Bloch Waves, Magnetization and Domain Walls: The Case of the Gluon Propagator

Attilio Cucchieri; Tereza Mendes

REVIEW 3 major objections 4 minor 1 cited by

Bloch Waves, Magnetization and Domain Walls: The Case of the Gluon Propagator

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper establishes a selection rule for the allowed momenta of the gluon propagator on replicated lattices and reduces their evaluation to the original lattice.

desk verdict A careful, honest derivation of the Bloch-wave allowed-momenta rule, conditional on an explicitly flagged uniqueness hypothesis; the internal check is real, but the load-bearing assumption needs a direct large-lattice test. read the letter →

arxiv 2506.07730 v1 pith:7NT2TDS6 submitted 2025-06-09 hep-lat

Attilio Cucchieri , Tereza Mendes This is my paper

classification hep-lat MSC 81T1381T25 PACS 11.15.Ha

keywords BlochwaveslatticegaugefixingminimalLandaugluonpropagatorreplicatedlatticesallowedmomentacolormagnetizationgauge-fixingdomains

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies minimal Landau gauge on a lattice made by replicating a small 'unit cell' $m$ times in each direction, and shows that the gauge-fixing transformation on this extended lattice has a Bloch-wave form: shifting by one cell multiplies the transformation by a fixed commuting $SU(N_c)$ matrix. From this form it derives a selection rule for momentum-space amplitudes: a gauge-field coefficient at wave vector $k'$ is nonzero only when $k'_\nu+n^j_\nu-n^i_\nu$ is a multiple of $m$. The gluon propagator $D(k')$ is therefore nonzero only at these 'allowed momenta', which always include the momenta of the original lattice but also some configuration-dependent ones. In the large-$m$ limit the stationarity condition forces the zero-momentum modes to vanish, so $D(0)$ is strongly suppressed; this is presented as an artifact of the extended gauge transformations. If the argument is correct, large-volume propagator data can be produced entirely from small-lattice variables, with cost independent of the replica factor $m$.

What carries the argument

The central object is the Bloch-wave ansatz for the gauge-fixing transformation, equivalently the eigenvalue condition $T(N e_\mu)g(z)=s_\mu g(z)$ with a site-independent matrix $s_\mu=\exp(i\Theta_\mu)$ in the Cartan sub-algebra of $su(N_c)$ (the maximal set of mutually commuting traceless Hermitian generators). This ansatz turns the extended-lattice minimization into the small-lattice functional $E_{U,\Theta}[h]$ and yields the selection rule that filters momentum-space coefficients. The zero-momentum suppression is carried by the stationarity condition on $\Theta_\mu$, which sets the Cartan components of the gauge-field zero mode to zero as $m\to\infty$.

What would settle it

Take a thermalized configuration, replicate it, and run the minimal-Landau-gauge minimization from several random starting gauge transformations; then compare the per-cell shift matrices $s_\mu$ defined by $g(z+N e_\mu)=s_\mu g(z)$. If, on a lattice of moderate size, two converged minima give different $s_\mu$ for the same physical configuration, the Bloch ansatz fails and with it the selection rule for allowed momenta.

Watch

Extended reading notes

Core claim

On the extended lattice $\Lambda_z$ formed by $m$ copies of a thermalized $SU(N_c)$ configuration, the minimal-Landau-gauge solution $g(z)$ takes the Bloch form $g(z)=\exp(i\sum_\nu \Theta_\nu z_\nu/N)h(x)$ with commuting Cartan-subalgebra matrices $\Theta_\nu$, and the extended-lattice Fourier coefficient of the gauge-fixed link is nonzero exactly when the wave vector satisfies the selection rule $k'_\nu+n^j_\nu-n^i_\nu=m(\ldots)$ of Eq. (5.19). As a consequence, the gluon propagator $D(k')$ is nonzero only at those allowed momenta, and each nonzero amplitude reduces to a Fourier transform on the original lattice $\Lambda_x$. The allowed set always contains the momenta of the original discretization and also some configuration-dependent momenta. In the limit $m\to\infty$ the stationarity condition on $\Theta_\nu$ forces the Cartan zero modes of the gauge field to vanish, so $D(0)$ is strongly suppressed for every local minimum, explaining a finding previously made only numerically.

Load-bearing premise

The argument rests on assuming that each local minimum of the gauge-fixing problem on the replicated lattice is unique up to one overall rotation applied everywhere, so that shifting a solution by one cell always multiplies it by the same constant group element.

Editorial extensions

If this is right

The full numerical evaluation, including thermalization, gauge fixing, and propagator computation, can be done on the original lattice $\Lambda_x$, so the computational cost no longer grows with the replica factor $m$.
Infrared gluon propagators on very large volumes can be produced from small unit cells with large $m$, allowing ensembles that direct large-lattice simulations cannot reach.
Only allowed momenta contribute to $D(k')$; momenta outside the original discretization are configuration-dependent and will appear with poor statistics, so they must be handled separately in any large-volume analysis.
The strong suppression of $D(0)$ at large $m$ is identified as an effect of the extended gauge transformations rather than a physical signal, consistent with earlier free-boundary results.
The same Bloch-wave setup is planned to be extended to the ghost propagator, using the same small-lattice reduction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One could test whether the configuration-dependent 'extra' allowed momenta behave like zone-folding or Umklapp terms, encoding information about the $\Theta$-domain structure rather than the physical infrared dynamics.
The color-magnetization domains suggest defining a domain-wall observable: if the magnetization jumps are localized, a low-cost order parameter for the number of inequivalent cells may be measurable using only variables on $\Lambda_x$.
If the uniqueness hypothesis fails on volumes larger than those numerically tested, the fixed shift matrix $s_\mu$ would become cell-dependent, and the clean selection rule would likely be replaced by a band-like structure over the replica index lattice.
A decisive extension would compare the full $D(k')$ spectrum, including configuration-dependent momenta, against an independent direct large-lattice simulation across many configurations to confirm the small-lattice reduction quantitatively.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

What you should know: this paper actually does the math behind the Bloch-wave replicated-lattice method. The genuinely new pieces are the allowed-momenta selection rule, Eq (5.19), the reduction of the whole computation to the small lattice, and the argument that D(0) suppression follows for local minima, not just absolute minima. The selection rule is checked against about 16,000 nonzero propagator momenta from six configurations and it holds in every case. The derivations in Secs 3 through 5 are detailed and internally consistent, and the Cartan-subalgebra machinery is handled cleanly. I also think the paper is honest: it explicitly labels Eq (3.30), the Bloch-wave hypothesis, as an assumption.

The soft spot is real and it is exactly the one you identified. Eq (3.30) requires that a translation of the gauge-fixing solution by one lattice period be equivalent to left multiplication by a single site-independent group element s_mu. That follows if local minima on the replicated lattice are unique up to global gauge transformations. The paper supports that with ref [27], which is small-volume evidence on the original lattice, not on the replicated lattice. The replicated lattice has extra translational symmetry, so additional Gribov copies unrelated by global transformations are a live possibility. If those exist, the Bloch form, the selection rule, and the D(0) suppression all go. The authors flag this as a hypothesis, so the flaw is not hidden, but it is load-bearing.

The numerical check in Sec 6 is also a self-consistency check: it verifies Eq (5.19) on configurations generated by the Bloch-wave algorithm itself, not on direct large-lattice gauge fixing. The earlier PRL did some direct comparison of D(k) on large lattices, but not for the selection rule. So the conditional verdict is fair: the argument is solid given the hypothesis, but the hypothesis itself needs a direct test, ideally by checking whether large-lattice direct gauge fixing produces nonzero D(k') outside the allowed set.

The citation pattern is fine. Self-citations to [1,5] are background, not a way of importing the result. No circularity. I would like to see the code or data released, but that is a request, not a fatal flaw.

This paper deserves a serious referee. It is a technical advance in a niche area, and the authors know exactly where the weak point is. A referee should push on the uniqueness hypothesis and ask for either a direct large-lattice check or a sharper argument that additional Gribov copies cannot appear.

Referee Report

3 major / 4 minor

Summary. The paper studies gauge fixing of replicated lattice Yang-Mills configurations: a thermalized link configuration on a small lattice Λx is copied m times per direction to form an extended lattice Λz, and minimal Landau gauge is imposed on Λz. The authors argue, by analogy with Bloch's theorem, that the gauge transformation g(z) minimizing the extended-lattice functional has the Bloch form g(z) = exp(i Σ_ν Θ_ν z_ν/N) h(x), where h is periodic on Λx and the commuting matrices Θ_ν lie in a Cartan subalgebra. From this ansatz they derive a selection rule, Eq. (5.19), singling out the nonzero Fourier coefficients of the gauge field on Λz, and hence the allowed momenta where the gluon propagator D(k') can be nonzero. They also derive a mechanism for suppression of D(0) in the m → ∞ limit and present numerical spectra of allowed momenta for SU(2), together with a visualization of color-magnetization domains. The central claim is that the entire simulation—thermalization, gauge fixing, and gluon-propagator evaluation—can be performed on the small lattice Λx, with a known, configuration-dependent set of allowed momenta.

Significance. If the Bloch ansatz holds, the paper provides a substantial technical advance: it reduces the cost of evaluating the large-volume gluon propagator to small-lattice computations, gives a concrete selection rule for the allowed momenta (Eq. (5.19)), and explains the previously observed suppression of D(0). The analytic derivations in Secs. 3–5 are detailed, internally consistent, and the selection rule was checked against about 16,000 nonzero propagator momenta from six configurations. The visualization in terms of color-magnetization domains is conceptually helpful. However, all of these results rest on the 'main hypothesis' of Eq. (3.30), which the paper itself identifies as an assumption inherited from refs. [1,5]. The central claim is therefore conditional, and the numerical verification in Sec. 6 checks the selection rule only on configurations produced by the Bloch-wave algorithm itself, not against an independent large-lattice gauge-fixing calculation.

major comments (3)

[Sec. 3.3, Eq. (3.30)] Equation (3.30), T(N e_μ) g(z) = s_μ g(z) with site-independent s_μ, is the load-bearing step: unless the translational phase s_μ is constant, the Bloch form (3.19), the selection rule (5.19), and the D(0) suppression of Sec. 5.4 do not follow. The justification given in Sec. 3.3 is that a local minimum is unique up to a global gauge transformation, citing ref. [27] for small lattice volumes. That support does not cover the replicated lattice Λz: standard gauge fixing on larger volumes is known to produce Gribov copies that are not related by global transformations, and ref. [27] concerns the original small lattice, not the replicated one. If the translation by N maps one minimum to a different Gribov equivalence class, s_μ cannot be constant and the Bloch ansatz fails. A concrete test would be to gauge-fix the same replicated configuration by direct minimization on Λz and check whether g(z+N e_μ) g(z)^† is independent of z; the paper does not provide such a test.
[Sec. 6, numerical check of Eq. (5.19)] The reported check that Eq. (5.19) is satisfied by all nonzero propagator momenta is necessary but not sufficient. The configurations were produced by the Bloch-wave algorithm, which by construction presupposes the Bloch form of Eq. (3.19); the check therefore verifies that the algorithm's own output is consistent with the selection rule, but it does not test whether a direct large-lattice gauge fixing would produce nonzero D(k') at momenta outside the predicted allowed set. The strong claim in Sec. 6—that D(k') is nonzero only for the allowed momenta—requires a comparison with standard gauge fixing on Λz for the same physical configurations. Without such a comparison, the selection rule remains a property of the ansatz, not an empirically tested property of the gauge-fixed ensemble.
[Sec. 5.4, Eqs. (4.47) and (4.51)] The derivation of D(0) suppression for arbitrary local minima, as opposed to absolute minima, relies on the stationarity condition with respect to the Θ_μ parameters and, through Eq. (4.47), on the Bloch-wave structure of the solution. If the main hypothesis of Eq. (3.30) is not established, the conclusion that D(0) → 0 as m → ∞ is conditional on the same unproven assumption. The paper correctly labels Eq. (3.30) as a hypothesis, but the conclusions section states the D(0) suppression as part of the 'main finding' without carrying that caveat. The conditional status should be made explicit in the abstract and conclusions, or the hypothesis should be tested directly on the replicated lattice.

minor comments (4)

[Sec. 3.1] The text reads 'state-solid physics'; this should be 'solid-state physics'.
[Secs. 5.2–5.3] The notation k'_ν = k_ν + K_ν m is clear, but the same symbol k is later used both for the Brillouin-zone index and for the common component in Eq. (5.30); please distinguish these, e.g. by using k̃ for the common value.
[Figs. 1–5] The captions of Figs. 3, 4, and 5 are nearly identical and could be shortened; it would help to state once that the color components M^b_3(y) are shown along the spatial directions, and then describe the projection in each figure.
[Sec. 6] The statement that 'slightly more than 16,000 allowed momenta' were checked is useful, but the manuscript does not report the statistics of the check (e.g., how many momenta were zero, how the nonzero threshold was set, or the precision of the numerical zero). Adding this information would make the numerical verification more reproducible.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: the selection rule (Eq. 5.19), the coefficient structure (Eq. 5.56) and the D(0) suppression (Eqs.

self definitional [Sec. 6 (Conclusions), with the operative definition of 'allowed momenta' in Sec. 1 (Introduction)]
"'the allowed momenta, i.e. the momenta for which a nonzero D(⃗k ) is obtained, include, but are not limited to, the momenta determined by the discretization on the original (small) lattice Λx.' ... 'Our main finding is that the gluon propagator D(⃗k ′ ) is nonzero only for the allowed momenta and, in these cases, its value comes from some of the coefficients ~Aij µ (g;⃗k ′ ), with all the other coefficients being equal to zero.'"

The paper defines 'allowed momenta' as, exactly, 'the momenta for which a nonzero D(k) is obtained.' The headline conclusion that 'D(k') is nonzero only for the allowed momenta' is therefore a restatement of that definition, not an independent empirical result. The substantive content that rescues the paper is adjacent to this sentence: Eq. (5.19) gives a derived, non-vacuous characterization of the allowed momenta in terms of the integer phases n^i_μ, Eq. (5.56) fixes which coefficients contribute, and Eqs. (4.47)/(5.60) derive the D(0) suppression. Those are consequences of the Bloch hypothesis (3.30), not of the definition, so the circularity is confined to the summary phrasing rather than the reasoning chain.

full rationale

The derivation chain is self-contained once the Bloch hypothesis is granted. From the explicitly stated hypothesis T(Ne_μ)g(z) = s_μ g(z) (Eq. 3.30), the paper derives the Bloch form (3.19), the cell-wise structure (4.2), the momentum-space selection rule (5.19), and the propagator reduction (5.47)/(5.56). No gluon-propagator value is used to fix any constant: the matrices Θ_μ are chosen by minimizing the functional (3.23)/(6.2), not by fitting D(k'). The numerical check in Sec. 6 verifies that nonzero propagator values satisfy Eq. (5.19) using the integers produced by the minimization; this is a consistency test of a derived condition, not a fitted input renamed as a prediction. Self-citations to [1] (the authors' prior work) and [5] (Zwanziger) supply the Bloch hypothesis and the uniqueness argument for absolute minima, but the paper labels Eq. (3.30) honestly as 'the main hypothesis considered in refs [1,5]' and supports its extension to local minima with the external numerical reference [27], while explicitly acknowledging the limitation ('in ref. [5], the proof of eq. (3.19) is presented only for the absolute minima... even in the case of local minima one can make the (reasonable) hypothesis'). That is a conditional derivation with an openly stated premise, which is a correctness risk about the replicated-lattice case, not a circular reduction of the conclusion to the premise. The one genuine circular element is the concluding sentence: since 'allowed momenta' is defined in Sec. 1 as 'the momenta for which a nonzero D(k) is obtained,' the assertion that 'D(k') is nonzero only for the allowed momenta' is definitionally true. This is a presentational self-definition in the summary, and it is offset by the genuinely derived content of Eq. (5.19), the coefficient-selection rule, and the D(0) suppression, none of which are assumed in the input. Accordingly the score is 2 rather than higher.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central derivation adds no numerically fitted constants; the parameters Theta_mu and integer phases n^j_mu are variational objects determined by the minimization, not free parameters fitted to observables. The load-bearing input is the uniqueness hypothesis behind Eq (3.30), plus standard lattice and Bloch background. No new physical entities are introduced; the color-magnetization domains are a visualization of gauge-fixed configurations.

assumptions (3)

ad hoc to paper Local minima of the extended-lattice gauge-fixing functional define a unique gauge transformation up to a global gauge transformation.
Invoked in Sec 3.3 to justify Eq (3.30), T(Ne_mu)g = s_mu g; the paper labels this the main hypothesis from refs [1,5] and cites numerical verification only for small volumes (ref [27]).
domain assumption Shifting the lattice by N in any direction maps a gauge-fixed solution to an equivalent solution of the same local minimum.
Used in Sec 3.3 from origin-redefinition invariance; requires periodic boundary conditions and periodic original links, Eqs (3.4) and (3.5).
standard math Commuting SU(Nc) matrices can be simultaneously diagonalized; with periodic boundary conditions their eigenvalues are discrete phases exp(2 pi i n/m).
Standard Lie algebra and Bloch-theorem background invoked in Secs 3.2, 3.3, and Appendix A for the Cartan-subalgebra parametrization.

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Cite this review

Pith. "Pith review of Bloch Waves, Magnetization and Domain Walls: The Case of the Gluon Propagator." pith.science (2026). https://pith.science/paper/7NT2TDS6

@misc{pith2026250607730,
  author       = {Pith},
  title        = {Pith review of: Bloch Waves, Magnetization and Domain Walls: The Case of the Gluon Propagator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7NT2TDS6}},
  note         = {Machine review of arXiv:2506.07730}
}

read the original abstract

We expand our previous study [1] of replicated gauge configurations in lattice SU(Nc) Yang-Mills theory -- employing Bloch's theorem, from condensed-matter physics -- to construct gauge-fixed field configurations on significantly larger lattices than the original, or primitive, one. We present a comprehensive discussion of the general gauge-fixing problem, identifying advantages of the replicated-lattice approach. In particular, the consideration of Bloch waves leads us to a visualization of the extended gauge-fixed configurations in terms of (color) magnetization domains. Moreover, we are able to explore features of the method to optimize the evaluation of gauge fields in momentum space, furthering our knowledge of the ``allowed momenta'', an issue that has hindered wider applications of this approach up to now. Interestingly, our analysis yields both a better conceptual understanding of the problem and a more efficient way to compute the desired large-volume observables.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Works this paper leans on

37 extracted references · 36 canonical work pages · cited by 1 Pith paper

[27]

Marinari, C

E. Marinari, C. Parrinello and R. Ricci, Evidence for the existence of Gribov copies in Landau gauge lattice QCD , Nucl. Phys. B362, 487 (1991)

work page 1991
[1]

Cucchieri and T

A. Cucchieri and T. Mendes, Bloch Waves in Minimal Landau Gauge and the Inﬁnite-Volume Limit of Lattice Gauge Theory , Phys. Rev. Lett. 118, no.19, 192002 (2017)

work page 2017
[2]

Cucchieri and T

A. Cucchieri and T. Mendes, Numerical test of the Gribov-Zwanziger scenario in Landau gauge, PoS QCD-TNT09, 026 (2009)

work page 2009
[3]

Cucchieri, D

A. Cucchieri, D. Dudal and N. Vandersickel, The No-Pole Condition in Landau gauge: Properties of the Gribov Ghost Form-Factor and a Constraint on t he 2d Gluon Propagator , Phys. Rev. D85, 085025 (2012)

work page 2012
[4]

Cucchieri, D

A. Cucchieri, D. Dudal, T. Mendes and N. Vandersickel, Modeling the Gluon Propagator in Landau Gauge: Lattice Estimates of Pole Masses and Dimensio n-Two Condensates , Phys. Rev. D85, 094513 (2012)

work page 2012
[5]

Zwanziger, Fundamental modular region, Boltzmann factor and area law i n lattice gauge theory, Nucl

D. Zwanziger, Fundamental modular region, Boltzmann factor and area law i n lattice gauge theory, Nucl. Phys. B412, 657 (1994)

work page 1994
[6]

Giusti, M

L. Giusti, M. L. Paciello, C. Parrinello, S. Petrarca and B. Taglienti, Problems on lattice gauge ﬁxing , Int. J. Mod. Phys. A16, 3487 (2001)

work page 2001
[7]

Greensite, The Conﬁnement problem in lattice gauge theory , Prog

J. Greensite, The Conﬁnement problem in lattice gauge theory , Prog. Part. Nucl. Phys. 51, 1 (2003)

work page 2003

Show all 37 references

[8]

V. N. Gribov, Quantization of Nonabelian Gauge Theories , Nucl. Phys. B139, 1 (1978)

1978
[9]

Zwanziger, Nonperturbative Modiﬁcation of the Faddeev-popov Formula and Banishment of the Naive Vacuum , Nucl

D. Zwanziger, Nonperturbative Modiﬁcation of the Faddeev-popov Formula and Banishment of the Naive Vacuum , Nucl. Phys. B209, 336 (1982)

1982
[10]

Dell’Antonio and D

G. Dell’Antonio and D. Zwanziger, Ellipsoidal Bound on the Gribov Horizon Contradicts the Perturbative Renormalization Group , Nucl. Phys. B326, 333 (1989)

1989
[11]

Vandersickel and D

N. Vandersickel and D. Zwanziger, The Gribov problem and QCD dynamics , Phys. Rept. 520, 175 (2012). – 64 –

2012
[12]

Cucchieri, T

A. Cucchieri, T. Mendes and A. Mihara, Ghost condensation on the lattice , Phys. Rev. D72, 094505 (2005)

2005
[13]

J. F. Cornwell, Group Theory in Physics: An Introduction , Academic Press, San Diego, U.S.A. (1997)

1997
[14]

D. B. Leinweber et al. [UKQCD], Asymptotic scaling and infrared behavior of the gluon propagator, Phys. Rev. D60, 094507 (1999) [Erratum: Phys. Rev. D61, 079901 (2000)]

1999
[15]

Smit, Introduction to Quantum Fields on a Lattice , Cambridge University Press, Cambridge, U.K

J. Smit, Introduction to Quantum Fields on a Lattice , Cambridge University Press, Cambridge, U.K. (2002)

2002
[16]

Gattringer and C

C. Gattringer and C. B. Lang, Quantum chromodynamics on the lattice (An Introductory Presentation), Lect. Notes Phys. 788, Springer, Berlin, Germany (2010)

2010
[17]

H. J. Rothe, Lattice Gauge Theories : An Introduction (Fourth Edition) , World Sci. Lect. Notes Phys. 82, 1-606 (2012)

2012
[18]

Suman and K

H. Suman and K. Schilling, A Comparative study of gauge ﬁxing procedures on the connection machines CM2 and CM5 , [arXiv:hep-lat/9306018 [hep-lat]]

arXiv
[19]

Cucchieri and T

A. Cucchieri and T. Mendes, Critical slowing down in SU(2) Landau gauge ﬁxing algorithms , Nucl. Phys. B471, 263 (1996)

1996
[20]

Cucchieri and T

A. Cucchieri and T. Mendes, Study of critical slowing down in SU(2) Landau gauge ﬁxing , Nucl. Phys. Proc. Suppl. 53, 811 (1997)

1997
[21]

Cucchieri and T

A. Cucchieri and T. Mendes, Critical slowing down in SU(2) Landau gauge ﬁxing algorithms at β = ∞, Comput. Phys. Commun. 154, 1 (2003)

2003
[22]

J. M. Leal, M. Cerqueira and T. Mendes, Eﬃciency Study of Overrelaxation and Stochastic Overrelaxation Algorithms for SU(3) Landau Gauge-Fixing , PoS LA TTICE2021, 057 (2022)

2022
[23]

Zwanziger, Vanishing of zero momentum lattice gluon propagator and col or conﬁnement , Nucl

D. Zwanziger, Vanishing of zero momentum lattice gluon propagator and col or conﬁnement , Nucl. Phys. B364, 127 (1991)

1991
[24]

de Soto, Restoring rotational invariance for lattice QCD propagators , JHEP 10, 069 (2022)

F. de Soto, Restoring rotational invariance for lattice QCD propagators , JHEP 10, 069 (2022)

2022
[25]

Cucchieri, Gribov copies in the minimal Landau gauge: The Inﬂuence on gluo n and ghost propagators, Nucl

A. Cucchieri, Gribov copies in the minimal Landau gauge: The Inﬂuence on gluo n and ghost propagators, Nucl. Phys. B508, 353 (1997)

1997
[26]

N. W. Ashcroft and N. D. Mermin, Solid State Physics , Harcourt Brace College Publishers, New York, U.S.A. (1976)

1976
[28]

J. M. Ziman, PRINCIPLES OF THE Theory of Solids , Cambridge University Press, Cambridge, UK, second edition (1972)

1972
[29]

Gonzalez-Arroyo, Yang-Mills ﬁelds on the four-dimensional torus

A. Gonzalez-Arroyo, Yang-Mills ﬁelds on the four-dimensional torus. Part 1.: Class ical theory, in *Peniscola 1997, Advanced school on non-perturbative q uantum ﬁeld physics* 57-91, and [arXiv:hep-th/9807108 [hep-th]]. – 65 –

1997 arXiv
[30]

https://www.escherinhetpaleis.nl/story-of-escher/metamorphosis-i-ii-iii/ (last accessed 2nd of May 2025)

2025
[31]

Grabenstein, Analysis and development of stochastic multigrid methods in lattice ﬁeld theory, [arXiv:hep-lat/9401024 [hep-lat]]

M. Grabenstein, Analysis and development of stochastic multigrid methods in lattice ﬁeld theory, [arXiv:hep-lat/9401024 [hep-lat]]

arXiv
[32]

C. A. Floudas, Nonlinear and Mixed-Integer Optimization. Fundamentals an d Applications, Oxford University Press (Topics in Chemical Engineering), New York, NY, USA (1995)

1995
[33]

Schaden and D

M. Schaden and D. Zwanziger, Horizon condition holds pointwise on ﬁnite lattice with free boundary conditions, in *Paris 1994, Proceedings, Quantum infrared physics* 10 -17, and [hep-th/9410019]

1994 arXiv
[34]

Zwanziger, Vanishing color magnetization in lattice Landau and Coulom b gauges , Phys

D. Zwanziger, Vanishing color magnetization in lattice Landau and Coulom b gauges , Phys. Lett. B257, 168-172 (1991)

1991
[35]

Cucchieri and T

A. Cucchieri and T. Mendes, Constraints on the IR behavior of the gluon propagator in Yang-Mills theories, Phys. Rev. Lett. 100, 241601 (2008)

2008
[36]

D. M. van Egmond and U. Reinosa, SU(N ) Cartan-Weyl bases and color factors , [arXiv:2104.12139 [hep-th]]

arXiv
[37]

Zuber, Invariances in Physics and Group Theory , https://www.lpthe.jussieu.fr/ ~zuber/Cours/InvariancesGroupTheory-2014.pdf (last accessed 2nd of May 2025)

J.-B. Zuber, Invariances in Physics and Group Theory , https://www.lpthe.jussieu.fr/ ~zuber/Cours/InvariancesGroupTheory-2014.pdf (last accessed 2nd of May 2025). – 66 –

2014

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [27]

Marinari, C

E. Marinari, C. Parrinello and R. Ricci, Evidence for the existence of Gribov copies in Landau gauge lattice QCD , Nucl. Phys. B362, 487 (1991)

work page 1991

[2] [1]

Cucchieri and T

A. Cucchieri and T. Mendes, Bloch Waves in Minimal Landau Gauge and the Inﬁnite-Volume Limit of Lattice Gauge Theory , Phys. Rev. Lett. 118, no.19, 192002 (2017)

work page 2017

[3] [2]

Cucchieri and T

A. Cucchieri and T. Mendes, Numerical test of the Gribov-Zwanziger scenario in Landau gauge, PoS QCD-TNT09, 026 (2009)

work page 2009

[4] [3]

Cucchieri, D

A. Cucchieri, D. Dudal and N. Vandersickel, The No-Pole Condition in Landau gauge: Properties of the Gribov Ghost Form-Factor and a Constraint on t he 2d Gluon Propagator , Phys. Rev. D85, 085025 (2012)

work page 2012

[5] [4]

Cucchieri, D

A. Cucchieri, D. Dudal, T. Mendes and N. Vandersickel, Modeling the Gluon Propagator in Landau Gauge: Lattice Estimates of Pole Masses and Dimensio n-Two Condensates , Phys. Rev. D85, 094513 (2012)

work page 2012

[6] [5]

Zwanziger, Fundamental modular region, Boltzmann factor and area law i n lattice gauge theory, Nucl

D. Zwanziger, Fundamental modular region, Boltzmann factor and area law i n lattice gauge theory, Nucl. Phys. B412, 657 (1994)

work page 1994

[7] [6]

Giusti, M

L. Giusti, M. L. Paciello, C. Parrinello, S. Petrarca and B. Taglienti, Problems on lattice gauge ﬁxing , Int. J. Mod. Phys. A16, 3487 (2001)

work page 2001

[8] [7]

Greensite, The Conﬁnement problem in lattice gauge theory , Prog

J. Greensite, The Conﬁnement problem in lattice gauge theory , Prog. Part. Nucl. Phys. 51, 1 (2003)

work page 2003