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$Z_{N}$ stability and continuity in the twisted Eguchi-Kawai model with two-flavor adjoint fermions

Department of Physics, Kindai University, Osaka 577-8502, Japan

^*Contact author: 2433310108h@kindai.ac.jp
^†Contact author: misumi@phys.kindai.ac.jp

Phys. Rev. D 112, 094503 – Published 7 November, 2025

DOI: https://doi.org/10.1103/dbg6-9d71

Abstract

We investigate the twisted Eguchi-Kawai (TEK) reduced model of four-dimensional $S U (N)$ gauge theory in the presence of two-flavor adjoint fermions (adjoint TEK model). Using Monte Carlo simulations with $N = 121$ , twist parameter $k = 1$ , hopping parameter $κ = 0.01 - 0.03$ ( $≪ κ_{c}$ ) and inverse ’t Hooft coupling $b = 0.30 - 0.45$ , we show that heavy adjoint fermions stabilize the $(Z_{N})^{4}$ center-symmetric vacuum even for the minimal twist satisfying $k / \sqrt{N} < 1 / 9$ , where the $(Z_{N})^{4}$ symmetry is spontaneously broken in the absence of adjoint fermions. This result also suggests that the adjoint TEK model with the minimal twist is equivalent to $S U (N)$ gauge theory over a broader $(κ, b)$ parameter region than the adjoint EK model without twist. We further extend our analysis to a partially reduced model to realize a geometry akin to $R^{3} \times S^{1}$ and study the theory under $S^{1}$ compactification with periodic adjoint fermions. Numerical simulations with $N = 16 - 49$ , $b = 0.30 - 0.46$ , and $κ = 0.03 - 0.16$ are consistent with the adiabatic continuity conjecture; with periodic adjoint fermions, the theory remains in a center-symmetric (confined) phase as the $S^{1}$ circle size is reduced, in contrast to the deconfining transition observed in the pure TEK model or in the TEK model with antiperiodic adjoint fermions. We present the Polyakov loop measurements and consistency checks supporting these findings.

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References (75)

G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B72, 461 (1974).
T. Eguchi and H. Kawai, Reduction of dynamical degrees of freedom in the large $N$ gauge theory, Phys. Rev. Lett. 48, 1063 (1982).
B. Lucini and M. Teper, $S U (N)$ gauge theories in four-dimensions: Exploring the approach to $N = infinity$ , J. High Energy Phys. 06 (2001) 050.
B. Lucini and M. Panero, $S U (N)$ gauge theories at large $N$ , Phys. Rep. 526, 93 (2013).
G. Bhanot, U. M. Heller, and H. Neuberger, The quenched Eguchi-Kawai model, Phys. Lett. 113B, 47 (1982).
A. Gonzalez-Arroyo and M. Okawa, A twisted model for large $N$ lattice gauge theory, Phys. Lett. 120B, 174 (1983).
A. Gonzalez-Arroyo and M. Okawa, The twisted Eguchi-Kawai model: A reduced model for large $N$ lattice gauge theory, Phys. Rev. D 27, 2397 (1983).
R. Narayanan and H. Neuberger, Large $N$ reduction in continuum, Phys. Rev. Lett. 91, 081601 (2003).
B. Bringoltz and S. R. Sharpe, Non-perturbative volume-reduction of large- $N$ QCD with adjoint fermions, Phys. Rev. D 80, 065031 (2009).
B. Bringoltz, M. Koren, and S. R. Sharpe, Large- $N$ reduction in QCD with two adjoint Dirac fermions, Phys. Rev. D 85, 094504 (2012).
A. Gonzalez-Arroyo and M. Okawa, Large $N$ reduction with the twisted Eguchi-Kawai model, J. High Energy Phys. 07 (2010) 043.
A. Gonzalez-Arroyo and M. Okawa, The string tension from smeared Wilson loops at large $N$ , Phys. Lett. B 718, 1524 (2013).
A. González-Arroyo and M. Okawa, Large $N$ meson masses from a matrix model, Phys. Lett. B 755, 132 (2016).
M. G. Pérez, A. González-Arroyo, and M. Okawa, Meson spectrum in the large $N$ limit, J. High Energy Phys. 04 (2021) 230.
C. Bonanno, P. Butti, M. García Peréz, A. González-Arroyo, K.-I. Ishikawa, and M. Okawa, The large- $N$ limit of the chiral condensate from twisted reduced models, J. High Energy Phys. 12 (2023) 034.
A. Gonzalez-Arroyo and M. Okawa, Twisted reduction in large $N$ QCD with two adjoint Wilson fermions, Proc. Sci. LATTICE2012 (2012) 046 [arXiv:1210.7881].
A. González-Arroyo and M. Okawa, Twisted space-time reduced model of large $N$ QCD with two adjoint Wilson fermions, Phys. Rev. D 88, 014514 (2013).
M. Garcia Perez, A. Gonzalez-Arroyo, L. Keegan, and M. Okawa, Mass anomalous dimension from large $N$ twisted volume reduction, Proc. Sci. LATTICE2013 (2014) 098 [arXiv:1311.2395].
M. García Pérez, A. González-Arroyo, L. Keegan, and M. Okawa, Mass anomalous dimension of Adjoint QCD at large $N$ from twisted volume reduction, J. High Energy Phys. 08 (2015) 034.
T. Azeyanagi, M. Hanada, M. Unsal, and R. Yacoby, Large- $N$ reduction in QCD-like theories with massive adjoint fermions, Phys. Rev. D 82, 125013 (2010).
A. González-Arroyo and M. Okawa, Twisted reduction in large $N$ QCD with adjoint Wilson fermions, in Proceedings of the KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC Perspective (World Scientific, Singapore, 2014), pp. 352–358.
N. M. Davies, T. J. Hollowood, V. V. Khoze, and M. P. Mattis, Gluino condensate and magnetic monopoles in supersymmetric gluodynamics, Nucl. Phys. B559, 123 (1999).
N. M. Davies, T. J. Hollowood, and V. V. Khoze, Monopoles, affine algebras and the gluino condensate, J. Math. Phys. (N.Y.) 44, 3640 (2003).
M. Unsal, Abelian duality, confinement, and chiral symmetry breaking in QCD(adj.), Phys. Rev. Lett. 100, 032005 (2008).
M. Unsal, Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions, Phys. Rev. D 80, 065001 (2009).
P. Kovtun, M. Unsal, and L. G. Yaffe, Volume independence in large $N_{c}$ QCD-like gauge theories, J. High Energy Phys. 06 (2007) 019.
M. Shifman and M. Unsal, QCD-like theories on $R_{3} \times S_{1}$ : A smooth journey from small to large $r (S_{1})$ with double-trace deformations, Phys. Rev. D 78, 065004 (2008).
M. Unsal and L. G. Yaffe, Large-N volume independence in conformal and confining gauge theories, J. High Energy Phys. 08 (2010) 030.
E. Poppitz and M. Unsal, Conformality or confinement: (IR)relevance of topological excitations, J. High Energy Phys. 09 (2009) 050.
M. M. Anber and E. Poppitz, Microscopic structure of magnetic bions, J. High Energy Phys. 06 (2011) 136.
E. Poppitz, T. Schäfer, and M. Ünsal, Continuity, deconfinement, and (Super) Yang-Mills theory, J. High Energy Phys. 10 (2012) 115.
T. Misumi and T. Kanazawa, Adjoint QCD on $R^{3} \times S^{1}$ with twisted fermionic boundary conditions, J. High Energy Phys. 06 (2014) 181.
M. Hongo, T. Misumi, and Y. Tanizaki, Phase structure of the twisted $S U (3) / U (1)^{2}$ flag sigma model on $R \times S^{1}$ , J. High Energy Phys. 02 (2019) 070.
T. Misumi, Y. Tanizaki, and M. Ünsal, Fractional $θ$ angle, ’t Hooft anomaly, and quantum instantons in charge- $q$ multi-flavor Schwinger model, J. High Energy Phys. 07 (2019) 018.
T. Fujimori, E. Itou, T. Misumi, M. Nitta, and N. Sakai, Confinement-deconfinement crossover in the lattice $C P^{N - 1}$ model, Phys. Rev. D 100, 094506 (2019).
T. Misumi, T. Fujimori, E. Itou, M. Nitta, and N. Sakai, Lattice study on the twisted $C P^{N - 1}$ models on $R \times S^{1}$ , Proc. Sci. LATTICE2019 (2019) 015 [arXiv:1911.07398].
T. Fujimori, E. Itou, T. Misumi, M. Nitta, and N. Sakai, Lattice $C P^{N - 1}$ model with $Z_{N}$ twisted boundary condition: Bions, adiabatic continuity and pseudo-entropy, J. High Energy Phys. 08 (2020) 011.
M. Ünsal, Strongly coupled QFT dynamics via TQFT coupling, J. High Energy Phys. 11 (2021) 134.
E. Poppitz, Notes on confinement on $R 3 \times S 1$ : From Yang–Mills, Super-Yang–Mills, and QCD (adj.) to QCD(F), Symmetry 14, 180 (2022).
Y. Tanizaki and M. Ünsal, Center vortex and confinement in Yang-Mills theory and QCD with anomaly-preserving compactifications, Prog. Theor. Exp. Phys. 2022, 04A108 (2022).
Y. Tanizaki and M. Ünsal, Semiclassics with ’t Hooft flux background for QCD with 2-index quarks, J. High Energy Phys. 08 (2022) 038.
Y. Hayashi, Y. Tanizaki, and H. Watanabe, Semiclassical analysis of the bifundamental QCD on $R^{2} \times T^{2}$ with ’t Hooft flux, J. High Energy Phys. 10 (2023) 146.
Y. Hayashi and Y. Tanizaki, Semiclassics for the QCD vacuum structure through $T^{2}$ -compactification with the baryon-’t Hooft flux, J. High Energy Phys. 08 (2024) 001.
Y. Hayashi, Y. Tanizaki, and H. Watanabe, Non-supersymmetric duality cascade of QCD(BF) via semiclassics on $R^{2} \times T^{2}$ with the baryon-’t Hooft flux, J. High Energy Phys. 07 (2024) 033.
Y. Hayashi and Y. Tanizaki, Unifying monopole and center vortex as the semiclassical confinement mechanism, Phys. Rev. Lett. 133, 171902 (2024).
Y. Hayashi, T. Misumi, and Y. Tanizaki, Monopole-vortex continuity of $N = 1$ super Yang-Mills theory on $R^{2} \times S^{1} \times S^{1}$ with ’t Hooft twist, J. High Energy Phys. 05 (2025) 194.
C. Güvendik, T. Schaefer, and M. Ünsal, The metamorphosis of semi-classical mechanisms of confinement: From monopoles on $R^{3} \times S^{1}$ to center-vortices on $R^{2} \times T^{2}$ , J. High Energy Phys. 11 (2024) 163.
A. Gocksch, P. Rossi, and F. Neri, A finite temperature twisted Eguchi-Kawai model, Phys. Lett. 130B, 407 (1983).
A. Gocksch, F. Neri, and P. Rossi, The deconfining phase transition in the finite temperature twisted Eguchi-Kawai model, Nucl. Phys. B238, 674 (1984).
F. Chamizo and A. Gonzalez-Arroyo, Tachyonic instabilities in $2 + 1$ dimensional Yang–Mills theory and its connection to number theory, J. Phys. A 50, 265401 (2017).
Y. Hayashi, Y. Tanizaki, and M. Ünsal, Center-vortex semiclassics with non-minimal ’t Hooft fluxes on $R^{2} \times T^{2}$ and center stabilization at large $N$ , arXiv:2505.07467.
T. Azeyanagi, M. Hanada, T. Hirata, and T. Ishikawa, Phase structure of twisted Eguchi-Kawai model, J. High Energy Phys. 01 (2008) 025.
L. Del Debbio, The conformal window on the lattice, Proc. Sci. Lattice2010 (2014) 004 [arXiv:1102.4066].
A. Athenodorou, E. Bennett, G. Bergner, P. Butti, J. Lenz, and B. Lucini, SU(2) gauge theory with one and two adjoint fermions towards the continuum limit, arXiv:2408.00171.
A. Gonzalez-Arroyo and M. Okawa, Testing volume independence of $S U (N)$ pure gauge theories at large $N$ , J. High Energy Phys. 12 (2014) 106.
M. Unsal and L. G. Yaffe, Center-stabilized Yang-Mills theory: Confinement and large $N$ volume independence, Phys. Rev. D 78, 065035 (2008).
A. González-Arroyo and M. Okawa (private communication).
In four dimensions $U_{0} (τ)$ and $W_{123}$ are non-commute, which enforces the condition $Tr W_{123} = 0$ even with the symmetric twist.
The twist in Eq. (14) is a further modified one from originally proposed one in Refs. [48, 49].
A. Cherman, S. Sen, M. Unsal, M. L. Wagman, and L. G. Yaffe, Order parameters and color-flavor center symmetry in QCD, Phys. Rev. Lett. 119, 222001 (2017).
H. Kouno, Y. Sakai, T. Makiyama, K. Tokunaga, T. Sasaki, and M. Yahiro, Quark-gluon thermodynamics with the $Z_{N_{c}}$ symmetry, J. Phys. G 39, 085010 (2012).
Y. Sakai, H. Kouno, T. Sasaki, and M. Yahiro, The quarkyonic phase and the $Z_{N_{c}}$ symmetry, Phys. Lett. B 718, 130 (2012).
H. Kouno, T. Makiyama, T. Sasaki, Y. Sakai, and M. Yahiro, Confinement and $Z_{3}$ symmetry in three-flavor QCD, J. Phys. G 40, 095003 (2013).
H. Kouno, T. Misumi, K. Kashiwa, T. Makiyama, T. Sasaki, and M. Yahiro, Differences and similarities between fundamental and adjoint matters in $S U (N)$ gauge theories, Phys. Rev. D 88, 016002 (2013).
H. Kouno, K. Kashiwa, J. Takahashi, T. Misumi, and M. Yahiro, Understanding QCD at high density from a $Z_{3}$ -symmetric QCD-like theory, Phys. Rev. D 93, 056009 (2016).
T. Iritani, E. Itou, and T. Misumi, Lattice study on QCD-like theory with exact center symmetry, J. High Energy Phys. 11 (2015) 159.
T. Misumi, T. Iritani, and E. Itou, Finite-temperature phase transition of $N_{f} = 3$ QCD with exact center symmetry, Proc. Sci. LATTICE2015 (2016) 152 [arXiv:1510.07227].
H. Shimizu and K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition, Phys. Rev. D 97, 105011 (2018).
Y. Tanizaki, T. Misumi, and N. Sakai, Circle compactification and ’t Hooft anomaly, J. High Energy Phys. 12 (2017) 056.
Y. Tanizaki, Y. Kikuchi, T. Misumi, and N. Sakai, Anomaly matching for phase diagram of massless $Z_{N}$ -QCD, Phys. Rev. D 97, 054012 (2018).
G. V. Dunne, Y. Tanizaki, and M. Ünsal, Quantum distillation of Hilbert spaces, semi-classics and anomaly matching, J. High Energy Phys. 08 (2018) 068.
The adjoint QCD with $N_{f} = 1 / 2$ (a single Weyl fermion) corresponds to $N = 1$ super Yang-Mills theory (SYM), where the adiabatic continuity of confined phase is believed to exist. The adjoint TEK model with $N_{f} = 1 / 2$ has been studied. [73, 74, 75].
P. Butti, M. Perez Garcia, A. Gonzalez-Arroyo, K.-I. Ishikawa, and M. Okawa, Scale setting for large-N SUSY Yang-Mills on the lattice, J. High Energy Phys. 07 (2022) 074.
C. Bonanno, P. Butti, M. García Pérez, A. González-Arroyo, K.-I. Ishikawa, and M. Okawa, Nonperturbative determination of the $N = 1$ supersymmetric Yang-Mills gluino condensate at large $N$ , Phys. Rev. D 110, 074507 (2024).
C. Bonanno, M. García Pérez, A. González-Arroyo, K.-I. Ishikawa, and M. Okawa, The mass of the gluino-glue bound state in large- $N$ $N = 1$ Supersymmetric Yang-Mills theory, J. High Energy Phys. 03 (2025) 174.

Phys. Rev. D 112, 094503– Published 7 November, 2025

Vol. 112, Iss. 9 — 1 November 2025

Received 5 August 2025
Accepted 17 October 2025

DOI: https://doi.org/10.1103/dbg6-9d71

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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$Z_{N}$ stability and continuity in the twisted Eguchi-Kawai model with two-flavor adjoint fermions

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ZN stability and continuity in the twisted Eguchi-Kawai model with two-flavor adjoint fermions

Abstract

Physics Subject Headings (PhySH)

Article Text

References (75)

$Z_{N}$ stability and continuity in the twisted Eguchi-Kawai model with two-flavor adjoint fermions