Isotope effects in the Hubbard-Holstein model within dynamical   mean-field theory
  We study the isotope effects arising from the coupling of correlated electrons with dispersionless phonons by considering the Hubbard-Holstein model at half-filling within the dynamical mean-field theory. In particular we calculate the isotope effects on the quasi-particle spectral weight $Z$, the renormalized phonon frequency, and the static charge and spin susceptibilities. In the weakly correlated regime $U/t \lesssim 1.5$, where $U$ is the Hubbard repulsion and $t$ is the bare electron half-bandwidth, the physical properties are qualitatively similar to those characterizing the Holstein model in the absence of Coulomb repulsion, where the bipolaronic binding takes place at large electron-phonon coupling, and it reflects in divergent isotope responses. On the contrary in the strongly correlated regime $U/t \gtrsim 1.5$, where the bipolaronic metal-insulator transition becomes of first order, the isotope effects are bounded, suggesting that the first order transition is likely driven by an electronic mechanism, rather then by a lattice instability. These results point out how the isotope responses are extremely sensitive to phase boundaries and they may be used to characterize the competition between the electron-phonon coupling and the Hubbard repulsion. 
P. Paci, M. Capone, E. Cappelluti, S. Ciuchi, C. Grimaldi