The Ginsparg-Wilson relation and overlap fermions

TL;DR

Problem: lattice fermions face chiral-symmetry breaking and fermion doubling; the GW framework provides an exact lattice chiral symmetry without doublers. Approach: the overlap construction realizes GW on the lattice, derived from a five-dimensional domain-wall picture, yielding a Dirac operator $D = (r_0/a)[1 + V]$ with a unitary $V$ built from a kernel and ensuring a modified chiral transformation; the massive form is $D(m) = (1 - m/(2 r_0/a)) D + m$ and there is a shifted inverse with good chiral properties. Contributions: formal GW properties (circle spectrum, index theorem), connection to domain wall fermions, explicit overlap operator and practical numerical strategies (sign-function approximations, low-mode projection, and HMC treatment of topology changes). Significance: provides exact lattice chiral symmetry at nonzero lattice spacing, enabling precise chiral observables and guiding nonperturbative chiral gauge theory research, while remaining computationally demanding and influencing later formulations such as Möbius domain-wall fermions.

Abstract

I review the physics of lattice fermions obeying the Ginsparg-Wilson relation. I describe their relation to domain wall fermions. I give a description of methodology for performing numerical simulations with overlap fermions. This is a chapter contributed to the on-line book ``Lattice QCD at 50 years,'' (LQCD@50), edited by Tanmoy Bhattacharya, Maarten Golterman, Rajan Gupta, Laurent Lellouch, and Steve Sharpe.

Figures (2)

• Figure 1: An example of a tenth-order Zolotarev approximation to the step function, where the desired range is for $0.1<x<1$. We plot $\epsilon_{10}(x)-1$ to display the error in the approximation.
• Figure 2: A comparison of the spectrum of eigenvalues of the ordinary Wilson fermion action (crosses) on a finite lattice, with the corresponding spectrum of an $r_0/a=1$ overlap action (the line).