Quantum versus thermal fluctuations in the harmonic chain and experimental implications

TL;DR

This paper compares quantum and thermal fluctuations in harmonic chains, showing how quantum effects lead to slower, logarithmic growth of fluctuations and influence observable properties like x-ray scattering patterns.

Contribution

It elucidates the distinct behaviors of quantum versus thermal fluctuations in harmonic chains and discusses their experimental implications.

Findings

Quantum fluctuations grow logarithmically with chain length.

Thermal fluctuations increase linearly with chain length.

Absence of sharp Bragg peaks at zero temperature due to quantum effects.

Abstract

The nonzero ground-state energy of the quantum mechanical harmonic oscillator implies quantum fluctuations around the minimum of the potential with the mean square value proportional to Planck's constant. In classical mechanics thermal fluctuations occur when the oscillator is coupled to a heat bath of temperature $T$. At finite temperature quantum statistical mechanics allows the description of the transition from pure quantum fluctuations at $T=0$ to classical thermal fluctuations in the high temperature limit. It was early pointed out by Peierls that the mean square thermal fluctuations in a {\it harmonic chain} increase {\it linearly} with the distance of the atoms in the chain, destroying long range crystalline order. The corresponding pure quantum fluctuations lead to a much slower {\it logarithmic} increase with the distance from the fixed end of the chain. It is also shown that this implies, for example, the absence of sharp Bragg peaks in x-ray scattering in an infinite chain at zero temperature, which instead show power law behaviour typical for one dimensional quantum liquids (called {\it Luttinger liquids}).