The two-particle vertex function is crucial for the diagrammatic extensions beyond DMFT for the nonlocal fluctuation. However, estimating the two-particle quantities is still a challenging task. In this study, we propose a simplification of the local two-particle full vertex and, using the simplified full vertex, we develop two methods to take into account the nonlocal fluctuation. We apply these methods to several models and confirm that our methods can capture important behaviors such as the pseudo gap in the DMFT + nonlocal calculation. In addition, the numerical costs are largely reduced compared to the conventional methods.
The two-particle vertex function is crucial for the diagrammatic extensions beyond DMFT for the nonlocal fluctuation. However, estimating the two-particle quantities is still a challenging task. In this study, we propose a simplification of the local two-particle full vertex and, using the simplified full vertex, we develop two methods to take into account the nonlocal fluctuation. We apply these methods to several models and confirm that our methods can capture important behaviors such as the pseudo gap in the DMFT + nonlocal calculation. In addition, the numerical costs are largely reduced compared to the conventional methods.
Estimating the local two-particle vertex functions, which are crucial for capturing the spatial fluctuation of the effective field beyond the single-site dynamical mean field theory (DMFT), is still challenging. In our previous work [R. Mizuno et al., J. Phys. Soc. Jpn. 91, 034002 (2022)], we developed a computationally efficient method for estimating the local full vertex in DMFT, where we can obtain the local two-particle full vertex from the one-particle self-energy. In this study, we further enhance our method by refining its formulation to be more faithful to the diagrammatic structure of the full vertex. With this improvement, we can qualitatively reproduce the characteristic frequency structures of the full vertex obtained by the numerically exact methods. In particular, the improved version of the simplified full vertex captures a sharp value change in the cross structure.
We apply a recently introduced hybridization flow functional renormalization group scheme for Anderson-like impurity models as an impurity solver in a dynamical mean-field theory (DMFT) approach to lattice Hubbard models. We present how this scheme is capable of reproducing metallic and insulating solutions of the lattice model. Our setup also offers a numerically rather inexpensive method to calculate two-particle correlation functions. For the paramagnetic Hubbard model on the Bethe lattice in infinite dimensions we calculate the local two-particle vertex for the metallic and the insulating phase. Then we go to a two-site cluster DMFT scheme for the two-dimensional Hubbard model that includes short-range antiferromagnetic fluctuations and obtain the local and nonlocal two-particle vertex functions. We discuss the rich frequency structures of these vertices and compare with the vertex in the single-site solution.
In systems of indistinguishable particles, correlations between the quantum degrees of freedom can give rise to collective modes where particles behave coherently together over some correlation length and time, thereby forming quasiparticles. One example of such quasiparticle would be the π-ton [66], describing the binding of two particle-hole pairs via the exchange of momentum corresponding to the antiferromagnetic (AFM) wave vector. These quasiparticles would emerge in the vicinity of an AFM ordering instability [66, 119, 120] and nonlocal correlations are crucial to describe them. Especially, in systems out of equilibrium, nonlocality can give rise to prethermal states wherein the distribution function can be momentum-dependent [92, 119], and nonthermal exotic physics can be observed and manipulated [40, 68, 79, 84, 93, 132, 147, 157–159]. To be able to model and simulate systems out of equilibrium is a challenge. A few numerical methods such as nonequilibrium Dynamical Mean Field Theory (DMFT) [44] have been used extensively [5]. Nonequilibrium DMFT is designed to treat nonperturbatively the local time-dependent single-particle correlations. Since phenomena such as quantum criticality, high-Tc superconductivity or the creation of π-tons via light-matter coupling stem from nonlocal two-particle correlations, one needs to resort to diagrammatic extensions to DMFT [103] and other methods to effectively describe them, such as the post-processing DMFT treatment of vertex corrections, which will be introduced. The Two-Particle Self-Consistent approach (TPSC) and its variants figure among those methods that include nonlocal two-particle and single-particle correlations in a self-consistent way [121, 122]. TPSC and its variants have the ability to capture the effect of the growing spin fluctuations (paramagnons) in the self-energy when close to amagnetic phase transition leading to the breakdown of the Fermi-liquid quasiparticles at the Fermi level: this would mark the onset of a pseudo-gap [145, 152]. It fulfils the Mermin-Wagner theorem in two dimensions, the Pauli principle, and various two-particle local sum-rules [152]. In the works embodied in this thesis, numerical methods have been designed to access nonequilibrium regimes, by changing model parameters across time, allowing for nonlocal correlations to be properly included. With these new algorithms at hand, the π-tons were studied in the optical conductivity and magnetic response [119, 120] as well as spin and charge fluctuations in the 2D and 3D single-band Hubbard model [121, 122]. It is demonstrated by means of those methods that nonlocal two-particle vertex corrections can underpin prethermal states and could generate transient nonthermal states such as low-energy charge excitations associated with nonthermal negative temperatures. It is also shown that states lying close to the Fermi level control the thermalization timescale and that lattice hopping quenches changing the unit-cell dimensionality could allow one to play around with the charge and spin scattering channels.
We apply a recently introduced hybridization-flow functional renormalization group scheme for Anderson-like impurity models as impurity solver in a dynamical mean field theory (DMFT) approach to lattice Hubbard models. We present how this scheme is capable of reproducing metallic and insulating solutions of the lattice model. Our setup also offers a numerically rather inexpensive method to calculate two-particle correlation functions. For the paramagnetic Hubbard-model on the Bethe lattice in infinite dimensions we calculate the local two-particle-vertex for the metallic and the insulating phase. Then we go to a two-site cluster-DMFT-scheme for the two-dimensional Hubbard-model that includes short-range antiferromagnetic fluctuations and obtain the local and non-local two-particle-vertex-functions. We discuss the rich frequency structures of these vertices and compare with the vertex in the single-site solution.
Local n-particle vertex functions represent the crucial ingredient for diagrammatic extensions of dynamical mean field theory (DMFT). Hitherto their application has been restricted---with a few exceptions---to the n=2-particle vertex while higher-order vertices have been neglected. In this paper we derive a general analytical expression for the local n-particle (one-particle-reducible) vertex of the Falicov-Kimball model (FKM). We observe that the magnitude of such vertex functions itself strongly increases with the number of particles n. On the other hand, their effect on generic Feynman diagrams remains rather moderate due to the damping effect of the Green's functions present in such diagrams. Nevertheless, they yield important contributions to the self-energy corrections calculated in diagrammatic extensions of DMFT as we explicitly demonstrate using the example of dual-fermion calculations for the two-dimensional FKM at quarter filling of the stationary f electrons. Here corrections to the self-energy obtained from the three-particle vertex are indeed comparable in magnitude to corresponding corrections stemming from the two-particle vertex.
In this last chapter, very recent promising developments beyond GW+EDMFT by taking into account nonlocal vertex corrections beyond the RPA are described. The first development is the dual fermion and boson approach in which DMFT is used as a starting point for a perturbation expansion of the nonlocal self-energy. This approach is very general and in its lowest similar to the GW+EDMFT scheme. The second development is the TRILEX approach based on the extension of the GW+EDMFT scheme to include higher-order correlation or vertex functions in the self-consistency condition. Like the GW+EDMFT scheme, this approach maintains the basic concept of DMFT in that local projected quantities are made self-consistent. In this way, nonlocal spin fluctuations missing in the GW+EDMFT scheme are incorporated. The third development is the dynamical vertex approximation method, which can be considered as a generalization of DMFT assuming the locality of the two-particle irreducible vertex. This chapter closes with a summary.