Critical Properties of the Half-Filled Hubbard Model in Three Dimensions

Abstract

By means of the dynamical vertex approximation (DΓA) we include spatial correlations on all length scales beyond the dynamical mean field theory (DMFT) for the half-filled Hubbard model in three dimensions. The most relevant changes due to non-local fluctuations are: (i) a deviation from the mean-field critical behavior with the same critical exponents as for the three dimensional Heisenberg (anti)-ferromagnet and (ii) a sizable reduction of the Néel temperature (TN ) by ∼ 30% for the onset of antiferromagnetic order. Finally, we give a quantitative estimate of the deviation of the spectra between DΓA and DMFT in different regions of the phase-diagram.PACS numbers: 71.10. Fd, 71.27.+a Almost 50 years after the invention of the Hubbard model [1] and despite modern petaflop supercomputers, a precise analysis of the criticality of this most basic model for electronic correlations has not been achieved so far, at least not in three dimensions. Dynamical mean field theory (DMFT) [2-4] was a big step forward to calculate the three dimensional Hubbard model since the major contribution of electronic correlations, i.e, the local one, is well captured within this theory. Local correlations give rise to quasiparticle renormalization, the Mott-Hubbard transition, magnetism, and even more subtle issues such as kinks in purely electronic models [5]. However, nonlocal spatial correlations are also naturally generated by a purely local Hubbard interaction, and, as it is well known, they become of essential importance in the vicinity of second-order phase transitions. As these correlations are neglected in DMFT, this scheme provides only for a conventional mean-field (MF) description of the critical properties.To overcome this shortcoming cluster extensions to DMFT such as the dynamical cluster approximation (DCA) and cluster-DMFT have been proposed [6]. In these approaches spatial correlations beyond DMFT are taken into account, however only within the range of the cluster size; and due to computational limitations the actual size of d = 3-clusters is restricted to about 100 sites. Hence, short-range correlations are included by these approaches, whereas long-range ones are not (e.g. for spacings larger than 5 lattice sites in d = 3). Nonetheless, Kent et al. [7] were able to extrapolate the cluster size of so-called Betts clusters to infinity, albeit assuming from the beginning the critical exponents to be those of the Heisenberg model. This way they extrapolated the Néel temperature of the paramagneticto-antiferromagnetic phase transition which was found in agreement with earlier lattice quantum Monte Carlo (QMC) results by Staudt et al. [8].As an alternative to cluster extensions and, in particular, to include long-range correlations on an equal footing, more recently diagrammatic expansions of DMFT have been proposed: (i) the DMFT plus spin-fermion model [9], (ii) the dynamical vertex approximation (DΓA) [10][11][12] which approximates the fully irreducible n-particle vertex to be local [10] or that of a DCA cluster [12...

One-particle irreducible functional approach: A route to diagrammatic extensions of the dynamical mean-field theory

“…In Ref. 22 T N was found smaller in D A than the one estimated in DCA 34 or in lattice quantum Monte Carlo 35 at weak coupling, indicating a possible overestimation of the nonlocal correlation effects. As it was argued in Ref.…”

“…Examples are the pseudogap 12 in d = 2 or the critical exponents in d = 3 dimensions. 22 In Fig. 9 we present our results for one-shot calculations of the nonlocal corrections to the DMFT self-energy, axis.…”

“…For the sake of conciseness, we will mainly discuss the numerical results obtained with ladder calculations, since they are most frequently adopted in previous papers, 12,17,22,30 and the inclusion of ladder diagrams proved to be essential to correctly describe crucial features of the two-and threedimensional physics. Examples are the pseudogap 12 in d = 2 or the critical exponents in d = 3 dimensions.…”

“…While the numerical solution of the parquet equations has been recently achieved 21 for single-band two-dimensional models, the computational effort is still considerable. Hence, most of the D A results obtained hitherto 17,22 employed the ladder approximation, where, taking into account the most important channel(s) only, the assumption of locality is made for the two-particle vertices, which are irreducible in these channels.…”

Towards ab initio Calculations with the Dynamical Vertex Approximation

“…34,75,76) Short-and long-ranged correlations are treated on an equal footing. Among others, critical exponents could be calculated for the first time for the Hubbard model 77,78) as well as the quantum critical point reached upon doping. 79) In two dimensions, spin fluctuations suppress antiferromagnetic order and give rise to pseudogap physics.…”