[Paper Review] Adjoint zero-modes as a tool to understand the Yang-Mills vacuum

Margarita Garcı́a Pérez, Antonio González-Arroyo+1 more|ArXiv.org|Oct 2, 2007

Quantum Chromodynamics and Particle Interactions1 references4 citations

TL;DR

This paper proposes using adjoint zero-modes of the overlap Dirac operator to probe the topological structure of the Yang-Mills vacuum, demonstrating that these modes efficiently reconstruct gauge action density and reveal lattice artifacts. The key contribution is showing that spurious zero modes—violating the Atiyah-Singer index theorem—arise from lattice roughness and small instantons below ~1.65 lattice spacings, which can be mitigated by tuning the overlap operator's mass parameter.

ABSTRACT

The use of adjoint (quasi) zero-modes of the Dirac operator to probe the Yangs-Mills vacuum has been recently advocated by Gonzalez-Arroyo and Kirchner. The construction relies on the use of the super-symmetric zero mode which, for classical configurations, provides a direct estimate of the gauge action density. In the lattice implementation of this idea, we show how the results improve considerably if the overlap operator is used instead of the Wilson-Dirac one. Before proceeding to the detailed study of Monte Carlo ensembles, we studied here a series of potentially complicated situations which can be encountered. In particular, we study the case of instanton anti-instanton pairs and analyse how the results depend upon separation. The effect of lattice artifacts is also of concern. Indeed, a statistical analysis of zero modes of thermalised SU(2) configurations at beta=2.57 shows a significant fraction having 4N+2 adjoint zero modes, in contradiction with the index theorem. This violation must be associated to the roughness of the lattice configurations. Indeed, we show that this situation occurs for instantons of size of the order of the lattice spacing.

Motivation & Objective

• To develop a computationally efficient method for probing the topological content of the Yang-Mills vacuum using adjoint zero-modes of the Dirac operator.
• To address the challenge of lattice artifacts and UV fluctuations in Monte Carlo configurations that distort topological measurements.
• To investigate the reliability of adjoint zero-mode filtering in non-self-dual configurations such as instanton-anti-instanton pairs and small instantons.
• To identify the origin of spurious zero modes (e.g., 4N+2 modes) in thermalized SU(2) lattice ensembles and link them to lattice roughness.
• To optimize the overlap Dirac operator's mass parameter to minimize spurious modes while preserving topological fidelity.

Proposed method

• Use the super-symmetric zero-mode construction from classical solutions as a template to identify physical zero modes on generic lattice configurations.
• Apply a reality condition on the first component of positive-chirality modes to isolate super-symmetric modes, minimizing ∑x,a[Im ψ₊¹,a(x)]².
• Reconstruct self-dual and anti-self-dual action densities from |ψ₊ᵃ(x)|² and |ψ₋ᵃ(x)|², respectively.
• Employ the overlap Dirac operator instead of the Wilson-Dirac operator to reduce lattice artifacts and improve mode localization.
• Use the admissibility condition ||1 - U(p)|| ≤ ε(mWD) to detect and quantify lattice roughness affecting the spectrum.
• Perform cooling sweeps on instanton configurations to vary instanton size and study the evolution of the adjoint spectrum.

Experimental results

Research questions

• RQ1 Can adjoint zero-modes of the overlap Dirac operator accurately reconstruct the action density in non-self-dual configurations like instanton-anti-instanton pairs?
• RQ2 Why do thermalized SU(2) lattice ensembles exhibit a significant fraction of 4N+2 adjoint zero modes, contradicting the Atiyah-Singer index theorem?
• RQ3 How does the presence of spurious zero modes correlate with lattice roughness and the violation of the admissibility condition for the overlap operator?
• RQ4 What is the critical instanton size below which the adjoint spectrum deviates from the expected 4N zero modes, and how does it depend on the overlap operator's mass parameter?
• RQ5 Can tuning the overlap operator's mass parameter minimize spurious zero modes while preserving topological fidelity in Monte Carlo simulations?

Key findings

• The adjoint zero-mode filtering method successfully reconstructs the action density in instanton-anti-instanton pairs, even at small separations, demonstrating robustness against quantum fluctuations.
• A significant fraction of thermalized SU(2) configurations at β=2.57 exhibit 4N+2 adjoint zero modes, violating the Atiyah-Singer index theorem due to lattice artifacts.
• Spurious zero modes are linked to instantons with sizes below ρc ≈ 1.65a, where lattice roughness causes violations of the admissibility condition for the overlap operator.
• The critical size ρc for the transition from 4 to 2 zero modes depends weakly on the overlap operator's mass parameter, with ρc ≈ 1.9a for mWD > 1.4.
• Configurations with spurious modes show a high density of plaquettes violating the admissibility condition ||1 - U(p)|| ≤ ε(mWD), localized near the action density maximum.
• Using mWD > 1.4 minimizes the window for spurious eigenvalues and is optimal for filtering topological structure in Monte Carlo ensembles.