Non-Perturbative Many-Body Approach to the Hubbard Model and Single-Particle Pseudogap

Abstract

obtained by numerically solving self-consistent integral equations. Detailed comparisons with experimental results on transition-metal oxides have shown that three-dimensional materials can be well described by the infinite-dimensional self-consistent mean-field approach. [11] Other methods, such as slave-boson [12] or slave-fermion [13] approaches, have also allowed one to gain insights into the Hubbard model through various mean-field theories corrected for fluctuations. In this context however, the mean-field theories are not based on a variational principle. Instead, they are generally based on expansions in the inverse of a degeneracy parameter, [14] such as the number of fermion flavors N , where N is taken to be large despite the fact that the physical limit corresponds to a small value of this parameter, say N = 2. Hence these theories must be used in conjunction with other approaches to estimate their limits of validity. [15] Expansions around solvable limits have also been explored. [16] Finally, numerical solutions, [17] with proper account of finite-size effects, can often provide a way to test the range of validity of approximation methods independently of experiments on materials that are generally described by much more complicated Hamiltonians.Despite all this progress, we are still lacking reliable theoretical methods that work in arbitrary space dimension. In two dimensions in particular, it is believed that the Hubbard model may hold the key to understanding normal state properties of high-temperature superconductors. But even the simpler goal of understanding the magnetic phase diagram of the Hubbard model in two dimensions is a challenge. Traditional mean-field techniques, or even slaveboson mean-field approaches, for studying magnetic instabilities of interacting electrons fail in two dimensions. The Random Phase Approximation (RPA) for example does not satisfy the Pauli principle, and furthermore it predicts finite temperature antiferromagnetic or spin density wave (SDW) transitions while this is forbidden by the Mermin-Wagner theorem. Even though one can study universal critical behavior using various forms of renormalization group treatments [18] [19] [20] [21] [22] or through the self-consistent-renormalized approach of Moriya [23] which all satisfy the Mermin-Wagner theorem in two dimensions, cutoff-dependent scales are left undetermined by these approaches. This means that the range of interactions or fillings for which a given type of ground-state magnetic order may appear is left undetermined.Amongst the recently developed theoretical methods for understanding both collective and single-particle properties of the Hubbard model, one should note the fluctuation exchange approximation [24] (FLEX) and the pseudo-potential parquet approach.[25] The first one, FLEX, is based on the idea of conserving approximations proposed by Baym and Kadanoff. [26] [27] This approach starts with a set of skeleton diagrams for the Luttinger-Ward functional [28] to generate a self-energy that is compute...

Migdal’s theorem and the pseudogap

“…We find that as the antiferromagnetic correlations become strong enough for vertex corrections to produce a qualitative change relative to the one-loop approximation, a pseudogap opens in the quasiparticle spectrum. In this respect, our results are similar to those obtained for the Hubbard model [17][18][19]. The pseudogap we find is strongly anisotropic in that it vanishes along the diagonal of the Brillouin zone and is large near the zone boundary.…”

“…The validity of Baym-Kadanoff many-body theories such as the fluctuation exchange approximation [16], which is in many ways similar to the Eliashberg theory of the magnetic interaction model, has been extensively studied by Tremblay and collaborators in the context of the Hubbard model [17][18][19]. They find that close to the magnetic boundary, Migdal's theorem qualitatively breaks down, in that a critical-fluctuation-induced pseudogap (or precursor pseudogap) is observed in the Quantum Monte Carlo simulations but is not found in the fluctuation exchange approximation.…”

Analytic impurity solver with Kondo strong-coupling asymptotics

“…This conclusion is not new and is also understandable. 11,16 Renormalizations due to the one-particle self-consistency unrealistically smear and unfold the low-frequency structure of the self-energy. Consequently, the high-frequency features of the one-particle propagator are washed out and no Hubbard satellite bands can be observed.…”

Towards ab initio Calculations with the Dynamical Vertex Approximation

“…Previous works resorting to this philosophy in the context of many-body models include Refs. 66,67,69,72,74,95,96) Here, we are going to describe AbinitioDΓA, 86) the dynamical vertex approximation for realistic electronic structure calculations. Before that, we discuss however recent advances for calculating the local F and Γ by CT-QMC.…”