Investigation of the highly non-perturbative nature of the Mott-Hubbard metal insulator transition: Vertex divergences in the coexistence region

Abstract

Recent developments have shown the emergence of divergences in the two-particle vertex function of numerous correlated electron systems. These vertex divergences are found along lines in the phase diagram of the respective systems. In spite of an initial interpretation of the divergences as a precursor of the Mott-Hubbard metal-insulator transition (MIT), they have been observed also in models where no MIT is present, such as the Anderson impurity model (AIM). At the same time, in the case of the Hubbard model, vertex divergences were hitherto analyzed in less challenging parameter regimes, i.e. never in the close proximity of the MIT itself. In order to fill this gap, this work focuses on the calculation of the vertex divergences of the charge channel in the coexistence region of the Mott MIT in the Hubbard model. Specifically, we study these divergences by computing the local two-particle Green’s function by means of dynamical mean field theory (DMFT). From these calculations, we extract the generalized susceptibility of the charge channel and we determine the number of negative eigenvalues, which correspond to the number of crossed divergences-lines for a specific temperature T and interaction strength U, starting from U = 0. Our results for the metallic phase in the coexistence region show a strong bending of divergency-lines towards the quantum critical "endpoint" of the MIT at T = 0, while the shape of the MIT itself is reflected, to a certain extent, in the behavior of the divergence-lines. Interestingly, the quantum critical endpoint of the MIT appears to be an accumulation point for the divergence-lines, due to rapid increase of their number while approaching the right border of the coexistence region (Uc2) of the MIT. We find also that in the insulating coexistence region the behavior of the divergence-lines qualitatively resembles the linear behavior of the atomic limit. In fact, the number of vertex divergences shows a jump at the first-order phase transition of the system, whereas at both second-order critical endpoints of the MIT a continuous behavior is restored. The number of vertex divergences along Uc2(T) as well as Uc(T), which marks the thermodynamic phase transition computed by Blümer et al., is observed to grow linearly along Uc2(T) and logarithmically along Uc(T) as a function of the inverse temperature β, eventually diverging for β → ∞ (T = 0). By assuming these trends we are able to predict the precise number of vertex divergences along Uc2(T) and Uc(T), which have been found in accordance with additional test calculations at lower temperatures. On the basis of these results, we can identify the precise relation between vertex divergences and the Mott-Hubbard MIT: The point where the MIT occurs at T = 0 corresponds to an accumulation of an infinite number of divergence lines. This highlights the highly non-perturbative nature of the Mott metal-insulator transition described by DMFT.