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Density-Functional Theory for the Dicke Hamiltonian.

Vebjørn H Bakkestuen1, Mihály A Csirik1,2, Andre Laestadius1,2

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|April 24, 2025
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This study details density-functional theory for quantum systems, proving a Hohenberg-Kohn theorem and introducing new functionals. Findings reveal low-lying eigenstates as optimizers, enabling an adiabatic-connection formula.

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Area of Science:

  • Quantum mechanics
  • Condensed matter physics
  • Quantum optics

Background:

  • Density-functional theory (DFT) is a powerful quantum mechanical method.
  • Applying DFT to quantum-electrodynamical (QED) systems presents unique challenges.
  • Existing DFT methods may not fully capture the complexities of QED models.

Purpose of the Study:

  • To develop and analyze density-functional theory for quantum-electrodynamical model systems.
  • To extend the Hohenberg-Kohn theorems to these QED systems.
  • To investigate the properties of constrained-search functionals for pure states and ensembles.

Main Methods:

  • Detailed analysis of density-functional theory.
  • Consideration of the quantum Rabi model, Dicke model, and multi-mode generalizations.
  • Proof of a Hohenberg-Kohn theorem with internal variables (magnetization and displacement).
  • Introduction and analysis of constrained-search functionals for pure states and ensembles.

Main Results:

  • A generalized Hohenberg-Kohn theorem is proven for QED model systems.
  • Magnetization and displacement are identified as internal variables.
  • Optimizers for the pure-state constrained-search functional are found to be low-lying eigenstates.
  • An adiabatic-connection formula is formulated based on these properties.
  • Differentiability of the universal density functional is shown for the Rabi model, implying unique pure-state v-representability.

Conclusions:

  • The study provides a rigorous DFT framework for QED model systems.
  • The findings extend the applicability of DFT to a broader range of quantum phenomena.
  • The developed methods and theorems pave the way for future investigations in quantum many-body physics and quantum optics.