This study explores a dark matter model in which a pseudo-Nambu-Goldstone boson arises as a viable dark matter candidate from the spontaneous and soft breaking of global \(U(1)\,\)symmetries and stabilized by a residual \(\mathbbZ_3\,\)discrete symmetry. The model introduces three complex scalar fields, singlets under the Standard Model gauge group, and charged under a dark \(U(1)_V\,\)gauge symmetry together with a permutative exchange symmetry among three scalars. These features naturally suppress the dark matter–nucleon scattering cross section by its Nambu-Goldstone boson nature. In addition to conventional annihilation channels, the \(\mathbbZ_3\,\)structure allows semi-annihilation processes, potentially leading to new phenomenological signatures. We analyze theoretical and experimental constraints, including relic abundance, Higgs invisible decays, and perturbative unitarity, and evaluate the elastic scattering cross section for boosted dark matter.

All possible theories of quantum gravity suggest the existence of a minimal length. As a consequence, the usual Heisenberg uncertainty principle (HUP) is replaced by a more general uncertainty principle known as the generalized uncertainty principle (GUP). The dynamics of all quantum mechanical system gets modified due to GUP. In this work, we consider both Einstein’s and Debye’s models to find the quantum gravity effect on the specific heat of solids. GUP modified specific heat in Einstein’s model shows usual exponential dominance at low temperatures. Further, the modification to Debye’s specific heat is calculated by considering the GUP modified dispersion relation, which becomes time dependent for elastic waves.