reducing . This is indeed what we observe, although the maximal number of clumps is smaller on this side of the funnel. For we only find a phase, for we see both a and phase (see Fig. 11), while for and we find , and phases. At larger values of , the arguments of Ref. AHUY () imply that the maximal number of clumps should increase. Indeed, we do find that, as increases, the phase appears at smaller values of .
iv.6 Results at large
The perturbative calculations of Refs. HollowoodMyers2009 (); BBPT () lead us to expect that the center-symmetry-unbroken funnel will close as , so that, in the continuum limit, reduction only holds for (for ). In addition, Ref. AHUY () makes a prediction for how rapidly the funnel should close: its width in (and thus in ) should be proportional to . This is because the width of each clump of link eigenvalues is predicted to scale to zero proportional to . A second prediction (which is explained in the previous subsection) is that, at the edge of the funnel, there should be multiple phases with differing numbers of clumps, and that the maximum number of clumps should increase with (as long as is large enough). In other words, a behavior similar to that we have already seen on the right side of the funnel (; see Fig. 2) extends to larger groups as increases.
We have investigated these predictions by doing scans in for (and, in some cases ) at , , and . We find that the HMC algorithm mostly performs well even at these very weak couplings. We did have to reduce the step-size as increases, such that increases roughly like . We also find that, for , thermalization for each new value of sometimes takes longer than our allotted 450 trajectories. On the other hand, the number of CG iterations gradually decreases with increasing . We also note that run histories of observables show no indication of correlation times that are close to the number of trajectories we were using for measurements (7500).
Results for the plaquette are collected in Fig. 15. The general shape of each curve is similar to those at (see Fig. 3) but the jump at the putative falls rapidly with increasing . This is qualitatively consistent with the expectations from chiral perturbation theory if this is the first-order scenario of Ref. SharpeSingleton ().
To verify the hypothesis of Ref. AHUY () that we analyze the lower edge of the funnel as a function of . It would obviously be advantageous to repeat the analysis of Sec. IV.4 for all values of ; unfortunately this is numerically too demanding. We do, however, have estimates of at and for a wide range of . These are shown in Fig. 16. We find that it is harder to determine for away from unity. For smaller values, e.g. , the transition becomes smoother. For much larger values, there is significant hysteresis (as seen in the middle panel of Fig. 15). The net result is that the errors in are much larger than those at .
Figure 16 also shows fits to , with set to for all values of for simplicity. The fit at is good, while that at is poorer. Better fits at can be obtained using estimates for the actual value of at each , but these estimates have sufficient uncertainty that the resulting errors in are substantially increased, so that the agreement with the theoretical form is less significant. Overall we conclude that our results are consistent with the predicted dependence on .
The figure also indicates that the narrowing of the funnel as increases holds for all values of . We do not attempt to extrapolate to from and , as our experience with indicates that is not in the asymptotic region. However, given that we do find a finite width when at , the observation of mild dependence on suggests that the funnel will remain of finite width also at other values of .
V Measurements inside the “funnel”
In this section we make a detailed study of the funnel region in which reduction appears to hold using results from . We consider in turn the spectrum of the single-site Wilson operator , the spectrum of , and, finally, attempt to extract a physical observable—the heavy-quark potential—from large Wilson loops.
v.1 Spectrum of
One way of viewing the equivalence of single-site and large-volume theories is that the space-time volume is being packaged inside the gauge matrices. It is thus useful to introduce an effective size, , and corresponding effective volume, , and study their scaling with . What we mean by is that the single-site theory leads to the same physical results as the theory on an volume with colors, up to corrections suppressed by powers of . Clearly there is a trade-off between increasing and increasing . Here we take the approach of holding fixed, but large enough that corrections to quantities of interest are small, and then asking how scales with .
Within this framework, the most conservative possibility is provided by orbifold-based demonstration of volume independence KUY2 (). This demonstration also provides an explicit example of the packaging of the volume into the gauge matrices: the link matrices are partitioned into blocks of size , with , only of which are non-zero, and the resultant orbifolded theory is an gauge theory on an lattice. Equivalence is demonstrated by holding fixed, and taking , and thus also , to infinity. Instead, using the approach espoused above, if we hold fixed, then increasing leads to .
A less conservative possibility is obtained, following Refs. AMNS (); U04 (), by assuming that all entries in the link matrices are used in the packaging of the large volume theory (not just 1 out of every as in the orbifold construction). This leads to or . There is also a more optimistic possibility, , which is motivated in the Appendix.
The spectrum of the fermion matrix, , can help distinguish these possibilities, as well as give insight into the nature of corrections to the large- limit. We expect, if reduction holds, that the spectrum should resemble that of a large-volume four-dimensional theory on an lattice. In particular, for weak couplings, , the spectrum should have the familiar five “fingers” which reach down to the real axis. The number of fingers is a direct indicator of the dimensionality (there are in dimensions), and the distance of the eigenvalues in the fingers from the real axis should scale as . These points are discussed in more detail in the Appendix.
We now show some representative results for the spectrum of from our simulations. The operator in the determinant is
so that eigenvalues of close to are suppressed. Since the spectrum is bounded, , the determinant suppression is important only for . We also note that, unlike on a lattice with an even number of sites in each direction, the spectrum is not symmetric under reflection about the axis, Thus the first (the leftmost) and fifth fingers are not related by symmetry, and neither are the second and fourth. If such a symmetry holds approximately, it indicates the presence of reduction.
In Fig. 17 we show how the spectrum changes as we vary at fixed and . At , where we are in the phase (see Figs. 3 and 5), we see one main finger and a small indication of a second. This is consistent with the eigenvalues forming a single clump, so that the momenta, given by eigenvalue differences, are all small. At we have moved into the phase, with two clumps of eigenvalues. We see that this allows the spectrum to spread into all five fingers, because eigenvalue differences can now range up to . The details of the spectrum differ from those of a large-volume free fermion, however, in particular having a low density of points in the central three fingers and a “rectangular” shaped envelope. Nevertheless, it is clear that one must interpret the spectrum with care—the presence of five fingers alone does not imply that reduction holds.
The next value, , is well inside the funnel, and we see a distribution which is qualitatively similar to that of a free fermion, with a rounded top and five fingers. These features are present for all inside the funnel. Particularly noteworthy is the presence of the comet-shaped clump of eigenvalues near the origin. We find that there are exactly eigenvalues per configuration in this clump. We thus interpret them as would-be zero modes, i.e. eigenvalues that would be zero if . These modes are dropped in weak coupling calculations, both because they do not impact the dynamics (as they do not depend on the ) and because they form only an fraction of the total number of modes. The spectrum indicates, however, that they could have an important impact on the long-distance dynamics which might overcome their relative paucity. We recall that for a large-volume Wilson operator it is the small eigenvalues which determine long-distance behavior such as chiral symmetry breaking. For very large we expect small eigenvalues to come dominantly from the first finger, which should approach the real axis. What we see from the figure is that is quite far from this limit. Thus we conclude that the would-be zero modes are a potential source of the corrections observed above in the plaquette and other quantities, and that their contribution could be sizable (given how far the “true” low-energy modes in the first finger are from the real axis).
The spectrum within the funnel but just above the transition is illustrated by the result for . Eigenvalues near are suppressed by the determinant. The would-be zero modes cluster to the left of this excluded point, while the first finger now approaches closer to the real axis. The latter feature indicates that the funnel-region above the transition is more continuum-like, which is consistent with its larger average plaquette. On the other hand, the spectrum as a whole is less symmetric about the line than that below the transition.
Moving to , which is still inside the funnel, the would-be zero modes have spread out again (perhaps because the excluded point has now moved to ), while the first finger has become longer and denser. The second finger, however, has almost disappeared.
Finally, at we are in the phase. This is reflected by the spectrum breaking into three distinct regions (only two being visible since the third has negative imaginary part), resulting from eigenvalue differences distributed around and .
We have done similar scans at lower , but the results are less illuminating, because the bulk of the spectrum moves closer to the real axis, such that, at