Validity of the single-point quenched reduction of large-$N$ lattice gauge theory: equivalence with the infinite-volume Wilson theory at intermediate couplings and in the approach to the continuum limit, and exact solution of the reduced few-matrix models

In 1983 Parisi and Zhang formulated 'single point field theories': large-$N$ matrix field theories, chiral spin models, and $U(N)$ lattice gauge theory reduced to one-site few-matrix models in which space-time dependence is replaced by quenched (frozen) random phases, following Parisi's repair of the Eguchi-Kawai reduction, whose center symmetry breaks spontaneously at weak coupling. They verified the reduction only at lowest perturbative orders — for the gauge theory, the leading weak-coupling internal energy of the axial-gauge reduced action with $D-1$ link matrices coupled to quenched phase matrices — and explicitly left open whether the reduced gauge model is equivalent to the infinite-volume Wilson theory at intermediate couplings and in the approach to the continuum limit. Linked open aspects of the same reduction program, treated at the perturbative level in the source paper or attested in the later literature, concern the behavior of open Wilson-line observables, the number of quenched phase samples needed, a controlled replica continuation of the quenched average, exact analytic solutions of the reduced few-matrix models, and the relation to the chiral spin model in $D-1$ dimensions. The subsequent history is as follows: early numerical tests supported the quenching conjecture; Bringoltz and Sharpe (2008) found strong evidence for its breakdown through eigenvalue momentum locking; Neuberger (2020) showed that the model tested was not the precisely defined original (the quenched measure must be quotiented by eigenvalue permutations) and declared the in-principle validity still open; large-$N$ volume reduction meanwhile works through the twisted Eguchi-Kawai prescription. The surviving open problem is the precise formulation and determination of the quenched single-point reduction, including its exact solvability.

Background

In the 't Hooft large-$N$ limit, a $U(N)$ or $SU(N)$ gauge theory is taken to $N\to\infty$ at fixed 't Hooft coupling $\lambda=g^2N$; only planar Feynman diagrams (those drawable without crossing in the double-line notation) survive, and expectation values of gauge-invariant traces factorize, i.e. $\langle \operatorname{tr}A\,\operatorname{tr}B\rangle-\langle\operatorname{tr}A\rangle\langle\operatorname{tr}B\rangle$ vanishes as $N\to\infty$. Lattice pure gauge theory puts a unitary matrix $U_\mu(x)$ on each link of a hypercubic lattice and integrates the Wilson action $S_W=-\frac{\beta}{N}\sum_{P}\operatorname{Re}\operatorname{tr}\prod_{\ell\in P}U_\ell$ (product around each elementary square, or plaquette) over the Haar measure, with $\beta\propto N/g^2$. A Wilson loop is the trace of the ordered product of link matrices around a closed contour; open (non-closed) Wilson lines are not gauge invariant in the original theory but reappear as momentum-space correlators in reduced models.

Eguchi and Kawai (1982) observed that the planar Schwinger-Dyson ('loop') equations for Wilson loops coincide on an infinite lattice and on a single-site lattice, since translation invariance and factorization close the equations on closed loops; this 'EK reduction' would replace the infinite-volume theory by a model of $D$ unitary matrices, one per Euclidean direction. However, for $D>2$ the center symmetry of the one-site model — invariance under $U_\mu\to z_\mu U_\mu$ with $z_\mu$ an element of the group center, equivalently the vanishing of the single-site Polyakov loops $\operatorname{tr}U_\mu$, the analogues of Wilson lines winding around a compactified direction — is spontaneously broken at weak coupling, so the naive one-site model does not represent the infinite-volume theory near the continuum limit.

Parisi (1982), and Bhanot, Heller and Neuberger (1982) in an explicit lattice realization, proposed to repair this by quenching (in the spin-glass sense): freeze the eigenvalue phases $\theta_\mu^a$ of the link matrices — the would-be momenta of the emergent momentum torus — compute expectation values at fixed phases, and average over the phases only afterwards with a prescribed distribution, rather than letting them fluctuate dynamically (annealed). Parisi and Zhang (1983) then systematically developed the 'single point field theories': for a $D$-dimensional complex matrix scalar theory with generalized periodic boundary conditions $M_{ij}(x+\hat\mu)=M_{ij}(x)e^{i(\theta_i^\mu-\theta_j^\mu)}$ they checked that the single-point model reproduces planar Feynman amplitudes at the lowest nontrivial order, preserves the renormalization structure on the examples examined, and realizes large-$N$ factorization after phase averaging; for a $U(N)\times U(N)$ chiral spin model (unitary matrices coupled by nearest-neighbor trace terms, a principal-chiral-type model) they matched the Dyson-Schwinger equations (the exact identities relating correlators of the fields) to those of its single-point reduction and analyzed weak-coupling eigenvalue stability; and for $U(N)$ gauge theory they fixed the axial gauge (setting all link matrices in one space-time direction to the identity) so the phases cannot be gauged away, obtaining the reduced action, their eq. (5.1), $-S=\beta\sum_{i\ne j}\operatorname{tr}U_iU_jU_i^{+}U_j^{+}+\beta\sum_j\operatorname{tr}\big(U_jD_\mu(\theta_\mu)U_j^{+}D_\mu(\theta_\mu)^{\dagger}\big)+\mathrm{c.c.}$ with $\mu$ the gauge-fixed direction, $D_\mu(\theta_\mu)=\operatorname{diag}(e^{i\theta_\mu^1},\dots,e^{i\theta_\mu^N})$ the quenched diagonal phase matrix of that direction, and $D-1$ remaining link matrices $U_j$ coupled to it — a structure suggestive of a chiral spin model in $D-1$ dimensions. For the gauge model they computed only the leading weak-coupling internal energy, $E=1-\frac{1}{2D}\frac{N}{\beta}$ (their eq. (5.3)), and left the core question open: the equivalence with the Wilson theory at intermediate couplings must be decided numerically, and open Wilson-loop-like observables were argued only plausibly to behave consistently. The remaining gaps that define this problem are linked open aspects of the same reduction program: the number of independent quenched samples $N_\theta$ needed to represent an effective space-time volume; a controlled replica route $\langle\ln Z\rangle_\theta=\left.\frac{d}{dn}\langle Z^n\rangle_\theta\right|_{n\to0}$ for the quenched average, which requires an analytic continuation from integer $n$; exact analytic solutions of the reduced few-matrix models — a constructive 'master field', the hypothetical single configuration reproducing all planar correlators, the analytic ideal of one-site reduction recorded in the review of Lucini and Panero; and a rigorous form of the suggested relation to the chiral spin model in $D-1$ dimensions.

The subsequent history frames the current status. Early Monte Carlo studies of the quenched Eguchi-Kawai (QEK) model (Okawa 1982; later studies including Lewis 1985) reported agreement with large-$N$ Wilson-theory observables. Bringoltz and Sharpe (2008) combined weak-coupling arguments with high-precision simulations at $20\le N\le 200$ using a Wang-Landau (flat-histogram) Monte Carlo algorithm and found strong evidence that the large-$N$ equivalence breaks down: dynamical fluctuations correlate the eigenvalue sets of the different $U_\mu$ ('momentum locking'), violating factorization for open-line-type observables such as $\frac{1}{N}\operatorname{tr}(U_\alpha U_\beta)$; the 2013 review of Lucini and Panero records this as a disproof of the earlier positive tests. Neuberger (2020), however, re-examined the construction and argued that the simulated model was not the precisely defined original: the change of variables to eigenvalues and eigenvectors requires quotienting the integration domain by simultaneous eigenvalue permutations, and with that quotient implemented no annealed average over permutations remains; on that precise formulation he concludes that the in-principle validity of quenched reduction remains open and requires future numerical work. In parallel, large-$N$ volume reduction has been achieved by other prescriptions — most successfully the twisted Eguchi-Kawai (TEK) model of González-Arroyo and Okawa, in which quenching is replaced by 't Hooft flux twist factors and which is an actively used working tool at $N$ of order $10^2$-$10^3$ — but that resolves the practical goal by a different mechanism and does not settle the validity of the quenched prescription that Parisi and Zhang proposed.

Problem Statement

Determine whether the single-point quenched reduction of large-$N$ lattice gauge theory, as proposed by Parisi and Zhang, is valid — i.e. whether the one-site model with quenched phases reproduces the planar ($N\to\infty$) content of the infinite-volume Wilson theory — and make the reduction an analytic and practical tool. Concretely, with the common objective of validating (or refuting, with a mechanism) the quenched single-point reduction, the question decomposes into linked components: the core equivalence question (1) and the open-line consistency requirement (2) arise from the source paper's own treatment, while (3) and (4) are linked open aspects of the same reduction program attested in the later literature:

(1) Core equivalence. For the axial-gauge single-site $U(N)$ model with reduced action $-S=\beta\sum_{i\ne j}\operatorname{tr}U_iU_jU_i^{+}U_j^{+}+\beta\sum_j\operatorname{tr}\big(U_jD_\mu(\theta_\mu)U_j^{+}D_\mu(\theta_\mu)^{\dagger}\big)+\mathrm{c.c.}$ where each remaining link matrix $U_j$ couples to $D_\mu(\theta_\mu)=\operatorname{diag}(e^{i\theta_\mu^1},\dots,e^{i\theta_\mu^N})$, the quenched diagonal phase matrix of the gauge-fixed direction $\mu$, establish analytically or by controlled simulation whether its quenched-phase-averaged expectation values reproduce the planar expectation values of the infinite-volume Wilson lattice gauge theory at intermediate couplings and in the approach to the continuum limit — beyond the leading weak-coupling check already performed and the strong-coupling regime already covered. The operative, precisely defined object is the equivalent permutation-quotiented quenched Eguchi-Kawai formulation: the measure of Neuberger (2020), in which the change of variables to eigenvalues and eigenvectors, $U_\mu=V_\mu D_\mu(\theta_\mu)V_\mu^{+}$ with $D_\mu(\theta_\mu)=\operatorname{diag}(e^{i\theta_\mu^1},\dots,e^{i\theta_\mu^N})$, carries a Vandermonde Jacobian $\prod_{i<j}\sin^2\!\big(\tfrac{\theta_\mu^i-\theta_\mu^j}{2}\big)$ and the integration domain is the quotient $[U(N)/U(1)^N\times U(1)^N]/S(N)$ by simultaneous eigenvalue permutations (with a canonical eigenvector ordering), the quenched partition function $Z(\theta)$ is evaluated at fixed ordered angles, and the angle average is performed only afterwards with the Jacobian weight. The equivalence question therefore turns on whether a mechanism protects large-$N$ factorization at intermediate coupling and in the approach to the continuum limit, or on which observables fail to match and how they scale in $N$.

(2) Open-line observables. Determine whether open Wilson-line-type observables in the reduced ensemble (non-closed matrix products whose phase-averaged values must vanish and factorize consistently, such as $\frac{1}{N}\operatorname{tr}(U_\alpha U_\beta)$ for $\alpha\ne\beta$) behave in accordance with the infinite-volume theory in the weak-coupling and intermediate-coupling regimes.

(3) Practical viability of the quench average. Quantify the number of independent quenched-phase samples $N_\theta$ needed to represent a given effective space-time volume at a stated accuracy (and the associated computational trade-offs), or provide a controlled alternative to direct sampling.

(4) Exact analytic structure. Obtain exact analytic solutions of the reduced few-matrix models (linear or non-linear) — a constructive master field for the planar limit — and establish rigorously the suggested relation between the reduced $U(N)$ gauge theory and the chiral spin model in $D-1$ dimensions.

Component (1) is the core question and settling it resolves the task; components (2)-(4) are optional linked extensions of the same reduction program.

References

1. G. Parisi and Y.-C. Zhang, 'On the single point field theories', Nuclear Physics B 216 (1983) 408-420, DOI 10.1016/0550-3213(83)90292-4, https://doi.org/10.1016/0550-3213(83)90292-4 (source paper)
2. T. Eguchi and H. Kawai, 'Reduction of dynamical degrees of freedom in the large-N gauge theory', Physical Review Letters 48 (1982) 1063-1066, DOI 10.1103/PhysRevLett.48.1063, https://doi.org/10.1103/PhysRevLett.48.1063
3. G. Bhanot, U. M. Heller and H. Neuberger, 'The quenched Eguchi-Kawai model', Physics Letters B 113 (1982) 47-50, DOI 10.1016/0370-2693(82)90106-X, https://doi.org/10.1016/0370-2693(82)90106-X
4. G. Parisi, 'A simple expression for planar field theories', Physics Letters B 112 (1982) 463-464, DOI 10.1016/0370-2693(82)90849-8, https://doi.org/10.1016/0370-2693(82)90849-8
5. M. Okawa, 'Monte Carlo study of the quenched Eguchi-Kawai model', Physical Review Letters 49 (1982) 705-707, DOI 10.1103/PhysRevLett.49.705, https://doi.org/10.1103/PhysRevLett.49.705
6. F. R. Klinkhamer, 'Heating the quenched Eguchi-Kawai model', Nuclear Physics B 218 (1983) 32-42, DOI 10.1016/0550-3213(83)90474-1, https://doi.org/10.1016/0550-3213(83)90474-1
7. M. R. Lewis, 'Monte-Carlo study of the Eguchi-Kawai and quenched Eguchi-Kawai models', Journal of Physics G: Nuclear Physics 11 (1985) 981-990, DOI 10.1088/0305-4616/11/9/008, https://doi.org/10.1088/0305-4616/11/9/008
8. B. Bringoltz and S. R. Sharpe, 'Breakdown of large-N quenched reduction in SU(N) lattice gauge theories', Physical Review D 78 (2008) 034507, DOI 10.1103/PhysRevD.78.034507, arXiv:0805.2146, https://arxiv.org/abs/0805.2146
9. H. Neuberger, 'Quenched Eguchi-Kawai model revisited', Physical Review D 102 (2020) 094503, DOI 10.1103/PhysRevD.102.094503, arXiv:2009.09539, https://arxiv.org/abs/2009.09539
10. M. García Pérez, 'Prospects for large N gauge theories on the lattice', Proceedings of Science LATTICE2019 (2020) 276, DOI 10.22323/1.363.0276, arXiv:2001.10859, https://arxiv.org/abs/2001.10859
11. A. González-Arroyo and M. Okawa, 'Large N reduction with the Twisted Eguchi-Kawai model', Journal of High Energy Physics 07 (2010) 043, DOI 10.1007/JHEP07(2010)043, arXiv:1005.1981, https://arxiv.org/abs/1005.1981
12. B. Lucini and M. Panero, 'SU(N) gauge theories at large N', Physics Reports 526 (2013) 93-163, DOI 10.1016/j.physrep.2013.01.001, arXiv:1210.4997, https://arxiv.org/abs/1210.4997