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arXiv:1608.01809 · Physical review. B./Physical review. B

Renormalized parameters and perturbation theory in dynamical mean-field theory for the Hubbard model

Alex C. Hewson

Published Nov 28, 2016• 3 citations
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Abstract

We calculate the renormalized parameters for the quasiparticles and their interactions for the Hubbard model in the paramagnetic phase as deduced from the low-energy Fermi-liquid fixed point using the results of a numerical renormalization-group calculation (NRG) and dynamical mean-field theory (DMFT). Even in the low-density limit there is significant renormalization of the local quasiparticle interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}$, in agreement with estimates based on the two-particle scattering theory of J. Kanamori [Prog. Theor. Phys. 30, 275 (1963)]. On the approach to the Mott transition we find a finite ratio for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}/\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}$, where $2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}$ is the renormalized bandwidth, which is independent of whether the transition is approached by increasing the on-site interaction $U$ or on increasing the density to half filling. The leading ${\ensuremath{\omega}}^{2}$ term in the self-energy and the local dynamical spin and charge susceptibilities are calculated within the renormalized perturbation theory (RPT) and compared with the results calculated directly from the NRG-DMFT. We also suggest, more generally from the DMFT, how an approximate expression for the $\mathbf{q},\ensuremath{\omega}$ spin susceptibility $\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ can be derived from repeated quasiparticle scattering with a local renormalized scattering vertex.

DOI
10.1103/physrevb.94.195152

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We calculate the renormalized parameters for the quasiparticles and their interactions for the Hubbard model in the paramagnetic phase as deduced from the low-energy Fermi-liquid fixed point using the results of a numerical renormalization-group calculation (NRG) and dynamical mean-field theory (DMFT). Even in the low-density limit there is significant renormalization of the local quasiparticle interaction U\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}U\ifmmode~\else˜\fi​, in agreement with estimates based on the two-particle scattering theory of J. Kanamori [Prog. Theor. Phys. 30, 275 (1963)]. On the approach to the Mott transition we find a finite ratio for U\ifmmode~\else˜\fi/D\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}/\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}U\ifmmode~\else˜\fi​/D\ifmmode~\else˜\fi​, where 2D\ifmmode~\else˜\fi2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}2D\ifmmode~\else˜\fi​ is the renormalized bandwidth, which is independent of whether the transition is approached by increasing the on-site interaction UUU or on increasing the density to half filling. The leading \ensuremathω2{\ensuremath{\omega}}^{2}\ensuremathω2 term in the self-energy and the local dynamical spin and charge susceptibilities are calculated within the renormalized perturbation theory (RPT) and compared with the results calculated directly from the NRG-DMFT. We also suggest, more generally from the DMFT, how an approximate expression for the q,\ensuremathω\mathbf{q},\ensuremath{\omega}q,\ensuremathω spin susceptibility \ensuremathχ(q,\ensuremathω)\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})\ensuremathχ(q,\ensuremathω) can be derived from repeated quasiparticle scattering with a local renormalized scattering vertex.
We present a study of the attractive Hubbard model based on the dynamical mean-field theory (DMFT) combined with the numerical renormalization group (NRG). For this study, the NRG method is extended to deal with self-consistent solutions of effective impurity models with superconducting symmetry breaking. We give details of this extension and validate our calculations with DMFT results with antiferromagnetic ordering. We also present results for static and integrated quantities for different filling factors in the crossover from weak to strong-coupling superfluidity. We study the evolution of the single-particle spectra throughout the crossover regime. Although the DMFT does not include the interaction of the fermions with the Goldstone mode, we find strong deviations from the mean-field theory in the intermediate and strong-coupling regimes. In particular, we show that low-energy charge fluctuations induce a transfer of spectral weight from the Bogoliubov quasiparticles to a higher-energy incoherent hump.
Away from half filling and above its Neel temperature, the three-dimensional Hubbard model is a paramagnetic metal with antiferromagnetic spin fluctuations, where controlled momentum-resolved results have been lacking. Using connected-determinant diagrammatic Monte Carlo, we compute the momentum-resolved self-energy for n = 0.4-0.95, T/t = 0.05-0.33 and U/t = 4-8, and the full-zone spin susceptibility at twelve of these points, focusing on U/t = 4. At U/t = 4 and toward half filling the spin fluctuations become strong: for n = 0.7-0.9 at T/t = 0.14 they enhance the susceptibility peak up to sevenfold and raise the momentum-dependent part of the scattering rate to four to five times its second-order value. They are driven by an effective spin interaction screened to about 2U/3 and nearly uniform over the zone. Yet the peak height is set by the slightly smaller value of this interaction at the peak wave vector, and near magnetic order this small difference changes the height by a large factor. Two-particle self-consistent theory reproduces the peak because its sum rule fixes its single constant interaction near this value, but for n = 0.7-0.9 at T/t = 0.14 its self-energy overestimates the local scattering rate and underestimates the momentum-dependent part. Though strong, the fluctuations are short ranged, with a correlation length of at most about two lattice spacings, and for n = 0.7-0.9 and T/t >= 0.14 the scattering rate near the Fermi surface varies by at most about 1% at fixed band energy, without hot spots. At U/t = 4 dynamical mean-field theory (DMFT) reproduces the local scattering rate and quasiparticle weight within a few percent, and the nonlocal part it misses is approximately a function of the band energy. Strong spin fluctuations thus coexist with nearly isotropic quasiparticles, and the results provide a benchmark for extensions of DMFT and for spin-fluctuation theories.
The paramagnetic phase of the one-band Hubbard model is studied at zero-temperature, within the framework of dynamical mean-field theory, and for general particle-hole asymmetry where a doping-induced Mott transition occurs. Our primary focus is the Mott insulator (MI) phase, and our main aim to establish what can be shown exactly about it. To handle the locally doubly-degenerate MI requires two distinct self-energies, which reflect the broken symmetry nature of the phase and together determine the standard single self-energy. Exact results are obtained for the local charge, local magnetic moment and associated spin susceptibilities, the interaction-renormalised levels, and the low-energy behaviour of the self-energy in the MI phase. The metallic phase is also considered briefly, and shown to acquire an emergent particle-hole symmetry as the Mott transition is approached. Throughout the metal, Luttinger's theorem is reflected in the vanishing of the Luttinger integral; for the generic MI by contrast this is shown to be non-vanishing, but again to have a universal magnitude. Numerical results are also obtained using NRG, for the metal/MI phase boundary, the scaling behaviour of the charge as the Mott transition is aproached from the metal, and associated universal scaling of single-particle dynamics as the low-energy Kondo scale vanishes.
We complement previous functional renormalization group (fRG) studies of the two-dimensional Hubbard model by mean-field calculations. The focus falls on Van Hove filling and the hopping amplitude ${t}^{\ensuremath{'}}/t=0.341$. The fRG data suggest a quantum critical point (QCP) in this region and in its vicinity a singular fermionic self-energy $\mathrm{Im}\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})/\ensuremath{\omega}\ensuremath{\sim}\ensuremath{-}{\left|\ensuremath{\omega}\right|}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\approx}0.26$ [K-U. Giering and M. Salmhofer, Self-energy flows in the two-dimensional repulsive Hubbard model, Phys. Rev. B 86, 245122 (2012)]. Here we start a more detailed investigation of this QCP using a bosonic formulation for the effective action, where the bosons couple to the order parameter fields. To this end, we use the channel decomposition of the fermionic effective action developed in C. Husemann and M. Salmhofer, Efficient Parametrization of the Vertex Function, Omega-Scheme, and the (t, t')-Hubbard Model at Van Hove Filling, Phys. Rev. B 79, 195125 (2009), which allows us to perform Hubbard-Stratonovich transformations for all relevant order parameter fields at any given energy scale $\mathrm{\ensuremath{\Omega}}$. We stop the flow at a scale $\mathrm{\ensuremath{\Omega}}$ where the correlations of the order parameter field are already pronounced, but the flow is still regular, and derive the effective boson theory. It contains ddd-wave superconducting, magnetic, and density-density interactions. We analyze the resulting phase diagram in the mean-field approximation. We show that the singular fermionic self-energy suppresses gap formation both in the superconducting and magnetic channel already at the mean-field level, thus rounding a first-order transition (without self-energy) to a quantum phase transition (with self-energy). We give a simple effective model that shows the generality of this effect. In the two-dimensional Hubbard model, the effective density-density interaction is peaked at a nonzero frequency, so that solving the mean-field equations already involves a functional equation instead of simply a matrix equation (on a technical level, similar to incommensurate phases). Within a certain approximation, we show that such an interaction leads to a short quasiparticle lifetime.
We consider the interaction-driven Mott transition at zero temperature from the viewpoint of microscopic Fermi liquid theory. To this end, we derive an exact expression for the Landau parameters within the dynamical mean-field theory (DMFT) approximation to the single-band Hubbard model. At the Mott transition, the symmetric and the antisymmetric Landau parameters diverge. The vanishing compressibility at the Mott transition directly implies the divergence of the forward-scattering amplitude in the charge sector, which connects the proximity of the Mott phase to a tendency toward phase separation. We verify the expected behavior of the Landau parameters in a DMFT application to the Hubbard model on the triangular lattice at finite temperature. Exact conservation laws and the Ward identity are crucial to capture vertex divergences related to the Mott transition. We furthermore generalize Leggett's formula for the static susceptibility of the Fermi liquid to the static fermion-boson response function. In the charge sector, the limits of small transferred momentum and frequency of this response function commute at the Mott transition.
In this paper, we investigate how nonlocal correlations affect, selectively, the physics of correlated electrons over different energy scales, from the Fermi level to the band edges. This goal is achieved by applying a diagrammatic extension of dynamical mean field theory (DMFT), the dynamical vertex approximation $(\mathrm{D}\mathrm{\ensuremath{\Gamma}}\mathrm{A})$, to study several spectral and thermodynamic properties of the unfrustrated Hubbard model in two and three dimensions. Specifically, we focus first on the low-energy regime by computing the electronic scattering rate and the quasiparticle mass renormalization for decreasing temperatures at a fixed interaction strength. This way, we obtain a precise characterization of the several steps through which the Fermi-liquid physics is progressively destroyed by nonlocal correlations. Our study is then extended to a broader energy range, by analyzing the temperature behavior of the kinetic and potential energy, as well as of the corresponding energy distribution functions. Our findings allow us to identify a smooth but definite evolution of the nature of nonlocal correlations by increasing interaction: They either increase or decrease the kinetic energy w.r.t. DMFT depending on the interaction strength being weak or strong, respectively. This reflects the corresponding evolution of the ground state from a nesting-driven (Slater) to a superexchange-driven (Heisenberg) antiferromagnet (AF), whose fingerprints are, thus, recognizable in the spatial correlations of the paramagnetic phase. Finally, a critical analysis of our numerical results of the potential energy at the largest interaction allows us to identify possible procedures to improve the ladder-based algorithms adopted in the dynamical vertex approximation.