Many monolayer transition metal dichalcogenides, including ${\mathrm{MoS}}_2$, ${\mathrm{MoSe}}_2$, ${\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$, are direct band gap two-dimensional (2D) semiconductors with sharp optical resonances at excitonic bound state frequencies. Recent adjacent-layer sensing experiments have demonstrated that excitonic resonance frequencies in multilayer van der Waals stacks are altered by long-range Coulomb interactions with electrons in nearby, but electrically isolated, 2D materials and have been used to detect transitions between distinct states of remote strongly correlated 2D electron fluids. In this Letter, we provide a long-wavelength theory of these frequency shifts that expresses the semiconductor band-gap renormalization as a weighted integral of the full interacting dynamic charge susceptibility of the remote layer $\chi (\mathit{q},\omega )$ separating a static exchange contribution from a retardation correction. We apply it to ${\mathrm{WSe}}_2$ proximate to rhombohedral trilayer graphene and to Bernal bilayer graphene in a finite magnetic field, and find that the retardation correction, while modest for each system individually, can be a sizable fraction of the differential signal when static contributions partially cancel, as for even–odd Landau-level filling in Bernal bilayer graphene.

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• Band gap
• Excitons
• 2-dimensional systems
• Graphene
• Strongly correlated systems
• Transition metal dichalcogenides
• Two-dimensional electron system
• Many-body techniques
• Optical spectroscopy