Noninteracting Fermions in infinite dimensions

TL;DR

This paper explores the behavior of noninteracting fermions as the spatial dimensionality approaches infinity, revealing that all fermions move at Fermi momentum, a result not commonly discussed in standard quantum statistics texts.

Contribution

It provides a detailed discussion and explanation of fermion behavior in infinite dimensions, highlighting a pedagogically interesting limit not covered in typical textbooks.

Findings

In infinite dimensions, the average energy per fermion reaches 100% of the Fermi energy.

The study shows all fermions move with Fermi momentum in infinite dimensions.

The paper offers educational insights into high-dimensional fermion systems.

Abstract

Usually, we study the statistical behaviours of noninteracting Fermions in finite (mainly two and three) dimensions. For a fixed number of fermions, the average energy per fermion is calculated in two and in three dimensions and it becomes equal to 50 and 60 per cent of the fermi energy respectively. However, in the higher dimensions this percentage increases as the dimensionality increases and in infinite dimensions it becomes 100 per cent. This is an intersting result, at least pedagogically. Which implies all fermions are moving with Fermi momentum. This result is not yet discussed in standard text books of quantum statistics. In this paper, this fact is discussed and explained. I hope, this article will be helpful for graduate students to study the behaviours of free fermions in generalised dimensionality.