Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Thermodynamic limit for the magnetic uniform electron gas and representability of density-current pairs

Mihály A. Csirik, Andre Laestadius, Erik I. Tellgren

Abstract

Although the concept of the uniform electron gas is essential to quantum physics, it has only been defined recently in a rigorous manner by Lewin, Lieb and Seiringer. We extend their approach to include the magnetic case, by which we mean that the vorticity of the gas is also held constant. Our definition involves the grand-canonical version of the universal functional introduced by Vignale and Rasolt in the context of current-density-functional theory. Besides establishing the existence of the thermodynamic limit, we derive an estimate on the kinetic energy functional that also gives a convenient answer to the (mixed) current-density representability problem.

Thermodynamic limit for the magnetic uniform electron gas and representability of density-current pairs

Abstract

Although the concept of the uniform electron gas is essential to quantum physics, it has only been defined recently in a rigorous manner by Lewin, Lieb and Seiringer. We extend their approach to include the magnetic case, by which we mean that the vorticity of the gas is also held constant. Our definition involves the grand-canonical version of the universal functional introduced by Vignale and Rasolt in the context of current-density-functional theory. Besides establishing the existence of the thermodynamic limit, we derive an estimate on the kinetic energy functional that also gives a convenient answer to the (mixed) current-density representability problem.
Paper Structure (19 sections, 21 theorems, 150 equations, 1 figure)

This paper contains 19 sections, 21 theorems, 150 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting
  3. States
  4. One-particle quantities
  5. Gauge transformation
  6. Representability
  7. Definition of the grand-canonical current-density functionals
  8. Main results
  9. The magnetic uniform electron gas
  10. Upper bound on the kinetic energy functional and representability
  11. Proofs
  12. Proof of \ref{['repthm']}
  13. Further bounds
  14. Transformation properties
  15. Decoupling lower bound
  16. ...and 4 more sections

Key Result

Proposition 2.1

Suppose that $\gamma$ is fermionic one-particle density matrix composed of sufficiently smooth orbitals. Then the following hold true.

Figures (1)

  • Figure 1: The tetrahedra corresponding to $\mathbf{R}=\mathbb{1}$ and ${\bm{\tau}}=0$ included in the sets $J_0$ and $J\smallsetminus J_0$ are depicted in blue and red, respectively. The tiling is composed of translates of $\ell\mathbb{\Delta}_j$.

Theorems & Definitions (31)

  • Proposition 2.1
  • Remark 1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 2.6: Correspondence with the nonmagnetic case
  • proof
  • Theorem 3.1: Uniform electron gas
  • Remark 2
  • ...and 21 more