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Most theoretical studies for correlated light-matter systems are performedwithin the long-wavelength limit, i.e., the electromagnetic field is assumed tobe spatially uniform. In this limit the so-called length-gauge transformationfor a fully quantized light-matter system gives rise to a dipole self-energyterm in the Hamiltonian, i.e., a harmonic potential of the total dipole mattermoment. In practice this term is often discarded as it is assumed to besubsumed in the kinetic energy term. In this work we show the necessity of thedipole self-energy term. First and foremost, without it the light-matter systemin the long-wavelength limit does not have a ground-state, i.e., the combinedlight-matter system is unstable. Further, the mixing of matter and photondegrees of freedom due to the length-gauge transformation, which also changesthe representation of the translation operator for matter, gives rise to theMaxwell equations in matter and the omittance of the dipole self-energy leadsto a violation of these equations. Specifically we show that without the dipoleself-energy the so-called 'depolarization shift' is not properly described.Finally we show that this term also arises if we perform the semi-classicallimit after the length-gauge transformation. In contrast to the standardapproach where the semi-classical limit is performed before the length-gaugetransformation, the resulting Hamiltonian is bounded from below and thussupports ground-states. This is very important for practical calculations andfor density-functional variational implementations of the non-relativistic QEDformalism. For example, the existence of a combined light-matter ground-stateallows one to calculate the Stark shift non-perturbatively.

Light-matter interaction in the long-wavelength limit: no ground-state without dipole self-energy