Localization and chiral symmetry in 2+1 flavor domain wall QCD

TL;DR

The paper analyzes how residual chiral symmetry breaking in 2+1 flavor domain wall QCD depends on the fifth-dimension extent $L_s$ through the spectrum and localization of $H_W$, and how this interacts with topology-change ergodicity under DBW2 and Iwasaki gauge actions. Using transfer-matrix formalism and a 5D axial current, it connects $m_{ m res}$ to both localized near-zero modes and extended states above the mobility edge, and demonstrates that $m_{ m res}$ scales roughly as $m_{ m res}(L_s) \approx \left( c_1 e^{-\lambda_c L_s} + c_2 \right)/L_s$, with $c_2$ tied to near-zero-mode density. Numerical results show acceptable locality and topology sampling at $a^{-1}\ge 1.6$ GeV, with Iwasaki maintaining better topology-changing ergodicity than DBW2 at finer lattices. The findings support practical 2+1 flavor DWF simulations with good chiral properties using Iwasaki (and DBW2) actions and guide choices of $L_s$ to balance chiral symmetry and topological sampling.

Abstract

We present results for the dependence of the residual mass of domain wall fermions (DWF) on the size of the fifth dimension and its relation to the density and localization properties of low-lying eigenvectors of the corresponding hermitian Wilson Dirac operator relevant to simulations of 2+1 flavor domain wall QCD. Using the DBW2 and Iwasaki gauge actions, we generate ensembles of configurations with a $16^3\times 32$ space-time volume and an extent of 8 in the fifth dimension for the sea quarks. We demonstrate the existence of a regime where the degree of locality, the size of chiral symmetry breaking and the rate of topology change can be acceptable for inverse lattice spacings $a^{-1} \ge 1.6$ GeV.

Figures (23)

• Figure 1: Schematic diagram of the quenched Aoki phase. A pionic condensate and corresponding non-zero density of near zero modes is developed throughout most of the negative mass region. In the coloured sectors, a non-zero mobility edge is developed and the contours could equally well represent either decreasing low mode density, decreasing pionic condensate or increasing $\lambda_c$ as one moves towards the continuum limit at $g^2=0$.
• Figure 2: Spectral flow of DBW2. The horizontal axis is the valence domain wall height, $0\le M_5\le3.0$ and the vertical axis is the eigenvalue of $H_W$, $-0.3\le\lambda_{H_W}\le0.3$. From left, $\beta=0.72, , \beta=0.764, \beta=0.78$ (first row) and $\beta=0.8,\beta=0.88$ (second row).
• Figure 3: Spectral flow of Iwasaki $\beta=2.13, \beta=2.2$, and $\beta=2.3$ (from left to right). Axes are the same as fig \ref{['fig:bubbleDBW2']}.
• Figure 4: Spectral density obtained on a subset of our data with no error analysis, and fixed bin width $0.02$. Thus, where the density is low the error could be large and the data should be considered only as a qualitative indication of the nature of our ensembles.
• Figure 5: Various measures can be used to define the degree of localization. Inverse participation ratio has been a popular order parameter for delocalization transitions and, while fine for that purpose, counts occupied sites without paying attention to mode shape. This one dimensional cartoon highlights the advantages of the more robust measure employed in this paper. Given the model of exponential localization, our tactic is to find a localization exponent that forms a tight bound for the eigenvector. This measure will show the influence of satellite peaks in a robust fashion by eliminating all forms of dilution by volume average. Other useful measures that take into account the shape would include appropriately weighted moments of the density function.