Large N reduction on a twisted torus

We consider SU(N) lattice gauge theory at infinite N defined on a torus with a CP invariant twist. Massless fermions are incorporated in an elegant way, while keeping them quenched. We present some numerical results which suggest that twisting can make numerical simulations of planar QCD more efficient.


INTRODUCTION
At an infinite number of colors, QCD on an Euclidean torus of size l 4 undergoes a staircase of transitions as l is reduced [1]. For l > l c , where in ordinary QCD units l c ∼ 1 f ermi, the system is in a phase where Wilson loop operators of arbitrary size have traces that are exactly l-independent. Thus, one can reduce the number of degrees of freedom from that of an infinite torus, without any loss of information at leading order in the 1 N expansion. This ought to be of help in getting at planar QCD using numerical simulation, as reduction holds on the lattice too. In practice this implies that increasing N reduces the finite size corrections, so that a balance between N and the size of the system can be reached which minimizes the computational effort required to get the planar limit of various physical QCD observables. It is likely that getting Monte Carlo numbers in the planar limit is cheaper than solving full QCD with the computer.
To obtain numbers appropriate for zero temperature infinite volume planar QCD one must make sure that all simulations are carried out at lattice spacings a and lattice sizes L which obey La > l c . For sufficiently fine lattices, L turns out to be of order 10. In this paper we aim to reduce this value even further by making use of an old idea due to González-Arroyo and Okawa [2]. We consider pure SU(N) YM theory on a twisted torus. At infinite N the large volume phase should be independent of the boundary conditions in as much as it does not depend on the size of the box. When the volume is decreased, the theory enters a phase where some dependence on boundary conditions is restored. This phase must be distinct from the corresponding phase in the untwisted case, where it corresponds to finite temperature deconfined planar QCD. Therefore, it is conceivable that the critical size l t c for the twisted box is smaller than l c and this follows from general arguments. It is also supported by the quantitative numerical work presented below.
TWISTED TORUS -PURE GAUGE SU(N) gauge fields are objects in SU(N)/Z(N) and therefore the allowed bundles are those of SU(N)/Z(N) over the torus [3]. Some of these bundles cannot be lifted to an SU(N) bundle and when this happens we'll say that we have a "twisted torus". In this work we are only interested in the CP invariant case where N is assumed to be even. We consider a nontrivial SU(N)/Z(2) bundle over the torus. To ensure that the classical limit is as simple as possible, we further restrict N to be divisible by four. This ensures that there are flat connections in the bundle. It is easy to transfer this continuum gauge bundle to the lattice. If N were not divisible by four, only by two, the bundle would admit only half integral topological charges and the minimal action configuration would have nontrivial space-time structure.
We use a single plaquette Wilson lattice action and the gauge group is SU(N), where N = 4M and M is chosen to be prime. Our choice of twist can be induced by choosing the sign of the lattice coupling β to be negative and taking a symmetrical hypercubic lattice of volume L 4 with L given by an odd integer. As is well known, the unusual sign of the coupling could be absorbed by a change of lattice gauge variables for even L, and there is no twist. The same change of variables is inconsistent at the boundaries when L is odd, where it becomes equivalent to the nontrivial boundary conditions one would use if one defined the twisted bundle in the continuum by starting from an open sub-hypercube of R 4 . Another way to see that a negative lattice coupling amounts to twisting by -1 for odd L is to observe that the Z(2) flux through any plane becomes (−1) L 2 .
The change of variables needed to bring the twisted action to a negative coupling Wilson action also affects observables. In particular, let C denote a closed finite curve C on an infinite lattice, mapped in the natural way to the torus. Associated with C there is a sign,  [4] to conclude that in the large N limit 1 N trW L (C; b, N) is L independent. Their proof can be extended [2] to twisted boundary conditions. This result can be also deduced from an analysis of the strong coupling expansion [5] directly.
Thus, in the strong coupling region one has: The above equation does not extend all the way to the continuum limit |b| → ∞ [6], but previous work has produced evidence in favor of its validity beyond the radius of convergence of the strong coupling expansion. The basic idea of this paper is to estimate As explained in the introduction, for reduction to work at a given b > 0, one needs L ≥ L c (b). Based upon past experience with twisting and on arguments to be given later Our numerical findings indicate that this is true, opening the way to more efficient numerical work on planar QCD, employing a CP invariant twist.
The proof of reduction in perturbation theory [2,7,8]  Thus, uniform averaging would be a correct procedure if we knew that we are in a phase where the entire Z 4 (N) stays unbroken even at infinite N. This average over the moduli space at infinite N effects an extension of the discrete momentum sums associated with ordinary Feynman diagrams on a torus to continuous Feynman integrals on the smooth space of crystal momenta normally associated with an infinite lattice, with the angles filling in the momentum gaps typical of a finite spacetime torus. As we have seen above, effectively, twisting fractionalizes the Brillouin zone into 16 identical hypercubes, facilitating the gapfilling role assumed by the remaining angular parameters. All in all, twisting "helps" the system to maintain the global Z 4 (N) at N = ∞ and this is required for reduction to work.
When |b| is increased too much, one expects the global Z 4 (N) to break spontaneously in the large N limit. We are not certain of the phase of the theory when L < L c (−b). The simplest guess would be a breaking of one of the Z(N) factors down to Z(2). Substantial numerical work would be needed to determine whether this is correct, or if another alter-native takes over. This is an issue we postpone to the future. In this work, we shall carry out tests mainly at one particular b-coupling. From these results, we shall be able to also conclude that in the test cases the entire Z 4 (N) symmetry group was preserved and that reduction held.

NUMERICAL TESTS -PURE GAUGE
Our first goal is to check whether the twist indeed helps in reducing the lattice size L at which we can attain the low temperature symmetric phase. For that purpose, we take a lattice spacing a for which we know that reduction works on a periodic lattice only for L larger than a specific L c . Then, we try to find out whether a simulation with twist on a smaller size L t < L c torus is able to reproduce the value of various large N observables.
Here we make use of the size independence of the results in the large N limit and in this phase.
In particular, we chose an inverse 't Hooft coupling, b set to |b| = 0.36; the corresponding lattice spacing is a ≈ (2.1 GeV ) −1 , quite typical of current QCD simulations on what is considered a fine lattice. Using regular boundary conditions at this lattice spacing would require L ≥ 9. We ran a series of tests which show that a twisted lattice of size L t = 5 at the same value of |b|, is able to remain in the phase where full reduction holds, but staying away from the lattice strong coupling phase. (The latter phase extends from |b| = 0 to about |b| = 0.36, but at large N the tunneling rate into the strong coupling phase can be kept so low that one does not need to worry even about going slightly below |b| = 0.36.) Our Wilson loops were built out ofŨ µ (x) matrices, rather than the original link matrices U µ (x). TheŨ µ (x) matrices are defined in term of the U µ (x) by an iterative "smearing" procedure [9]. Let Σ U (n) µ (x) denote the "staple" associated with the link U (n) µ (x) in terms of the entire set of U (n) ν (y) matrices. One step in the iteration takes one from a set U (n) µ (x) to a set U (n+1) µ (x), by the following equation: We chose α = 0.45 in the untwisted case and iterated twice: Given the change of variables mentioned previously this changes to α = −0.45 in the twisted case. Smearing has the effect of removing some of the ultraviolet fluctuations and produces more meaningful numbers for our comparison. Also, the test is made more stringent by including smearing because smearing is a relatively complicated operation involving longer loops and, although it should survive twisting theoretically, numerical effects might have marred the equivalence to the untwisted case.
In Table I These were linear extrapolations in 1 N 2 and are shown in Figure 1. We compare the action density (raw plaquette average), and smeared n by n Wilson loops for n = 1, 2, 3, 4, 5. The agreement is within error bars, but the large N extrapolation works better in the untwisted case; nevertheless, the coefficients of the 1 N 2 correction come out quite close in the twisted and untwisted cases, except in the twisted case of the 5 × 5 Wilson loop, where the linear fit in 1 N 2 does not work well. Theoretically, one expects that the 1 N 2 correction depend on the shape of the box, and since the twisted box is of size 5 4 , this is a natural place to see some larger corrections. There is no question that the twisted simulations took less computer time, but a more quantitative comparison of efficiency needs to take into account the errors.
The errors in Table I were estimated without taking correlations into account because we did varied kinds or runs. The quoted errors should be viewed just as rough indicators. We don't have enough information for a quantitative efficiency estimate, but there is little doubt that it pays to twist. When the fermion twisting trick is taken to the lattice more checks are needed. First of all, we certainly want to preserve the hard won option of maintaining exact global chiral symmetry at finite lattice spacing [11]. In the untwisted case we know how to do this for each fermion flavor independently. Now we need to make sure that the coupling of the flavors in the specific way associated with twisting can be taken to the lattice. We cannot first put each flavor on the lattice and then couple them, because we would loose explicit lattice translational invariance. So, we must first couple the flavors and then carry out the overlap construction, as it is obvious that the twisting procedure meshes well with the sparse Wilson lattice fermion action.
In the untwisted case, the bilinear fermionic action for one flavor is described by the massless overlap Dirac operator D o [11]: is the Wilson Dirac operator at mass m, which we shall choose as m = −1.5.
The T µ matrices are the lattice generators of parallel transport and depend parametrically and analytically on the lattice links U µ (x) which are SU(N) matrices at site x associated with the link connecting site x to site x +μ, whereμ is a unit vector in the positive µ-direction.
The internal fermion-line propagator, 2 1+V is not needed at infinite N, as the fermions are quenched at leading order in 1 N . For fermion lines continuing external fermion sources we are allowed to use a slightly different quark propagator [12,13] defined by: and anticommutes with γ 5 . The spectrum of A is unbounded, but is determined by the spectrum of V which is restricted to the unit circle. One should think of A as dimensionless, and of |m| as providing the needed dimension. Up to a dimensional unit, A should be thought of as a lattice realization of the continuum massless Dirac operator, D: Twisting involves the T µ operators, which would act now on two indices, color and flavor.
There is also the notational inconvenience since we have to deal with three kinds of gamma matrices: the Lorentz γ µ from above, the Γ µ of color space that enter the classical vacua, and now, in addition,γ µ acting on flavor. To include the twist for fermions, the new T µ operators are extended by: Given the rather intricate nature of this mechanism, a numerical check is highly desirable; as we shall see, it works very well. Before turning over to numerical results, we wish to point out that had we been interested in dynamical fermions, the case of fermions in the double indexed antisymmetric representation of SU(N) is an excellent candidate theoretically, as it is unaffected by our Z(2) twist. In this case, no extra flavors are needed. Thus, for projects trying to get at the planar limit of supersymmetric QCD [14], our Z(2) twist poses no fermionic problems.

NUMERICAL TESTS -FERMIONS
Previous work employing an untwisted action at infinite N in the physical phase, showed how the use of random matrix theory [15] to calculate the fermion condensate and establish spontaneous chiral symmetry breaking on the lattice. In the twisted case, at infinite N, we can do the same. We wish to check that after proper rescaling we find a condensate identical to the condensate we found using the regular method, just multiplied by four on account of the four flavors [16].
In the untwisted case we gathered a large amount of data at b = 0.35, which is a coarser lattice. To facilitate comparison of the bare quantities directly we now focus on the twisted This looks good in comparison, but we must keep in mind that the second eigenvalue in the twisted case also appears to obey its universal prediction after rescaling, but now we get 1 N ψ ψ = (0.1102(4)) 3 . Therefore, the second eigenvalue has not yet converged to its random matrix distribution, and we cannot be sure that the first already has, although the indication is that it did.

THREE DIMENSIONS
Very similar constructions hold in there dimensions, with the obvious difference that one has no topological charge to worry about and one can take N = 2M with M prime. Brief tests we have carried out in three dimensions also indicate that twisting allows one to deal with fine lattices at smaller lattice volumes than in the regular approach.

SUMMARY AND DISCUSSION
The outstanding question is to clearly characterize the phase that the twisted system decays into when l decreases to just below l t c and numerically check that this phase survives the continuum limit. Until this is done, one cannot claim to have an understanding of twisted simulations at the same level as of untwisted ones. Twisted simulations hold the promise of substantial savings in computer time due to the ability to work at even smaller volumes than when using conventional periodic boundary conditions on the torus.
For our twisted simulations we used our regular untwisted code, and simply used a negative b and a negative α, leaving everything else the same. Our regular code was tuned for the untwisted case, but seemed to perform reasonably also on the twisted case. Further work is needed to tune a code specifically for the twisted case, specifically for weaker couplings than the ones used in untwisted numerical simulations.