Mapping Fermion and Boson systems onto the Fock space of harmonic oscillators

TL;DR

This paper demonstrates that any linear response fermionic or bosonic system at finite temperature can be explicitly mapped onto a system of free harmonic oscillators, supporting their use in condensed matter physics.

Contribution

It provides the first explicit mapping of generic fermionic and bosonic systems onto harmonic oscillators, enhancing theoretical understanding and interpretation of the fluctuation-dissipation theorem.

Findings

Established explicit mapping for bosonic systems.

Extended the mapping to fermionic systems.

Clarified the origin of the Bose-Einstein factor in FDT.

Abstract

The fluctuation-dissipation theorem (FDT) is very general and applies to a broad variety of different physical phenomena in condensed matter physics. With the help of the FDT and following the famous work of Caldeira and Leggett, we show that, whenever linear response theory applies, any generic bosonic or fermionic system at finite temperature $T$ can be mapped onto a fictitious system of free harmonic oscillators. To the best of our knowledge, this is the first time that such a mapping is explicitly worked out. This finding provides further theoretical support to the phenomenological harmonic oscillator models commonly used in condensed matter. Moreover, our result helps in clarifying an interpretation issue related to the presence and physical origin of the Bose-Einstein factor in the FDT.