Accurate determination of the translational correlation function of two-dimensional solids

TL;DR

This paper clarifies how to accurately determine the translational correlation function in 2D solids, resolving previous misconceptions about its decay behavior by properly identifying the symmetry axis.

Contribution

It introduces a method to correctly identify the symmetry axis of 2D solids, ensuring accurate measurement of the translational correlation function.

Findings

Proper symmetry axis identification reveals algebraic decay in the correlation function.

Incorrect axis determination leads to false exponential decay observations.

The method improves phase identification in 2D systems.

Abstract

The identification of the different phases of a two-dimensional (2d) system, which might be in solid, hexatic, or liquid, requires the accurate determination of the correlation function of the translational and of the bond-orientational order parameters. According to the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory, in the solid phase the translational correlation function decays algebraically, as a consequence of the Mermin-Wagner long-wavelength fluctuations. Recent results have however reported an exponential-like decay. By revisiting different definitions of the translational correlation function commonly used in the literature, here we clarify that the observed exponential-like decay in the solid phase results from an inaccurate determination of the symmetry axis of the solid; the expected power-law behaviour is recovered when the symmetry axis is properly identified. We show that, contrary to the common assumption, the symmetry axis of a 2d solid is not fixed by the direction of its global bond-orientational parameter, and introduce an approach allowing to determine the symmetry axis from a real space analysis of the sample.