[Paper Review] Polyakov loops and SU(2) staggered Dirac spectra

Falk Bruckmann, Stefan Keppeler+2 more|arXiv (Cornell University)|Feb 5, 2008

Quantum chaos and dynamical systems10 references3 citations

TL;DR

This paper investigates the spectrum of the staggered Dirac operator in SU(2) lattice gauge theory near the free limit, showing that eigenvalues form clusters of eight due to Polyakov loop-induced level splitting. It derives an analytical formula predicting cluster positions using Polyakov loops and boundary conditions, with intra-cluster spacing matching the chiral symplectic ensemble (chSE) and inter-cluster spacing approaching Poisson statistics on prime lattices, confirming symmetry transitions in the continuum limit.

ABSTRACT

We consider the spectrum of the staggered Dirac operator with SU(2) gauge fields. Our study is motivated by the fact that the antiunitary symmetries of this operator are different from those of the SU(2) continuum Dirac operator. In this contribution, we investigate in some detail staggered eigenvalue spectra close to the free limit. Numerical experiments in the quenched approximation and at very large $β$-values show that the eigenvalues occur in clusters consisting of eight eigenvalues each. We can predict the locations of these clusters for a given configuration very accurately by an analytical formula involving Polyakov loops and boundary conditions. The spacing distribution of the eigenvalues within the clusters agrees with the chiral symplectic ensemble of random matrix theory, in agreement with theoretical expectations, whereas the spacing distribution between the clusters tends towards Poisson behavior.

Motivation & Objective

• To understand the spectral structure of the staggered Dirac operator in SU(2) gauge theories near the free limit, where quantum fluctuations are suppressed.
• To resolve the discrepancy between the antiunitary symmetries of the staggered Dirac operator (chSE) and the continuum Dirac operator (chOE) in SU(2) fundamental fermion systems.
• To investigate how Polyakov loops and boundary conditions influence eigenvalue clustering and spacing statistics in the free limit regime.
• To disentangle three distinct energy scales in the spectrum: overall plateau structure, cluster separation, and intra-cluster level splitting.
• To verify whether spacing distributions between clusters converge to Poisson statistics, as expected in the free limit, using prime lattice geometries to remove accidental degeneracies.

Proposed method

• Constructing vacuum configurations with uniform SU(2) links that reproduce the average traced Polyakov loops $P_ u$ of a given gauge configuration.
• Using analytical solutions of the free staggered Dirac operator on finite lattices to predict the positions of spectral plateaux, given by Eq. (4), which depend on lattice sizes $L_ u$ and boundary conditions.
• Applying a cluster-based unfolding procedure to compute nearest-neighbor spacing distributions $P(s)$ separately within clusters and between clusters.
• Employing a 'prime lattice' with $L_ u = 2 au_ u$ where $ au_ u$ are distinct primes to eliminate accidental degeneracies and improve statistical convergence.
• Comparing numerical $P(s)$ distributions from quenched lattice configurations at large $eta = 10000$ with theoretical predictions from chSE and Poisson ensembles.
• Using random matrix theory (RMT) to interpret spacing statistics: chSE for intra-cluster spacing and Poisson for inter-cluster spacing, with proper unfolding for each scale.

Experimental results

Research questions

• RQ1 How do Polyakov loops and boundary conditions determine the positions of eigenvalue clusters in the staggered Dirac spectrum near the free limit?
• RQ2 Does the spacing distribution within clusters of eight eigenvalues agree with the chiral symplectic ensemble (chSE), consistent with the antiunitary symmetries of the staggered Dirac operator?
• RQ3 Do the spacing distributions between clusters converge to Poisson statistics in the free limit, and what role do lattice geometry and accidental degeneracies play?
• RQ4 Can the spectral structure be disentangled into three distinct energy scales: plateau structure, cluster separation, and intra-cluster splitting?
• RQ5 What is the effect of using prime lattice sizes on the convergence of inter-cluster spacing statistics to the Poisson distribution?

Key findings

• Eigenvalues of the SU(2) staggered Dirac operator form distinct clusters of eight, with positions accurately predicted by an analytical formula involving Polyakov loops and boundary conditions.
• The nearest-neighbor spacing distribution within clusters agrees with the chiral symplectic ensemble (chSE), confirming the antiunitary symmetries of the staggered Dirac operator even at large $eta$.
• On a $10^4$ lattice, inter-cluster spacing does not fully converge to Poisson due to accidental degeneracies; however, on a $34\times38\times46\times58$ prime lattice, the spacing distribution between clusters matches Poisson statistics.
• The spacing distribution between the free Dirac eigenvalues (plateaux) also agrees with Poisson, as expected, and this is confirmed on the same prime lattice.
• The average spacing between plateaux is more than ten times larger than the average spacing between clusters, confirming the existence of three well-separated energy scales.
• The results support the expectation of a symmetry transition from chSE (staggered) to chOE (continuum) in the continuum limit for SU(2) fundamental fermions.