The effect of long range interactions on the stability of classical and quantum solids

TL;DR

This paper extends Peierls' argument to analyze how long-range interactions affect the stability of one-dimensional classical and quantum solids, revealing a critical decay exponent for maintaining crystalline order.

Contribution

It generalizes Peierls' argument to include long-range interactions and quantum fluctuations, providing new insights into the stability criteria of 1D solids.

Findings

For $ ext{α} < 2$, long-range interactions stabilize crystalline order at finite temperature.

For $ ext{α} extgreater 2$, crystalline order vanishes at any finite temperature.

Quantum fluctuations influence the melting behavior at zero temperature.

Abstract

We generalise the celebrated Peierls' argument to study the stability of a long-range interacting classical solid. Long-range interaction implies that all the atomic oscillators are coupled to each other via a harmonic potential, though the coupling strength decays as a power-law $1/x^{\alpha}$, where $x$ is the distance between the oscillators. We show that for the range parameter $\alpha <2$, the long-range interaction dominates and the one-dimensional system retains a crystalline order even at a finite temperature whereas for $\alpha \geq2$, the long-range crystalline order vanishes even at an infinitesimally small temperature. We also study the effect of quantum fluctuations on the melting behaviour of a one-dimensional solid at T=0, extending Peierls' arguments to the case of quantum oscillators.