Localization in Lattice QCD

Maarten Golterman; Yigal Shamir

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Localization in Lattice QCD

Maarten Golterman, Yigal Shamir

High Energy Physics - Lattice · Submitted 2 Jun 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This divergence occurs when we take a "twisted" quark mass to zero, and we conclude that quenched QCD with Wilson fermions is well-defined only with a non-vanishing twisted mass.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.68 · human participants 0.01 · controlled comparison 0.19 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.60 · mentions code or data availability 0.01.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

In this paper, we examine the phase diagram of quenched QCD with two flavors of Wilson fermions, proposing the following microscopic picture. The super-critical regions inside and outside the Aoki phase are characterized by the existence of a density of near-zero modes of the (hermitian) Wilson-Dirac operator, and thus by a non-vanishing pion condensate. Inside the Aoki phase, this density is built up from extended near-zero modes, while outside the Aoki phase, there is a non-vanishing density of exponentially localized near-zero modes, which occur in "exceptional" gauge-field configurations. Nevertheless, no Goldstone excitations appear outside the Aoki phase, and the existence of Goldstone excitations may therefore be used to define the Aoki phase in both the quenched and unquenched theories. We show that the density of localized near-zero modes gives rise to a divergent pion two-point function, thus providing an alternative mechanism for satisfying the relevant Ward identity in the presence of a non-zero order parameter. This divergence occurs when we take a "twisted" quark mass to zero, and we conclude that quenched QCD with Wilson fermions is well-defined only with a non-vanishing twisted mass. We show that this peculiar behavior of the near-zero-mode density is special to the quenched theory by demonstrating that this density vanishes in the unquenched theory outside the Aoki phase. We discuss the implications for domain-wall and overlap fermions constructed from a Wilson-Dirac kernel. We argue that both methods work outside the Aoki phase, but fail inside because of problems with locality and/or chiral symmetry, in both the quenched and unquenched theories.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper, we examine the phase diagram of quenched QCD with two flavors of Wilson fermions, proposing the following microscopic picture.	contribution 0.99	0.72	0.04	unclear 0.42
s1	The super-critical regions inside and outside the Aoki phase are characterized by the existence of a density of near-zero modes of the (hermitian) Wilson-Dirac operator, and thus by a non-vanishing pion condensate.	result 0.77	0.55	0.17	asserted 1.00
s2	Inside the Aoki phase, this density is built up from extended near-zero modes, while outside the Aoki phase, there is a non-vanishing density of exponentially localized near-zero modes, which occur in "exceptional" gauge-field configurations.	result 0.78	0.63	0.15	asserted 1.00
s3	Nevertheless, no Goldstone excitations appear outside the Aoki phase, and the existence of Goldstone excitations may therefore be used to define the Aoki phase in both the quenched and unquenched theories.	interpretation 0.63	0.47	0.19	hedged possibility 0.71
s4	We show that the density of localized near-zero modes gives rise to a divergent pion two-point function, thus providing an alternative mechanism for satisfying the relevant Ward identity in the presence of a non-zero order parameter.	result 0.92	0.93	0.13	asserted 1.00
s5	This divergence occurs when we take a "twisted" quark mass to zero, and we conclude that quenched QCD with Wilson fermions is well-defined only with a non-vanishing twisted mass.	result 0.66	0.91	0.34	asserted 1.00
s6	We show that this peculiar behavior of the near-zero-mode density is special to the quenched theory by demonstrating that this density vanishes in the unquenched theory outside the Aoki phase.	result 0.94	0.94	0.14	asserted 1.00
s7	We discuss the implications for domain-wall and overlap fermions constructed from a Wilson-Dirac kernel.	interpretation 0.46	0.38	0.03	unclear 0.76
s8	We argue that both methods work outside the Aoki phase, but fail inside because of problems with locality and/or chiral symmetry, in both the quenched and unquenched theories.	interpretation 0.68	0.58	0.16	asserted 0.67

What kind of paper the classifier reports this is: theoretical result 0.92 · evidence from proof or derivation 0.49.

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references and citing sentences
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The paper

Original abstract

In this paper, we examine the phase diagram of quenched QCD with two flavors of Wilson fermions, proposing the following microscopic picture. The super-critical regions inside and outside the Aoki phase are characterized by the existence of a density of near-zero modes of the (hermitian) Wilson-Dirac operator, and thus by a non-vanishing pion condensate. Inside the Aoki phase, this density is built up from extended near-zero modes, while outside the Aoki phase, there is a non-vanishing density of exponentially localized near-zero modes, which occur in "exceptional" gauge-field configurations. Nevertheless, no Goldstone excitations appear outside the Aoki phase, and the existence of Goldstone excitations may therefore be used to define the Aoki phase in both the quenched and unquenched theories. We show that the density of localized near-zero modes gives rise to a divergent pion two-point function, thus providing an alternative mechanism for satisfying the relevant Ward identity in the presence of a non-zero order parameter. This divergence occurs when we take a "twisted" quark mass to zero, and we conclude that quenched QCD with Wilson fermions is well-defined only with a non-vanishing twisted mass. We show that this peculiar behavior of the near-zero-mode density is special to the quenched theory by demonstrating that this density vanishes in the unquenched theory outside the Aoki phase. We discuss the implications for domain-wall and overlap fermions constructed from a Wilson-Dirac kernel. We argue that both methods work outside the Aoki phase, but fail inside because of problems with locality and/or chiral symmetry, in both the quenched and unquenched theories.

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References

Forward citations

Cited by 8 Pith papers

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

The chiral transition and U(1)_A symmetry restoration from lattice QCD using Domain Wall Fermions
hep-lat 2012-05 accept novelty 6.0 of 10

Domain wall fermion lattice QCD at a 200 MeV pion places the chiral crossover near 160-170 MeV and finds the anomalous axial symmetry still measurably broken up to 195 MeV, though weakly at the highest temperatures.
Two-flavor QCD simulation with exact chiral symmetry
hep-lat 2008-03 conditional novelty 6.0 of 10

Two-flavor lattice QCD with exactly chiral overlap quarks is now simulable at 0.12 fm spacing and sea masses down to a sixth of the strange mass, with the lattice spacing fixed at a = 0.1184(3)(17)(12) fm.
Non-perturbative renormalization of quark bilinear operators and B_K using domain wall fermions
hep-lat 2007-12 accept novelty 6.0 of 10

Non-perturbative RI/MOM renormalization constants for quark bilinears and B_K are determined on 2+1 flavor domain wall fermion lattices and matched to MS-bar at 2 GeV.
Domain Wall QCD with Near-Physical Pions
hep-lat 2012-08 conditional novelty 5.0 of 10

Adding a coarse-lattice domain wall ensemble with 171 MeV pions to two existing data sets reduces the chiral extrapolation systematic and yields f_pi = 127.1(3.8) MeV, f_K = 152.4(3.4) MeV, and B_K(MSbar, 3 GeV) = 0.535(16).
Lattice gauge action suppressing near-zero modes of H_W
hep-lat 2006-07 conditional novelty 5.0 of 10

A lattice action with extra negative-mass Wilson fermions produces a measured gap in the spectral density of the hermitian Wilson-Dirac operator near zero, removing the near-zero-mode bottleneck for overlap fermions.

Show 3 more citing papers

Lattice QCD with fixed topology
hep-lat 2006-03 conditional novelty 5.0 of 10

Fixed-topology lattice QCD with an admissible gauge action or a negative-mass Wilson determinant stabilizes the topological charge and reproduces known low-energy constants in the epsilon-regime.
Rooted staggered fermions: good, bad or ugly?
hep-lat 2006-10 accept novelty 3.0 of 10

Rooted staggered fermions are non-local and unphysical at nonzero lattice spacing, but the non-locality plausibly vanishes in the continuum limit, so the method is 'ugly' rather than 'bad'.
Domain wall fermions
hep-lat 2026-03 unverdicted novelty 2.0 of 10

Domain wall fermions recover exact chiral symmetry in the infinite fifth dimension limit and produce an effective four-dimensional operator satisfying the Ginsparg-Wilson relation.

How the literature uses it

15 arXiv papers cite this in 25 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 19 sentences without a confident label

Builds on it 1 paper

This proof may be generalised to cover the Shamir Kernel $K_S$ that corresponds to the $L_s→∞$ limit of DWF, and also the condition can be relaxed to require only a gap in the spectrum of delocalised eigenmodes [ref] .

2+1 flavour Domain Wall Fermion simulations by the RBC and UKQCD collaborations · Locality · 0.83

Compares with it 1 paper

The major differences between Refs. [ref] and this work and Refs. [ref] are on the one hand technical, due to the reduced amount of symmetry in finite-temperature quantum field theory in the imaginary-time formulation stemming from the presence of a compactified direction; and on the other hand of practical relevance (in principle…), as this work is concerned with physically more realistic theories.

Localisation of Dirac modes in gauge theories and Goldstone's theorem at finite temperature · Conclusions and outlook · 0.93

Cites it as background 4 papers

There is a wealth of literature on this topic, and I will use particularly the results and insights from Refs. [ref] .

On the consistency of the Aoki-phase · Consistency of the Aoki-phase with eigenvalue sums · 0.80
The interesting conjecture is whether the opening of a localised region in the spectrum is connected to the chiral phase transition [ref] .[ Anderson localisation has been considered in lattice QCD also in the context of construction of chiral fermions, domain-wall and overlap, regarding the localisation of the kernel eigenmodes and its effect on the chirality of the discretised action (see for example [ref] ).]

Anderson Localization in high temperature QCD: background configuration properties and Dirac eigenmodes · Introduction · 0.81
The connection between localisation and disappearance of Goldstone modes has been known for a long time in condensed matter physics [ref] , and for quite some time in zero-temperature lattice field theory as well, although in an unphysical setup [ref] -1.3pt (see also Ref. [ref] ).

Localisation of Dirac modes in gauge theories and Goldstone's theorem at finite temperature · Conclusions and outlook · 0.91

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