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arXiv:cond-mat/0311128 · Physical Review B

Time-dependent Gutzwiller theory of magnetic excitations in the Hubbard model

G. Seibold, Federico Becca, Pavel Rubin, J. Lorenzana

Published Apr 26, 2004• 38 citations
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Abstract

We use a spin-rotational invariant Gutzwiller energy functional to compute random-phase-approximation-like (RPA) fluctuations on top of the Gutzwiller approximation (GA). The method can be viewed as an extension of the previously developed $\mathrm{GA}+\mathrm{RPA}$ approach for the charge sector [G. Seibold and J. Lorenzana, Phys. Rev. Lett. 86, 2605 (2001)] with respect to the inclusion of the magnetic excitations. Unlike the charge case, no assumptions about the time evolution of the double occupancy are needed in this case. Interestingly, in a spin-rotational invariant system, we find the correct degeneracy between triplet excitations, showing the consistency of both computations. Since no restrictions are imposed on the symmetry of the underlying saddle-point solution, our approach is suitable for the evaluation of the magnetic susceptibility and dynamical structure factor in strongly correlated inhomogeneous systems. We present a detailed study of the quality of our approach by comparing with exact diagonalization results and show its much higher accuracy compared to the conventional $\mathrm{Hartree}\ensuremath{-}\mathrm{Fock}+\mathrm{RPA}$ theory. In infinite dimensions, where the GA becomes exact for the Gutzwiller variational energy, we evaluate ferromagnetic and antiferromagnetic instabilities from the transverse magnetic susceptibility. The resulting phase diagram is in complete agreement with previous variational computations.

DOI
10.1103/physrevb.69.155113

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We use a spin-rotational invariant Gutzwiller energy functional to compute random-phase-approximation-like (RPA) fluctuations on top of the Gutzwiller approximation (GA). The method can be viewed as an extension of the previously developed GA+RPA\mathrm{GA}+\mathrm{RPA}GA+RPA approach for the charge sector [G. Seibold and J. Lorenzana, Phys. Rev. Lett. 86, 2605 (2001)] with respect to the inclusion of the magnetic excitations. Unlike the charge case, no assumptions about the time evolution of the double occupancy are needed in this case. Interestingly, in a spin-rotational invariant system, we find the correct degeneracy between triplet excitations, showing the consistency of both computations. Since no restrictions are imposed on the symmetry of the underlying saddle-point solution, our approach is suitable for the evaluation of the magnetic susceptibility and dynamical structure factor in strongly correlated inhomogeneous systems. We present a detailed study of the quality of our approach by comparing with exact diagonalization results and show its much higher accuracy compared to the conventional Hartree\ensuremath−Fock+RPA\mathrm{Hartree}\ensuremath{-}\mathrm{Fock}+\mathrm{RPA}Hartree\ensuremath−Fock+RPA theory. In infinite dimensions, where the GA becomes exact for the Gutzwiller variational energy, we evaluate ferromagnetic and antiferromagnetic instabilities from the transverse magnetic susceptibility. The resulting phase diagram is in complete agreement with previous variational computations.
The description of electronic inhomogeneities in transition metal compounds requires a technique which is able to deal with the excitations of an inhomogeneous, strongly correlated system. In this talk I will outline the formalism of the timedependent Gutzwiller approximation [1] for the Hubbard model which allows for the computation of random phase approximation (RPA) like fluctuations on top of the Gutzwiller approximation (GA). The performance of the method is demonstrated by comparing static quantities like ground state energy and double occupancies as well as dynamical correlation functions in the charge, spin and pair channels with HartreeFock+RPA and exact results. I will further show that the method gives a good account of several spectroscopic quantities related to stripes in highTc superconductors. A major part of the talk will address the recent generalization of the method to the case of multiple bands [2,3]. Contrary to the standard RPA, the incorporation of the local multipletstructure within our approach leads to a much larger sensitivity of ferromagnetism on the the ratio between Hund coupling and intraorbital repulsion J/U. Besides, we find that the Brinkman Rice localization transition, which in the paramagnetic GA appears at integer doping is always masked by an antiferromagnetic or an orbitally ordered ground state. In addition, a quantum Lifshitz point is found at larger values of J/U which corresponds to the common boundary of ferro, para and incommensurate magnetic phases. [1] G. Seibold and J. Lorenzana, Phys. Rev. Lett. 86, 2605 (2001) [2] E. von Oelsen, G. Seibold, and J. Bunemann, New J. Phys. 13, 113031 (2011) [3] E. v. Oelsen, G. Seibold, and J. Bunemann, Phys. Rev. Lett. 107, 076402 (2011)
We present a detailed study of the time-dependent Gutzwiller approximation for the Hubbard model. The formalism, labeled GA+RPA, allows us to compute random-phase-approximation-like (RPA) fluctuations on top of the Gutzwiller approximation (GA). No restrictions are imposed on the charge and spin configurations which makes the method suitable for the calculation of linear excitations around symmetry-broken solutions. Well-behaved sum rules are obeyed as in the Hartree-Fock (HF) plus RPA approach. Analytical results for a two-site model and numerical results for charge-charge and current-current dynamical correlation functions in one and two dimensions are compared with exact and HF+RPA results, supporting the much better performance of GA+RPA with respect to conventional HF+RPA theory.
We present a method to compute pairing fluctuations on top of the Gutzwiller approximation (GA). Our investigations are based on a charge-rotational invariant GA energy functional which is expanded up to second order in the pair fluctuations. Equations of motion for the fluctuations lead to a renormalized ladder-type approximation. Both spectral functions and corrections to static quantities, such as the ground-state energy, are computed. The quality of the method is examined for the single-band Hubbard model, where we compare the dynamical pairing correlations for sss- and ddd-wave symmetries with exact diagonalizations. We find a significant improvement with respect to analogous calculations done within the standard Hartree-Fock ladder approximation. The technique has potential applications in the theory of Auger spectroscopy, superconductivity, and cold atom physics.
We develop a time-dependent Gutzwiller approximation (GA) for the Hubbard model analogous to the time-dependent Hartree-Fock (HF) method. This new formalism incorporates ground state correlations of the random phase approximation (RPA) type beyond the GA. Static quantities like ground state energy and double occupancy are in excellent agreement with exact results in one dimension up to moderate coupling and in two dimensions for all couplings. We find a substantial improvement over traditional GA and HF+RPA treatments. Dynamical correlation functions can be computed and are also substantially better than HF+RPA ones and obey well behaved sum rules.
We investigate the magnetic phase diagram of the half-filled Hubbard model on the anisotropic triangular lattice using the Gutzwiller approximation (GA) and its ghost generalization (ghost-GA). By combining a rotating spin-frame formulation with high-resolution momentum grids, we determine magnetic ground states through direct total-energy minimization over the ordering wavevector. We benchmark standard GA and ghost-GA against dynamical mean-field theory (DMFT) and dual-fermion results. We show that GA already captures the qualitative structure of the phase diagram, but systematically overestimates the stability of magnetic order due to the absence of dynamical fluctuations. We find that introducing a small number of auxiliary ''ghost'' orbitals is sufficient to recover most dynamical effects and significantly improves quantitative agreement with DMFT. Exploring the full Brillouin zone, we obtain a phase diagram comprising paramagnetic and various magnetic phases. In contrast to ladder dual-fermion susceptibility-based predictions, we find that the one-dimensional antiferromagnetic phase is never stabilized, despite being the leading instability in certain regimes. Our results establish ghost-GA as an efficient and systematically improvable framework for studying magnetism in frustrated systems, capable of achieving near-DMFT accuracy at a fraction of the computational cost. They also highlight that standard GA performs qualitatively well for capturing the general phase diagram, enabling the investigation of incommensurate magnetic orders in more complex systems.
A concise general expression for the dynamical charge susceptibility of the Hubbard model is presented. The results are obtained from the spin-rotation-invariant slave-boson theory, whose paramagnetic saddle point corresponds to Gutzwiller's variational approximation. The charge structure factor and charge susceptibility are calculated for a wide range of interactions and temperatures and are compared thoroughly with the available quantum Monte Carlo and variational Monte Carlo results for the one-dimensional Hubbard model. The comparison shows reasonable agreement. All the calculations are done in a totally self-consistent way, without adjustable parameters.