A strange property of lattices with an even number of sites

TL;DR

This paper investigates a peculiar property of even-site lattices related to the self-consistency of derivative operators, revealing an unexpected structural requirement not present in odd-site lattices, with implications still unclear.

Contribution

It uncovers a unique self-consistency requirement for even-site lattices' derivative operators, highlighting a novel structural aspect not previously understood.

Findings

Lattices with even sites have a special self-consistency condition.

Odd-site lattices do not require this extra structure.

Implications of this property are still being explored.

Abstract

By examining the behaviour of the "SLAC" lattice derivative operators, it is found that lattices with an even number of sites have a somewhat strange self-consistency requirement for extra structure in the spatial derivative operator, which is not needed by lattices having an odd number of sites, and which is not at all obvious from a first-principles derivation. The general implications of this extra required structure are not, as yet, completely clear.