Strength of the double-phonon state within an exactly solvable model

Abstract

The deviation of the energy-weight sum rule (EWSR) and the energy shift for the two-phonon state from the prediction of the independent-phonon picture (the harmonic limit) are studied within the exactly solvable Lipkin-Meskov-Glick model. The exact results are used to compare with the estimations given within the random-phase approximation (RPA) and the renormalized RPA (RRPA). The analysis of the numerical results shows that the source of the ``enhancement'' of the two-phonon ESWR compared to the value given by the harmonic limit is the violation of the condition $[\mathrm{D\ifmmode \hat{}\else \^{}\fi{}},[\mathcal{V},\mathrm{D\ifmmode \hat{}\else \^{}\fi{}}]]=0$ for the interaction part $\mathcal{V}$ of the model Hamiltonian and the operator $\mathrm{D\ifmmode \hat{}\else \^{}\fi{}}$ generating the electromagnetic transition. As a result, the EWSR for the two-phonon excitation exceeds its value in the harmonic limit by a factor of $\ensuremath{\sim}1.8$ at $N\ensuremath{\simeq}136$ and $\ensuremath{\chi}=0.8.$ It is also shown that the energy shift of the two-phonon energy compared to its value in the harmonic limit decreases with increasing the particle number N following a power law, which is more complicated than the simple approximation $\ensuremath{\sim}{N}^{\ensuremath{-}x}.$ The RPA and RRPA underestimate the exact EWSR of the two-phonon excitation by about $30%$ at a given interaction in the region where the RPA is valid.