Wilson–Dirac Spectrum in Lattice QCD

• Wilson–Dirac Spectrum is the statistical distribution of eigenvalues from the lattice discretization of the Dirac operator, highlighting effects like chiral symmetry breaking and lattice artifacts.
• The methodology employs Wilson chiral perturbation theory and chiral random matrix theory to match analytical predictions with lattice data, enabling precise extraction of low-energy constants.
• Analysis of microscopic spectral density, including graded partition functions and histogram fitting, provides insights into continuum extrapolation and the impact of discretization effects.

The Wilson–Dirac spectrum refers to the statistical properties of the eigenvalues of the Wilson–Dirac operator and its hermitized forms in lattice gauge theory. Detailed analysis of this spectrum provides direct access to nonperturbative aspects of chiral symmetry breaking, the impact of lattice discretization, and critical low-energy constants (LECs) within the Wilson chiral perturbation theory (WχPT) framework. It is foundational to precision extrapolation of lattice QCD results to the continuum and understanding lattice-induced chiral symmetry breaking.

1. The Wilson–Dirac Operator and Hermitian Forms

The Wilson–Dirac operator $D_W$ is the standard lattice discretization of the Dirac operator, augmented by a Wilson term to remove doubler modes at finite lattice spacing $a$. In $d=4$, and after inclusion of a bare mass $m_0$: $$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$ where $\gamma_\mu$ are Euclidean Dirac matrices, and $\nabla_\mu$, $\nabla^*_\mu$ are forward/backward covariant derivatives.

$D_W$ obeys $\gamma_5$-Hermiticity, $a$0, but is nonnormal for $a$1—hence not all eigenvalues are purely imaginary. For spectral analysis, the Hermitian Wilson–Dirac operator is introduced: $a$2

which has a real spectrum, making microscopic spectral analysis and comparison with field theory tractable (Akemann et al., 2010).

In twisted-mass lattice QCD, the Hermitian operator is generalized to

$a$3

where $a$4 is the untwisted mass and $a$5 the twisted mass (Cichy et al., 2015).

2. Microscopic Spectral Density and Wilson χPT

The microscopic spectral density refers to the rescaled spectral density in the so-called $a$6-regime ($a$7 with $a$8, $a$9, $d=4$0 fixed). For index $d=4$1: $d=4$2 where $d=4$3 is the infinite-volume chiral condensate, and the $d=4$4 describe cutoff effects with the LECs $d=4$5, $d=4$6, $d=4$7 entering at $d=4$8 in the chiral Lagrangian (Akemann et al., 2010, Damgaard et al., 2010, Deuzeman et al., 2011).

At leading order, the zero-momentum sector of WχPT yields a group integral for the partition function at fixed $d=4$9: $m_0$0 with

$m_0$1

Extracting the microscopic spectral density involves constructing a (graded) generating functional and taking imaginary parts of the partially quenched resolvent or its supersymmetric extension (Akemann et al., 2010, Damgaard et al., 2010).

3. Analytical Results: Diffusion Representation and Random Matrix Theory Matching

For $m_0$2 (twisted-mass), the microscopic spectral density in sector $m_0$3 is given by: $m_0$4 where $m_0$5 is derived from the graded chiral partition function (Cichy et al., 2015, Splittorff et al., 2012).

A key result is the diffusion representation (Splittorff et al., 2011): $m_0$6 with $m_0$7 a Bessel–diffusion kernel reflecting the smearing of zero modes due to $m_0$8 effects.

Wilson chiral Random Matrix Theory (WRMT) provides an alternative, mathematically equivalent formulation for the microscopic spectrum via ensembles of random matrices matching the symmetry class and index structure of the Wilson–Dirac operator (Akemann et al., 2010, Kieburg et al., 2013, Damgaard et al., 2010, Deuzeman et al., 2011).

The mapping between chiral Lagrangian and WRMT parameters is: $m_0$9 This mapping ensures that microscopic observables—eigenvalue densities, spacing distributions, and individual eigenvalue distributions—agree precisely between WχPT and WRMT in the microscopic scaling limit (Akemann et al., 2010, Damgaard et al., 2010).

4. Extraction of Low-Energy Constants and Spectral Observables

Microscopic spectral properties, particularly near the origin, are sensitive to the chiral condensate $$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$0 and the Wilson LECs ($$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$1, $$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$2, $$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$3). Several practical methods have been established:

• The width of the near-zero eigenvalue peak (smeared former zero modes) is $$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$4, directly fixing $$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$5 (Splittorff et al., 2011, Cichy et al., 2015).
• The spectral density of the Hermitian operator $$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$6 and distribution of chirality over real $$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$7 eigenvalues allow for fits of $$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$8, $$D_W = m_0 + \frac12 \sum_\mu \left[ \gamma_\mu (\nabla_\mu + \nabla^*_\mu) - a\,\nabla^*_\mu\nabla_\mu \right]$$9, and (where resolved) $\gamma_\mu$0, $\gamma_\mu$1 (Deuzeman et al., 2011, Kieburg et al., 2013).
• Volume- and mass-scaling tests of spectral data confirm that fitting the theoretical microscopic density to lattice histograms (usually in fixed-$\gamma_\mu$2 sectors) yields robust extractions of LECs. The mapping of sector scaling for parameters $\gamma_\mu$3 is confirmed by data (Damgaard et al., 2011, Deuzeman et al., 2011, Cichy et al., 2015).

Typical values extracted for $\gamma_\mu$4 and $\gamma_\mu$5 in $\gamma_\mu$6 twisted-mass QCD, using the ETM lattice data, are $\gamma_\mu$7 and $\gamma_\mu$8 (in suitably normalized units) (Cichy et al., 2015). The analytic determination of all $\gamma_\mu$9 LECs is possible in principle by fitting to sufficiently rich, unquenched spectra.

5. Topology, Twisted Mass, and Physical Regime Dependence

The Wilson–Dirac spectrum exhibits strong dependence on the topological index $\nabla_\mu$0. For $\nabla_\mu$1, the continuum limit shows exact zero modes—at finite $\nabla_\mu$2, these are broadened into Gaussian peaks whose properties are governed by $\nabla_\mu$3. The peak structure, repulsion from the origin, and overall normalization of the microscopic density encode information about topology and discretization effects (Cichy et al., 2015, Splittorff et al., 2011).

Large twisted mass $\nabla_\mu$4 suppresses dynamical fermion contributions, tending the spectrum towards the quenched (determinantless) form. For smaller $\nabla_\mu$5 the dynamical effects are pronounced, influencing the density near the origin (Cichy et al., 2015, Splittorff et al., 2012).

The combined scaling regime ($\nabla_\mu$6, $\nabla_\mu$7) ensures that the lowest eigenvalues probed are below the Thouless energy and thus dominated by chiral zero-mode dynamics, with finite-volume corrections suppressed (Cichy et al., 2015, Deuzeman et al., 2011).

6. Spectral Features, Continuum Limit, and Lattice Artifacts

Nonzero lattice spacing causes the “hard-edge” structure of the continuum spectrum to be smoothed. For small $\nabla_\mu$8, $\nabla_\mu$9 corrections parametrized by $\nabla^*_\mu$0 dominate, while $\nabla^*_\mu$1, $\nabla^*_\mu$2 encode double-trace lattice artifacts and are suppressed for $\nabla^*_\mu$3.

Key features:

• The zero-mode peak for $\nabla^*_\mu$4 becomes a finite-width Gaussian for nonzero $\nabla^*_\mu$5.
• Tail states appear within the continuum gap as Lifshitz tails, scaling as $\nabla^*_\mu$6.
• At larger $\nabla^*_\mu$7, the spectrum exhibits square-root edges analogous to Tracy–Widom distributions.
• Dynamical quarks further suppress the singularity at the origin, removing the $\nabla^*_\mu$8 divergence present in the quenched density (Splittorff et al., 2011).

Fits to the spectrum of $\nabla^*_\mu$9 (including the lowest several eigenvalues across topological sectors) can thus separate continuum physics from lattice artifacts and facilitate continuum extrapolation. The methodology—analytical prediction from WχPT/WRMT, followed by sector-resolved histogram fitting—is standard (Cichy et al., 2015, Damgaard et al., 2011, Deuzeman et al., 2011).

7. Impact, Applications, and Extensions

The Wilson–Dirac spectrum is central to several directions in lattice gauge theory:

• It enables precision extraction of the chiral condensate and $D_W$0 LECs directly from spectral data.
• Control of lattice artifacts through analysis of the microscopic spectrum informs improved action development, tuning of simulation parameters (e.g., fixing $D_W$1, $D_W$2 to avoid the Aoki phase), and direct assessment of universality (Damgaard et al., 2013).
• The approach extends to enriched lattice actions (e.g., twisted-mass, clover improvement), different gauge groups, and probes of topological charge distributions.
• The analytic realization and matching with WRMT provide efficient computational tools for analysis and interpolation, now standard in extracting nonperturbative low-energy constants (Cichy et al., 2015, Damgaard et al., 2010, Splittorff et al., 2011, Kieburg et al., 2013).

In summary, the detailed understanding of the Wilson–Dirac spectrum at the microscopic level—grounded in WχPT and confirmed via lattice data—forms the backbone of modern spectral analyses in lattice QCD, chiral dynamics, and the study of discretization effects in strongly coupled gauge theories.