Electrostatic nature of cavity-mediated interactions between low-energy matter excitations

The use of cavity quantum electrodynamical effects, i.e., of vacuum electromagnetic fields, to modify material properties in cavities has rapidly gained popularity and interest in the last few years. However, there is still a scarcity of general results that provide guidelines for intuitive understanding and limitations of what kind of effects can be achieved. We provide such a result for the effective interactions between low-energy matter excitations induced either directly by their mutual coupling to the cavity electromagnetic (EM) field or indirectly through coupling to mediator modes that couple to the EM field. We demonstrate that the induced interactions are purely electrostatic in nature and are thus fully described by the EM Green's function evaluated at zero frequency. Our findings imply that reduced models with one or a few cavity modes can easily give misleading results.