[Paper Review] Controlling Residual Chiral Symmetry Breaking in Domain Wall Fermion Simulations

Dwight Renfrew, Thomas Blum+3 more|arXiv (Cornell University)|Feb 16, 2009

Quantum Chromodynamics and Particle Interactions6 references15 citations

TL;DR

This paper proposes a novel determinant-based weighting factor using chirally twisted masses in the Wilson-Dirac fermion determinant to suppress residual chiral symmetry breaking in domain wall fermion (DWF) lattice QCD simulations. By tuning the mass parameters, the method significantly reduces the residual mass $m_{\rm res}$ while preserving adequate topological tunneling, enabling larger physical volumes and improved control over chiral symmetry violation at strong couplings.

ABSTRACT

At stronger gauge-field couplings, the domain wall fermion (DWF) residual mass, a measure of chiral symmetry breaking, grows rapidly. This measure is largely due to near zero fermion eigenmodes of logarithm of the 4D transfer matrix along the fifth dimension, and these eigenmodes increase rapidly at strong coupling. To suppress these eigenmodes, we have added to the DWF path integral a multiplicative weighting factor consisting of a ratio of determinants of Wilson-Dirac fermions having a chirally twisted mass with a large negative real component and a small imaginary chiral component. Numerical results show that this weighting factor with an appropriate choice of twisted masses significantly suppresses the residual mass while allowing adequate topological tunneling.

Motivation & Objective

• To reduce the residual mass $m_{\rm res}$ in domain wall fermion simulations, which increases rapidly at strong gauge couplings due to near-zero eigenmodes of the 4D transfer matrix $T$.
• To maintain sufficient topological tunneling during Monte Carlo simulations, which requires access to gauge configurations with near-zero Dirac eigenvalues.
• To develop a computationally feasible weighting factor that suppresses near-zero eigenmodes of the Hermitian Wilson-Dirac operator as a proxy for suppressing those of $H_4 = -\log(T)$.
• To enable larger physical lattice volumes (e.g., $L \approx 4.5$ fm) with controlled $m_{\rm res} \approx 0.002$ at $1/a \approx 1.4$ GeV.

Proposed method

• Introduces a multiplicative weighting factor in the DWF path integral composed of a ratio of determinants of Wilson-Dirac fermions with chirally twisted masses: one with a large negative real mass and one with a small imaginary chiral component.
• Uses the squared determinant of the Hermitian Wilson-Dirac operator $\gamma_5 D_{\mathcal{W}}(-M)$ as a proxy to suppress near-zero eigenmodes of $H_4 = -\log(T)$, which drive $m_{\rm res}$.
• Employs molecular dynamics (MD) evolution with the weighting factor to generate a repulsive force away from gauge-field configurations hosting near-zero eigenmodes.
• Tunes the twisted mass parameters ($\epsilon_f$, $\epsilon_b$) to balance suppression of $m_{\rm res}$ and preservation of topological charge tunneling.
• Numerically evaluates $m_{\rm res}$ and topological charge $\langle |\nu_{\rm top}| \rangle$ across different parameter sets to assess effectiveness.
• Compares results at zero and non-zero temperature to confirm $m_{\rm res}$ independence from thermal effects and validates suppression across couplings.

Experimental results

Research questions

• RQ1 Can a ratio of determinants with chirally twisted masses effectively suppress the residual mass $m_{\rm res}$ in domain wall fermion simulations?
• RQ2 Does the proposed weighting factor preserve sufficient topological tunneling to ensure proper sampling of gauge-field topology?
• RQ3 What is the optimal range of twisted mass parameters for balancing $m_{\rm res}$ suppression and topological charge preservation?
• RQ4 How does the suppression of near-zero eigenmodes of the Hermitian Wilson-Dirac operator correlate with suppression of $m_{\rm res}$ and topological charge?
• RQ5 Can this method enable simulations with $m_{\rm res} \approx 0.002$ on $32^3 \times 64$ lattices at $1/a \approx 1.4$ GeV, corresponding to $L \approx 4.5$ fm?

Key findings

• The method suppresses $m_{\rm res}$ most effectively at the parameter set $\epsilon_f/\epsilon_b = 0.005/0.50$, reducing it significantly compared to simulations without any weighting factor.
• At $\epsilon_f/\epsilon_b = 0.005/0.50$, the topological charge $\langle |\nu_{\rm top}| \rangle$ is nearly eliminated, indicating strong suppression of very-close-to-zero eigenmodes critical for tunneling.
• The suppression of $m_{\rm res}$ is most effective at strong couplings, with relative suppression increasing below the QCD transition temperature and decreasing at the transition.
• Numerical results confirm that $m_{\rm res}$ is largely independent of temperature, as shown by consistent values at $\beta = 1.75$ on $16^4 \times 32$ and $16^3 \times 8 \times 32$ lattices.
• The eigenvalue range $|\lambda| \lesssim 0.1-0.5$ is identified as critical for $m_{\rm res}$ suppression, while $|\lambda| \lesssim 0.01-0.05$ governs topological charge suppression.
• The results suggest that $m_{\rm res} \approx 0.002$ on $32^3$ lattices at $1/a \approx 1.4$ GeV and $L \approx 4.5$ fm may be achievable, enabling larger physical volumes than previously possible.