Non-affine displacements in crystalline solids in the harmonic limit

TL;DR

This paper provides an exact statistical analysis of non-affine displacements in harmonic crystalline solids, revealing their coupling with affine deformations and identifying a tunable field that can induce a transition to a divergent non-affinity state.

Contribution

It introduces an exact calculation of the joint distribution and correlations of non-affine and affine displacements in low-temperature harmonic solids, highlighting their coupling and potential phase transition.

Findings

Non-affine and affine deformations are coupled in harmonic solids.

The correlation length of non-affinity can diverge under a tunable field.

The joint distribution of non-affinity and deformation is explicitly derived.

Abstract

A systematic coarse graining of microscopic atomic displacements generates a local elastic deformation tensor ${\mathsf D}$ as well as a positive definite scalar $\chi$ measuring non-affinity, i.e. the extent to which the displacements are not representable as affine deformations of a reference crystal. We perform an exact calculation of the statistics of $\chi$ and ${\mathsf D}$ and their spatial correlations for solids at low temperatures, within a harmonic approximation and in one and two dimensions. We obtain the joint distribution $P(\chi, {\mathsf D})$ and the two point spatial correlation functions for $\chi$ and ${\mathsf D}$. We show that non-affine and affine deformations are coupled even in a harmonic solid, with a strength that depends on the size of the coarse graining volume $\Omega$ and dimensionality. As a corollary to our work, we identify the field, $h_{\chi}$, conjugate to $\chi$ and show that this field may be tuned to produce a transition to a state where the ensemble average, $<\chi>$, and the correlation length of $\chi$ diverge. Our work should be useful as a template for understanding non-affine displacements in realistic systems with or without disorder and as a means for developing computational tools for studying the effects of non-affine displacements in melting, plastic flow and the glass transition.