Density-functional theory for the Dicke Hamiltonian

TL;DR

This work develops a rigorous density-functional framework for quantum electrodynamical Dicke-type models, establishing a Hohenberg–Kohn mapping that uses magnetization $\boldsymbol{\sigma}$ and displacement $\boldsymbol{\xi}$ as internal densities to external potentials $(\mathbf{v},\mathbf{j})$. It introduces pure-state Levy–Lieb and mixed-state Lieb functionals for the multi-mode Dicke Hamiltonian, proves optimizer existence, and derives an adiabatic-connection formula capturing the coupling dependence via $G^{\boldsymbol{\Lambda}}(\boldsymbol{\sigma})$. The authors show convexity, differentiability on the regular set, and $N=M=1$ special-case results including unique pure-state $v$-representability and a bijection between densities and potentials. The framework clarifies $v$-representability and density-to-potential mappings in QEDFT for Dicke-type models, enabling first-principles studies of light–matter coupling effects on material properties. Overall, the paper extends Lieb-type DFT concepts to a tractable QED setting, revealing how density functionals, adiabatic connections, and representability behave in a multi-mode light-mmatter context.

Abstract

A detailed analysis of density-functional theory for quantum-electrodynamical model systems is provided. In particular, the quantum Rabi model, the Dicke model, and a generalization of the latter to multiple modes are considered. We prove a Hohenberg-Kohn theorem that manifests the magnetization and displacement as internal variables, along with several representability results. The constrained-search functionals for pure states and ensembles are introduced and analyzed. We find the optimizers for the pure-state constrained-search functional to be low-lying eigenstates of the Hamiltonian and, based on the properties of the optimizers, we formulate an adiabatic-connection formula. In the reduced case of the Rabi model we can even show differentiability of the universal density functional, which amounts to unique pure-state v-representability.

Figures (2)

• Figure 1: Left: The set $\mathcal{R}_2\subset(-1,1)^2$ is the union of 4 congruent open triangles. Right: The set $\mathcal{R}_3\subset(-1,1)^3$ is the union of 24 congruent open tetrahedra.
• Figure 2: The universal density functional $F_\mathrm{LL}(\sigma,0) = F_\mathrm{L}(\sigma,0)$ in the reduced setting ($N=M=1$) with $t=1$ and for different values of $\lambda$.

Theorems & Definitions (48)

• Theorem 2.1: Virial
• Example 1
• Example 2
• Example 3
• Example 4
• Proposition 3.1
• Theorem 3.2: Hohenberg--Kohn
• Theorem 3.3: $N$-representability
• Theorem 3.4: Existence of an optimizer for $F_\mathrm{LL}$
• Theorem 3.5: Properties of $F_\mathrm{LL}$