The quenched Eguchi-Kawai model revisited

The motivation and construction of the original Quenched Eguchi-Kawai model are reviewed, providing much greater detail than in the first, 1982 QEK paper. A 2008 article announced that QEK fails as a reduced model because the average over permutations of eigenvalues stays annealed. It is shown here that the original quenching logic naturally leads to a formulation with no annealed average over permutations.

four dimensional SU (N ) pure gauge theory [4]. At leading order in 1/N the expectation values of Wilson loops on an infinite Euclidean four-torus are the same as "folded" counterparts on 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 Coulomblike law realized in a finite periodic Euclidean four volume which has no room for "Faraday's flux lines" to spread?
Continuum EK reduction is some times referred to as "partial reduction" because the lattices one has to use in a numerical simulation in order to approach the continuum limit to reasonable accuracy must be substantially larger than 1 4 . "Partial" is a bad epithet because it 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 4 lattice was the quenched EK model, QEK [2].
The main physics question was how the largeness of the SU (N ) group at infinite N could provide a place-holder for an infinite lattice while consisting of just four unitary link matrices. A simple calculation at one loop order showed that the eigenvalue phases of the link matrices in the four directions could play the role of continuous lattice momenta in (−π, π] 4 -an "emerging" toroidal momentum space -and produce the standard Coulomb's force law on the lattice if we summed up the contributions of a large number of saddles and ignored their instabilities. As was very well understood from other semi-classical calculations, fluctuations in the "flat" directions, connecting the saddles, produced zero modes and were easily 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 4 lattice the eigenvalue sets of each link matrix are gauge invariant angles. They are unique candidates for lattice momenta.
The QEK fix consisted of a removal of the link matrices' eigenvalues from the set of annealed variables. They were quenched instead. This difference did not matter at large N by a degrees-of-freedom counting argument: There were 4N angles, and order 4N 2 matrix elements: The angles ought to be governed by uniform and uncorrelated distributions in each direction if some obvious symmetries remain preserved as N → ∞.
For strong lattice coupling (small "β"), where EK had been proven to work, the quenching prescription would have no effect to leading order in 1 N . Quenched or annealed, the angle distributions would be frozen to continuous uniform densities in each direction and those would be uncorrelated. The requirement to match onto the original EK version, which was proven to hold at strong coupling, left little freedom for constructing QEK. The "loop equations", on which the EK proof relied, have trivial boundary conditions in the strong coupling limit and determine the entire strong coupling series. That series has a finite radius of convergence. The precise boundary conditions for the lattice loop equations at weak coupling remain unknown to date. They would be needed for constructing the Feynman series for Wilson loops. The loop equations themselves have only a relatively formal continuum limit. 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 is uncertain whether it is valid throughout the bridge connecting short and long distance pure gauge theory physics. Even if it does, there remains doubt whether it would be practical in comparison to the safer continuum EK method which relies both on the lattice loop equations 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 original integration measure, is After the change of variables the integration measure is locally given by up to a constant factor determined by a normalization convention. The Jacobian does not depend on V , only on the angles. This is crucial for the quenching proposal because the integration over the angles factorizes. In EK the instability resides in the the angle dynamics.
The QEK model addresses the instability directly.
The When acting on a column from the left, it permutes its row entries by P . P † acts from the right on a row and permutes its column entries by P −1 . Inserting 1 = P P † on the right hand side of of eq. (1) in IV.1 between the V and the θ-diagonal matrix replaces V by V P on both sides. The effect of P † .....P on the diagonal θ matrix is to permute its columns by P and its rows by P † . The result is that the θ i µ get permuted by P along the diagonal.
Thus, elements of P ∈ S(N ) act simultaneously on the columns of V and on their associated eigenvalues, preserving the eigenvalue-eigenvector association.
Suppose one is given a U matrix. By a probability argument, it has distinct eigenvalues.
A procedure to identify a unique decomposition in terms of V and diag(θ) matrices is defined Permutations of eigenvectors must be eliminated in order to perform a correct variable change in EK with no over-counting at any coupling. That is just applied multivariate calculus.
To motivate the ordering prescription for the V matrices, consider the simple case of U (2)/S(2). The two U (1) 2 terms in the "numerator" of eq. IV.5 are ignored. Up to an irrelevant overall phase, any V ∈ U (2) can be written as where |w| 2 + |z| 2 = 1. S(2) = {1 2 , σ 1 } using Pauli's notation. Let V be given by the same expression with w, z replaced by w , z . The equation V = V σ 1 defines an equivalent pair V ∼ V . In components, z = −w * and w = −z * . For V to be "diagonally right row entry dominant" we need |w| > |z|; then V is not "diagonally right row entry dominant" because |w | < |z |. The split of U (2) into "halves" is explicit. The half containing the identity makes up the QEK integration domain for V . As usual, ambiguous cases are ignored for probability reasons.
U (N ) can be restricted to SU (N ) by adding a factor of δ 2π ( i θ i µ ) in the θ integral for each direction. δ 2π is the 2π-periodic δ function. Mentally, one can imagine the V 's to be also fixed by an overall phase, restricting to SU (N ); this phase does not enter observables dependent only on the U 's.
Once the change of variables in the EK integrand is correctly implemented one can replace each U µ by equality (2) in eq. IV.1 in the EK model and nothing has changed. The integral for the partition function can be done successively, first integrating over all the columns of the V µ in the pair [θ i µ , i-th column of V µ ] with local Haar measure at fixed, ordered, angles at 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 function is replaced by an integral with the same measure and action, but at fixed angles in all directions. The QEK partition function is a function of these angles, Z(θ). For a Wilson loop observable one uses Z(θ) as normalization, now in the denominator, obtaining averages of the observable at a fixed ordered θ set. Next, these annealed V -averages are integrated over the angles with weight given by the Jacobian in eq. IV.4. The θ variables are treated as 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 in the segment (−π, π]. Such an ordering makes the distribution equal to the derivative of a smooth 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 eigenvalue on the unit circle, distributed just according to the Jacobian factor. The Jacobian measure is invariant under direction dependent permutations. They are not induced by annealed generation of permutations among the columns of V because two distinct "diagonally right row 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 that new large N transitions occur, to phases where the hypercubic symmetry is spontaneously broken. Such phases indeed do occur in the EK model [4]. If they persist to the quenched case, QEK fails. It also could happen that it is practically impossible to attain high enough values of N because prohibitively large samples of the eigenvalue sets are needed for a reasonable 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 the angles in the µ, ν directions. The common canonical ordering of the V µ 's induces some preference for the V µν 's to be closer to identity with no direct feedback on the anglesunlike in the EK situation. The hope is that angles are now free to take on the role of an "emergent" lattice momentum space. This symmetry acts on the V -matrices from the left and therefore commutes with the action on V by the S(N ) we had to mod out by. A permutation gauge transformation will permute the rows of each V . After its action each V needs to be reordered back to canonical order. The end result is that gauge transformations which happen to be permutations do not 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, QEK model. There are many ways and points of view in which QEK can fail. To the limited extent I understand it, the Bringolz-Sharpe paper [3] has not analyzed a precise enough version of the originally intended QEK model. If I am right, the problem of in-principle validity of QEK remains open. Numerical tests might discover a new candidate problem with 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 HMC version 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 and one needs to go to truly large values of N . By the law of "conservation of difficulty" QEK may 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 of calculus would tell you to eliminate permutations in the quenching approach and how to do it.