# Virial theorem, boundary conditions, and pressure for massless Dirac electrons

## TL;DR

This paper derives and analyzes the virial and Hellmann--Feynman theorems for massless Dirac electrons, revealing how boundary conditions affect the Hamiltonian's Hermiticity and the resulting pressure components in solid-state systems.

## Contribution

It introduces generalized theorems for massless Dirac electrons, accounting for boundary effects and boundary-induced pressures, with applications to graphene and comparison to massive electrons.

## Key findings

- Boundary conditions break Hamiltonian Hermiticity, leading to boundary integral terms.
- Electron reflections contribute to the kinetic pressure component.
- Distinct anomalous pressure arises in solid-state Dirac electron systems.

## Abstract

The virial and the Hellmann--Feynman theorems for massless Dirac electrons in a solid are derived and analyzed using generalized continuity equations and scaling transformations. Boundary conditions imposed on the wave function in a finite sample are shown to break the Hermiticity of the Hamiltonian resulting in additional terms in the theorems in the forms of boundary integrals. The thermodynamic pressure of the electron gas is shown to be composed of the kinetic pressure, which is related to the boundary integral in the virial theorem and arises due to electron reflections from the boundary, and the anomalous pressure, which is specific for electrons in solids. Connections between the kinetic pressure and the properties of the wave function on the boundary are drawn. The general theorems are illustrated by examples of uniform electron gas, and electrons in rectangular and circular graphene samples. The analogous consideration for ordinary massive electrons is presented for comparison.