Cornwall-Jackiw-Tomboulis effective field theory and the nonuniversal equation  of state of an ultracold Bose gas

The equation of state (EOS) serves as a cornerstone in elucidating the properties of quantum many-body systems. A recent highlight along this research line consists of the derivation of the nonuniversal Lee-Huang-Yang (LHY) EOS for an ultracold quantum bosonic gas with finite-range interatomic interactions using one-loop effective path-integral field theory [Phys. Rev. Lett. 118, 130402 (2017)]. The purpose of this work is to extend Salasnich's pioneering work to uncover beyond-LHY corrections to the EOS by employing the Cornwall-Jackiw-Tomboulis (CJT) effective-field theory, leveraging its two-loop approximation. To this end, we expand Salasnich's remarkable findings of EOS to the next leading order characterized by ${(\ensuremath{\rho}{a}_{\text{s}}^{3})}^{2}$, with $\ensuremath{\rho}$ and ${a}_{\text{s}}$ being the density and the $s$-wave scattering length. Notably, we derive analytical expressions for quantum depletion and chemical potential, representing the next-to-LHY corrections to nonuniversal EOS induced by finite-range effects. Moreover, we propose an experimental protocol of observing the nonuniversal next-to-LHY corrections to the EOS by calculating fractional frequency shifts in the breathing modes. The nonuniversal beyond-LHY EOS in this work paves the way of using LHY effects in quantum simulation experiments and for investigations beyond the LHY regime.