A new proposal for the fermion doubling problem II. Improving the operators for finite lattices

In a previous paper I showed how the ideal SLAC derivative and second-derivative operators for an infinite lattice can be obtained in simple closed form in position space, and implemented very efficiently in a stochastic fashion for practical calculations on finite lattices. In this second paper I show how the small (order 1/N) errors introduced by truncating the operators to a finite lattice may be removed by a small adjustment of coefficients, without incurring any additional computational cost. The derivation of these results is again presented in a simple, pedagogical fashion. 1. Truncating the SLAC derivative operators to fit on a finite lattice In a previous paper [1] we looked at the ideal “SLAC” specification of Drell, Weinstein and Yankielowicz [2] for the spatial derivative operator on an infinite one-dimensional lattice, which demands that −i times the Fourier transform of the derivative operator take on its ideal functional form, pideal = p, within the first Brillouin zone. Such an operator avoids the pathologies of the fermion doubling problem or the violation of chiral invariance that have plagued other sub-optimal definitions of the derivative operator, but has a representation in position space that is very “nonlocal”, ∆idealf(x) = 1 a { . . .− 1 4 f(x+ 4a) + 1 3 f(x+ 3a)− 1 2 f(x+ 2a) + f(x+ a)− f(x− a) + 1 2 f(x− 2a)− 1 3 f(x− 3a) + 1 4 f(x− 4a)− . . . }