The Virial Theorem in Graphene and other Dirac Materials

TL;DR

This paper applies the virial theorem to Dirac materials like graphene, deriving exact energy relations and showing that linear dispersion prevents Wigner crystallization, with implications for energy and compressibility calculations.

Contribution

It provides exact formulas for total energy, chemical potential, pressure, and compressibility in Dirac materials, highlighting the absence of Wigner crystallization under linear dispersion.

Findings

Total energy scales as 1/r_s with a dimension-independent constant

Linear dispersion prevents Wigner crystallization in these systems

Exact expressions for chemical potential, pressure, and compressibility are derived

Abstract

The virial theorem is applied to graphene and other Dirac Materials for systems close to the Dirac points where the dispersion relation is linear. From this, we find the exact form for the total energy given by $E = \mathcal{B}/r_s$ where $r_s a_0$ is the mean radius of the $d$-dimensional sphere containing one particle, with $a_0$ the Bohr radius, and $\mathcal{B}$ is a constant independent of $r_s$. This result implies that, given a linear dispersion and a Coulombic interaction, there is no Wigner crystalization and that calculating $\mathcal{B}$ or measuring at any value of $r_s$ determines the energy and compressibility for all $r_s$. In addition to the total energy we calculate the exact forms of the chemical potential, pressure and inverse compressibility in arbitrary dimension.