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Register a new userNon-perturbative landscape of the Mott-Hubbard transition: Multiple divergence lines around the critical endpoint
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We analyze the highly non-perturbative regime surrounding the Mott-Hubbard metal-to-insulator transition (MIT) by means of dynamical mean field theory calculations at the two-particle level. By extending the results of Schafer, et al. [Phys. Rev. Lett. 110, 246405 (2013)] we show the existence of infinitely many lines in the phase diagram of the Hubbard model where the local Bethe-Salpeter equations, and the related irreducible vertex functions, become singular in the charge as well as the particle-particle channel. These divergence lines accumulate around the critical Mott endpoint in accordance with the interpretation as precursors of the MIT. By comparing our numerical data with analytical calculations of increasing complexity, such as for the disordered Binary Mixture and Falicov-Kimball (FK) models, as well as for the atomic limit (AL) case, (i) we identify two different kinds of divergences lines; (ii) we classify them in terms of the frequency-structure of the associated singular eigenvectors; (iii) we investigate their relation to the multiple branches in the Luttinger-Ward formalism. Moreover, we could distinguish the situations where the multiple divergences simply reflect the emergence of an underlying, unique energy scale $ u^*$ below which perturbation theory does no longer apply, from those where the breakdown of perturbation theory affects, not trivially, different energy regimes. Finally, we discuss the implications of our results on the theoretical understanding of the non-perturbative physics around the MIT and for future developments of many-body algorithms applicable in this regime.
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We apply the density-matrix renormalization group (DMRG) method to a one-dimensional Hubbard model that lacks Umklapp scattering and thus provides an ideal case to study the Mott-Hubbard transition analytically and numerically. The model has a linear dispersion and displays a metal-to-insulator transition when the Hubbard interaction~$U$ equals the band width, $U_{rm c}=W$, where the single-particle gap opens linearly, $Delta(Ugeq W)=U-W$. The simple nature of the elementary excitations permits to determine numerically with high accuracy the critical interaction strength and the gap function in the thermodynamic limit. The jump discontinuity of the momentum distribution $n_k$ at the Fermi wave number $k_{rm F}=0$ cannot be used to locate accurately $U_{rm c}$ from finite-size systems. However, the slope of $n_k$ at the band edges, $k_{rm B}=pm pi$, reveals the formation of a single-particle bound state which can be used to determine $U_{rm c}$ reliably from $n_k$ using accurate finite-size data.
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