TThe quenched Eguchi-Kawai model revisited.
Herbert Neuberger ∗ Department of Physics and Astronomy,Rutgers University,Piscataway, NJ 08854, U.S.A (Dated: 09–20–2020)The motivation and construction of the original Quenched Eguchi-Kawai model arereviewed, providing much greater detail than in the first, 1982 QEK paper. A 2008article announced that QEK fails as a reduced model because the average over per-mutations of eigenvalues stays annealed. It is shown here that the original quenchinglogic naturally leads to a formulation with no annealed average over permutations.
PACS numbers: 11.15.Ha, 11.15.PG
I. INTRODUCTION.
In 1982 Eguchi and Kawai (EK) [1] made the spectacular observation that SU ( N ) latticegauge theory with no matter fields gave Wilson loop expectation values that were repro-ducible on a minimal 1 lattice at leading order in 1 /N . Intensive research ensued. It wasfound that large N phase transitions caused difficulties in approaching the continuum limit.Some cures were suggested. The purpose of this paper is to make explicit the underlyinglogic and definitions of one of them: the QEK conjecture [2]. It will be shown that the 2008article [3] does not establish failure of the general idea of quenching. A determination of thevalidity or failure of quenched reduction requires future extensive numerical work. II. EGUCHI-KAWAI REDUCTION IS NOT JUST A LATTICE PECULIARITY.
At first sight EK reduction seemed to be an incredible lattice trick. Only much morerecently has numerical work indicated that EK reduction is a property of continuum planar ∗ Electronic address: [email protected] a r X i v : . [ h e p - l a t ] S e p four dimensional SU ( N ) pure gauge theory [4]. At leading order in 1 /N the expectation val-ues of Wilson loops on an infinite Euclidean four-torus are the same as “folded” counterpartson any finite torus with side size larger than about one Fermi in QCD terms.In retrospect, this sharpens the original question that motivated [2]: how is a Coulomb-like law realized in a finite periodic Euclidean four volume which has no room for “Faraday’sflux lines” to spread?Continuum EK reduction is some times referred to as “partial reduction” because thelattices one has to use in a numerical simulation in order to approach the continuum limit toreasonable accuracy must be substantially larger than 1 . “Partial” is a bad epithet becauseit emphasizes a detail of implementation and conceals the physical content of the result. III. LATTICE FIX OF EK REDUCTION.
The first conjectured fix for lattice EK reduction to an 1 lattice was the quenched EKmodel, QEK [2].The main physics question was how the largeness of the SU ( N ) group at infinite N couldprovide a place-holder for an infinite lattice while consisting of just four unitary link matri-ces. A simple calculation at one loop order showed that the eigenvalue phases of the linkmatrices in the four directions could play the role of continuous lattice momenta in ( − π, π ] – an “emerging” toroidal momentum space – and produce the standard Coulomb’s forcelaw on the lattice if we summed up the contributions of a large number of saddles and ig-nored their instabilities. As was very well understood from other semi-classical calculations,fluctuations in the “flat” directions, connecting the saddles, produced zero modes and wereeasily dealt with. But, the saddles in the integral were dominated by coalescing eigenvalues,so perturbation theory was unstable. As a whole, the matrix integral was benign.On an 1 lattice the eigenvalue sets of each link matrix are gauge invariant angles. Theyare unique candidates for lattice momenta.The QEK fix consisted of a removal of the link matrices’ eigenvalues from the set ofannealed variables. They were quenched instead. This difference did not matter at large N by a degrees-of-freedom counting argument: There were 4 N angles, and order 4 N matrixelements: The angles ought to be governed by uniform and uncorrelated distributions ineach direction if some obvious symmetries remain preserved as N → ∞ .For strong lattice coupling (small “ β ”), where EK had been proven to work, the quench-ing prescription would have no effect to leading order in N . Quenched or annealed, theangle distributions would be frozen to continuous uniform densities in each direction andthose would be uncorrelated. The requirement to match onto the original EK version, whichwas proven to hold at strong coupling, left little freedom for constructing QEK. The “loopequations”, on which the EK proof relied, have trivial boundary conditions in the strongcoupling limit and determine the entire strong coupling series. That series has a finite radiusof convergence. The precise boundary conditions for the lattice loop equations at weak cou-pling remain unknown to date. They would be needed for constructing the Feynman seriesfor Wilson loops. The loop equations themselves have only a relatively formal continuumlimit. They do not offer a reliable tool for analysis in continuum directly. IV. QUENCHING IN DETAIL
QEK was originally presented as a conjecture and this remains its status to date. It isuncertain whether it is valid throughout the bridge connecting short and long distance puregauge theory physics. Even if it does, there remains doubt whether it would be practicalin comparison to the safer continuum EK method which relies both on the lattice loopequations and on some numerical, nonperturbative tests.The QEK prescription is explained below in detail and at an elementary level.
A. Quenching “with calculus”
The lattice variables we shall deal with are phase angles and unitary matrices.The first step in constructing the quenching prescription consists of a proper changeof integration variables in the EK case. For simplicity, we consider U ( N ) – restricting to SU ( N ) later presents little difficulty. The change of variables requires a one-to-one relationbetween the old and new, together with a matching of integration domains. The domainsin the new variables are obviously important and will be discussed later below.The variable change is an EK → QEK map replacing each of the link variables U µ , µ =1 , , , θ iµ , i = 1 , ..., N and a unitary matrix V µ . For four fixed U µ ’s , thereare multiple solutions. To determine domains of integration in the new variables requiresselecting one unique branch among them.(1) U µ V µ = V µ D µ ( θ µ ); (2) U µ = V µ D µ ( θ µ ) V † µ . (IV.1)where D µ ( θ µ ) = diag( e iθ µ , e iθ µ , ....e iθ Nµ ) . (IV.2)The original integration measure, is (cid:89) µ =1 , , , dU µ where dU µ is Haar . (IV.3)After the change of variables the integration measure is locally given by (cid:89) µ =1 , , , [ dV µ (cid:89) i =1 ,...N dθ iµ (cid:89) i 2) location and soon. Real positivity fixes the phase of each column. It mods out the factor U (1) N in the“numerator” of IV.5. For the lack of a better term, let us call this ordering “diagonally rightrow entry dominant”. This clumsy name avoids confusion with the standard term “diagonaldominance”. With probability one, this choice is unambiguous.In the EK case, at any finite N , which specific choice of representative was made does notmatter because the integral over ordered angles and bases is done on one common integrandand the U (1) N and S ( N ) are symmetries of the integrand. In the QEK case however, theintegral over the angles occurs at a later stage, with a modified integrand and the choice doesmatter because S ( N ) acts simultaneously on θ ’s and V ’s. At finite N quenched expectationvalues depend on which representative of each equivalence class was chosen. This is a crucialfeature of quenching.The correct ranges of integration in QEK are over a set of four V ’s, all in “canonical”order. A criterion for choosing the canonical order is that the V representative be “perturba-tive”. There are no restrictions on the angles. Different orders of the same set are includedas separate contribution. There may exist other prescriptions that are equally valid. To besure, for large lattice β couplings the reduction validity of QEK remains a conjecture.Permutations of eigenvectors must be eliminated in order to perform a correct variablechange in EK with no over-counting at any coupling. That is just applied multivariatecalculus.To motivate the ordering prescription for the V matrices, consider the simple case of U (2) /S (2). The two U (1) terms in the “numerator” of eq. IV.5 are ignored. Up to anirrelevant overall phase, any V ∈ U (2) can be written as V = w z ∗ − z w ∗ . (IV.6)where | w | + | z | = 1. S (2) = { , σ } using Pauli’s notation. Let V (cid:48) be given by the sameexpression with w, z replaced by w (cid:48) , z (cid:48) . The equation V = V (cid:48) σ defines an equivalent pair V (cid:48) ∼ V . In components, z (cid:48) = − w ∗ and w (cid:48) = − z ∗ . For V to be “diagonally right row entrydominant” we need | w | > | z | ; then V (cid:48) is not “diagonally right row entry dominant” because | w (cid:48) | < | z (cid:48) | . The split of U (2) into “halves” is explicit. The half containing the identitymakes up the QEK integration domain for V . As usual, ambiguous cases are ignored forprobability reasons. U ( N ) can be restricted to SU ( N ) by adding a factor of δ π ( (cid:80) i θ iµ ) in the θ integral foreach direction. δ π is the 2 π -periodic δ function. Mentally, one can imagine the V ’s to bealso fixed by an overall phase, restricting to SU ( N ); this phase does not enter observablesdependent only on the U ’s.Once the change of variables in the EK integrand is correctly implemented one can replaceeach U µ by equality (2) in eq. IV.1 in the EK model and nothing has changed. The integralfor the partition function can be done successively, first integrating over all the columns ofthe V µ in the pair [ θ iµ , i -th column of V µ ] with local Haar measure at fixed, ordered, anglesat each µ . Next, one integrates over all of the possible ordered angle sets for each µ .Quenching replaces the EK by QEK. In QEK the integral for the partition functionis replaced by an integral with the same measure and action, but at fixed angles in alldirections. The QEK partition function is a function of these angles, Z ( θ ). For a Wilsonloop observable one uses Z ( θ ) as normalization, now in the denominator, obtaining averagesof the observable at a fixed ordered θ set. Next, these annealed V -averages are integratedover the angles with weight given by the Jacobian in eq. IV.4. The θ variables are treatedas a set of random couplings, akin to the J -couplings of a spin-glass model.The traditional choice for angle ordering is descending along the diagonal with values inthe segment ( − π, π ]. Such an ordering makes the distribution equal to the derivative of asmooth approximation to the angle dependence on the index in each direction separately [6].But, this is not permitted in QEK. The order of the angles cannot be restricted in any way.One has a diagonalizing ordered basis and one can assign to each eigenvector an eigenvalueon the unit circle, distributed just according to the Jacobian factor. The Jacobian measureis invariant under direction dependent permutations. They are not induced by annealedgeneration of permutations among the columns of V because two distinct “diagonally rightrow entry dominant” V -matrices cannot be related by a nontrivial permutation.There is no invariance under the hypercubic group at fixed θ . It could happen thatnew large N transitions occur, to phases where the hypercubic symmetry is spontaneouslybroken. Such phases indeed do occur in the EK model [4]. If they persist to the quenchedcase, QEK fails. It also could happen that it is practically impossible to attain high enoughvalues of N because prohibitively large samples of the eigenvalue sets are needed for areasonable accurate estimate of the final angle integral.The ordering of V determines that of V † . V † is not “diagonally right row entry dominant”.The action controlling the V average depends only on the six combinations V µν = V † νµ ≡ V † µ V ν for µ > ν [2]. These are overlap matrices of ordered eigenvector-sets corresponding to theangles in the µ , ν directions. The common canonical ordering of the V µ ’s induces somepreference for the V µν ’s to be closer to identity with no direct feedback on the angles –unlike in the EK situation. The hope is that angles are now free to take on the role of an“emergent” lattice momentum space. V. “GAUGE INVARIANCE” IN QEK. On an 1 lattice, gauge theory has a symmetry under simultaneous conjugation by thesame matrix of all link matrices. These matrices can be thought of as Polyakov loops.Evidently, on a one site lattice there can be no geometrically open contours.This symmetry acts on the V -matrices from the left and therefore commutes with theaction on V by the S ( N ) we had to mod out by. A permutation gauge transformation willpermute the rows of each V . After its action each V needs to be reordered back to canonicalorder. The end result is that gauge transformations which happen to be permutations donot change anything. We might as well forget about them altogether. A. Conclusion of section. This paper contains the full description of the original, with no shortcuts allowed, QEKmodel. There are many ways and points of view in which QEK can fail. To the limitedextent I understand it, the Bringolz-Sharpe paper [3] has not analyzed a precise enoughversion of the originally intended QEK model. If I am right, the problem of in-principlevalidity of QEK remains open. Numerical tests might discover a new candidate problemwith QEK in the future which could invalidate the quenching approach in principle. VI. FINAL COMMENTS. In this paper algorithmic issue in the QEK case have not been addressed. Clearly, the U (2) example was presented with traditional SU (2) Monte Carlo updates in mind. An HMCversion might be also worth looking into.From the extensive and ultimately successful work on the twisted EK model [7], TEK,it is known that for TEK to work, the large N limit needs to be approached with care andone needs to go to truly large values of N . By the law of “conservation of difficulty” QEKmay also need further nontrivial refinements. The problem of annealed permutations BS [3]found, at least at the theoretical level, seems harmless to me because the basic rules ofcalculus would tell you to eliminate permutations in the quenching approach and how to doit. Acknowledgments I am grateful to Barak Bringolz for providing me, many years ago, with a succinct for-mulation of the main point of the BS paper [3], namely, that eq. (10) in [2] showed thatthere was an issue with permutations that could jeopardize the validity of QEK. I thankRajamani Narayanan for valuable discussions and comments.