For the quantum Rabi model, the Levy-Lieb functional is almost fully explicit along the adiabatic connection, and all regular density pairs are uniquely v-representable.
Weinberg,The Quantum Theory of Fields (Cambridge University Press, 1995)
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Quantum-Electrodynamical Density-Functional Theory Exemplified by the Quantum Rabi Model
For the quantum Rabi model, the Levy-Lieb functional is almost fully explicit along the adiabatic connection, and all regular density pairs are uniquely v-representable.