Lattice gauge action suppressing near-zero modes of H_W

TL;DR

The study introduces an action with two flavors of heavy Wilson fermions of large negative mass $m_0$ and twisted-mass pseudo-fermions to suppress near-zero modes of the hermitian Wilson-Dirac operator $H_W$, creating a spectral gap in $\rho(\lambda_W)$ near $\lambda_W=0$ and thereby ensuring exponential locality of the overlap-Dirac operator. This approach also fixes topology during simulations, enabling controlled exploration in the $\epsilon$-regime while significantly reducing the computational cost of applying the matrix sign function required by overlap fermions. Numerical results on $16^3\times32$ lattices show vanishing near-zero density for $\mu>0$, with the strongest suppression achieved using the RG action, and a manageable finite renormalization of $\beta$ that can be offset by appropriate twisted-mass terms. The work argues that fixed topology introduces finite-volume effects that diminish as volume grows but may affect quenched observables, and it outlines ongoing dynamical overlap simulations, the role of mobility edges, and the potential for robust, efficient chiral fermion simulations.

Abstract

We propose a lattice action including unphysical Wilson fermions with a negative mass m_0 of the order of the inverse lattice spacing. With this action, the exact zero mode of the hermitian Wilson-Dirac operator H_W(m_0) cannot appear and near-zero modes are strongly suppressed. By measuring the spectral density rho(lambda_W), we find a gap near lambda_W=0 on the configurations generated with the standard and improved gauge actions. This gap provides a necessary condition for the proof of the exponential locality of the overlap-Dirac operator by Hernandez, Jansen, and Luescher. Since the number of near-zero modes is small, the numerical cost to calculate the matrix sign function of H_W(m_0) is significantly reduced, and the simulation including dynamical overlap fermions becomes feasible. We also introduce a pair of twisted mass pseudo-fermions to cancel the unwanted higher mode effects of the Wilson fermions. The gauge coupling renormalization due to the additional fields is then minimized. The topological charge measured through the index of the overlap-Dirac operator is conserved during continuous evolutions of gauge field variables.

Figures (6)

• Figure 1: History of $\Delta H$ for the lattice "Pl" with $1/\epsilon=0$. The left panel shows our main run with $\delta\tau=0.01$, while the right panel is for a reduced step size: $\delta\tau=0.005$.
• Figure 2: Low-lying eigenvalues of $H_W(m_0)$. Lattice actions are "Pl" with $1/\epsilon=0$ (top), "Pl" with $1/\epsilon=2/3$ (middle), and "RG" (bottom). Data for three values of $\mu$ ($\mu$ = 0.0, 0.2, and 0.4) are shown in each plot. The highest mode is also shown, whose value is $\sim$ 5.9, almost independent of the gauge configuration.
• Figure 3: Histogram of the spectral density of $H_W(m_0)$. Lattice actions are "Pl" with $1/\epsilon=0$ (top), "Pl" with $1/\epsilon=2/3$ (middle), and "RG" (bottom). Data for three values of $\mu$ ($\mu$ = 0.0, 0.2, and 0.4) are shown in each plot.
• Figure 4: Pseudo-scalar mass squared calculated on the quenched configurations at fixed topological charges $|Q|$ = 0, 2, 3, and 6.
• Figure 5: $\Pi(p)/N_f$ for various mass parameters.