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arXiv:1806.09645 · Physical review. B./Physical review. B

Self-energy dispersion in the Hubbard model

Thomas Mertz, Karim Zantout, Roser Valentí

Published Dec 3, 2018• 9 citations
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Abstract

We introduce the concept of self-energy dispersion as an error bound on local theories and apply it to the two-dimensional Hubbard model on the square lattice at half filling. Since the self-energy has no single-particle analog and is not directly measurable in experiments, its general behavior as a function of momentum is an open question. In this paper, we benchmark the momentum dependence with the two-particle self-consistent approach together with analytical and numerical considerations and we show that through the addition of a local single-particle potential to the Hubbard model the self-energy can be flattened, such that it is essentially described by only a frequency-dependent term. We use this observation to motivate that local theories, such as the dynamical mean-field theory, should be expected to give very accurate results in the presence of a potential of this kind. This is especially interesting in the context of topology, where such a term is present in many popular models, e.g., the Haldane and Kane-Mele models. Finally, we propose a simple energy argument as an estimator for the crossover from nonlocal to local self-energies, which can be computed even by local theories such as dynamical mean-field theory.

DOI
10.1103/physrevb.98.235105

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We introduce the concept of self-energy dispersion as an error bound on local theories and apply it to the two-dimensional Hubbard model on the square lattice at half filling. Since the self-energy has no single-particle analog and is not directly measurable in experiments, its general behavior as a function of momentum is an open question. In this paper, we benchmark the momentum dependence with the two-particle self-consistent approach together with analytical and numerical considerations and we show that through the addition of a local single-particle potential to the Hubbard model the self-energy can be flattened, such that it is essentially described by only a frequency-dependent term. We use this observation to motivate that local theories, such as the dynamical mean-field theory, should be expected to give very accurate results in the presence of a potential of this kind. This is especially interesting in the context of topology, where such a term is present in many popular models, e.g., the Haldane and Kane-Mele models. Finally, we propose a simple energy argument as an estimator for the crossover from nonlocal to local self-energies, which can be computed even by local theories such as dynamical mean-field theory.
We study the two-dimensional Hubbard model in the weak-coupling regime and compare the self-energy obtained from various approximate diagrammatic schemes to the result of diagrammatic Monte Carlo simulations, which sum up all weak-coupling diagrams up to a given order. While dynamical mean-field theory provides a good approximation for the local part of the self-energy, including its frequency dependence, the partial summation of bubble and/or ladder diagrams typically yields worse results than second-order perturbation theory. Even widely used self-consistent schemes such as GWGWGW or the fluctuation-exchange approximation (FLEX) are found to be unreliable. Combining the dynamical mean-field self-energy with the nonlocal component of GWGWGW in GW+DMFTGW + \mathrm{DMFT}GW+DMFT yields improved results for the local self-energy and nonlocal self-energies of the correct order of magnitude, but here, too, a more reliable scheme is obtained by restricting the nonlocal contribution to the second-order diagram. FLEX + DMFT is found to give accurate results in the low-density regime, but even worse results than FLEX near half-filling.
Nonlocal correlations play an essential role in correlated electron systems, especially in the vicinity of phase transitions and crossovers, where two-particle correlation functions display a distinct momentum dependence. In nonequilibrium settings, the effect of nonlocal correlations on dynamical phase transitions, prethermalization phenomena, and trapping in metastable states is not well understood. In this paper, we introduce a dynamical mean field theory (DMFT) extension to the nonequilibrium two-particle self-consistent (TPSC) approach, which allows to perform nonequilibrium simulations capturing short- and long-ranged nonlocal correlations in the weak-, intermediate-, and strong-correlation regimes. The method self-consistently computes local spin and charge vertices, from which a momentum-dependent self-energy is constructed. Replacing the local part of the self-energy by the DMFT result within this self-consistent scheme provides an improved description of local correlation effects. We explain the details of the formalism and the implementation, and demonstrate the versatility of DMFT+TPSC with interaction quenches and dimensional crossovers in the Hubbard model.
The two-dimensional Hubbard model on the square lattice is studied in the presence of lattice distortions in the adiabatic approximation. The self energy is computed within perturbation theory up to second order, which provides a way for studying the quasiparticle dispersion. We compute numerically the second order contribution to the self-energy using a standard Fast Fourier Transform Algorithm for finite system sizes. The stability of the lattice distortions is investigated and a schematic phase diagram is drawn. The Fermi surface is analyzed for densities near to half filling, the presence of lattice distortion change some spectral properties of the model and gives an anisotropic interacting FS. The spectral function is calculated along several lines in momentum space and the renormalized quasiparticle dispersion is obtained. The behavior of the density of states is shown for different values of the intrasite repulsion U in the different phases.
We compute the self-energy for the half-filled Hubbard model on a square lattice using lattice quantum Monte Carlo simulations and the dynamical vertex approximation. The self-energy is strongly momentum-dependent, but it can be parametrized via the noninteracting energy-momentum dispersion ${\ensuremath{\varepsilon}}_{\mathbf{k}}$, except for pseudogap features right at the Fermi edge. That is, it can be written as $\mathrm{\ensuremath{\Sigma}}({\ensuremath{\varepsilon}}_{\mathbf{k}},\ensuremath{\omega})$, with two energylike parameters $(\ensuremath{\varepsilon}, \ensuremath{\omega})$ instead of three (kx,ky{k}_{x}, {k}_{y}kx​,ky​, and $\ensuremath{\omega}$). The self-energy has two rather broad and weakly dispersing high-energy features and a sharp $\ensuremath{\omega}={\ensuremath{\varepsilon}}_{\mathbf{k}}$ feature at high temperatures, which turns to $\ensuremath{\omega}=\ensuremath{-}{\ensuremath{\varepsilon}}_{\mathbf{k}}$ at low temperatures. Altogether this yields a Z\mathcal{Z}Z- and reversed-Z\mathcal{Z}Z-like structure, respectively, for the imaginary part of $\mathrm{\ensuremath{\Sigma}}({\ensuremath{\varepsilon}}_{\mathbf{k}},\ensuremath{\omega})$. We attribute the change of the low-energy structure to antiferromagnetic spin fluctuations.
We investigate the two-dimensional Hubbard model using a real-frequency implementation of the TPSC+DMFT\mathrm{TPSC}+\mathrm{DMFT}TPSC+DMFT approach. This hybrid method combines the nonlocal correlations captured by the two-particle self-consistent (TPSC) approach with the local dynamical correlations of dynamical mean field theory (DMFT). The results demonstrate that TPSC+DMFT\mathrm{TPSC}+\mathrm{DMFT}TPSC+DMFT effectively describes pseudogap physics and nonlocal fluctuations in the moderately correlated regime, while also reproducing the Mott insulating state at larger interaction strengths. For doped Mott insulators, we find that TPSC+DMFT\mathrm{TPSC}+\mathrm{DMFT}TPSC+DMFT captures the evolution of Fermi pockets into Fermi arcs, consistent with the results from cluster DMFT and photoemission studies. These findings highlight the capability of TPSC+DMFT\mathrm{TPSC}+\mathrm{DMFT}TPSC+DMFT to bridge the gap between weak and strong coupling physics in Hubbard models, providing insights into spin and charge fluctuations, as well as their role in the pseudogap formation.