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arXiv:0711.3647 · Physical Review B

Dual fermion approach to susceptibility of correlated lattice fermions

S. Brener, Hartmut Hafermann, Alexey N. Rubtsov, Mikhail I. Katsnelson, Alexander I. Lichtenstein

Published May 9, 2008• 65 citations
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Abstract

In this paper, we show how the two-particle Green function can be obtained within the framework of the dual fermion approach. This facilitates the calculation of the susceptibility in strongly correlated systems where long-ranged nonlocal correlations cannot be neglected. We formulate the Bethe--Salpeter equations for the full vertex in the particle-particle and particle-hole channels and introduce an approximation for practical calculations. The scheme is applied to the two-dimensional Hubbard model at half-filling. The spin-spin susceptibility is found to strongly increase for the wave vector $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$, which indicates the antiferromagnetic instability. We find a suppression of the critical temperature compared to the mean-field result due to the incorporation of the nonlocal spin fluctuations.

DOI
10.1103/physrevb.77.195105

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In this paper, we show how the two-particle Green function can be obtained within the framework of the dual fermion approach. This facilitates the calculation of the susceptibility in strongly correlated systems where long-ranged nonlocal correlations cannot be neglected. We formulate the Bethe--Salpeter equations for the full vertex in the particle-particle and particle-hole channels and introduce an approximation for practical calculations. The scheme is applied to the two-dimensional Hubbard model at half-filling. The spin-spin susceptibility is found to strongly increase for the wave vector q=(\ensuremathπ,\ensuremathπ)\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})q=(\ensuremathπ,\ensuremathπ), which indicates the antiferromagnetic instability. We find a suppression of the critical temperature compared to the mean-field result due to the incorporation of the nonlocal spin fluctuations.
In this paper, we present the details of the dual fermion (DF) method to study the nonlocal corrections to the single-site dynamical mean-field theory (DMFT). The DMFT two-particle Green's function is calculated using continuous-time quantum Monte Carlo method. The momentum dependence of the vertex function is analyzed and its renormalization based on the Bethe-Salpeter equation is performed in the particle-hole channel. We found magnetic instability for both the dual and the lattice fermions. Furthermore, the lattice fermion susceptibility is calculated in this method and also in another recently proposed method, namely, dynamical vertex approximation $(\text{D}\ensuremath{\Gamma}\text{A})$. The comparisons between these two methods are presented in both weak- and strong-coupling regions. Compared to the results from the quantum Monte Carlo calculation, the susceptibility obtained from the DF and $\text{D}\ensuremath{\Gamma}\text{A}$ methods are satisfactory.
In this thesis, we tackle various problems in strongly correlated electron systems, which can be addressed properly via the non-perturbative dynamical mean field theory (DMFT) approach using the continuous time quantum Monte Carlo method as an impurity solver. First, we revisit the old Nagaoka ferromagnetism problem in the U=oo Hubbard model and study the stability of the ferromagnetic state as a function of the temperature, the doping level, and the next-nearest-neighbor hopping t'. We then address the nature of the Mott transition in the two-dimensional Hubbard model at half-filling using cluster DMFT. Cluster DMFT can incorporate the short-range correlations beyond DMFT by extending the spatial range in which correlations are treated exactly to a finite cluster size. The non-local correlations reduce substantially the critical interaction U and modify the shape of the transition lines in the phase diagram. We then concentrate on the calculation of two-particle response functions from the ab initio perspective by means of computing the one-particle excitation spectrum using the combination of the density functional theory (DFT) and DMFT and extracting the two-particle irreducible vertex function from a local two-particle Green's function computed within DMFT. In particular, we derive the equations for calculating the magnetic/charge susceptibility and the pairing susceptibility in superconductivity. This approach is applied to the Hubbard model and the periodic Anderson model and we determine the phase diagram of magnetism and superconductivity in these models. We show that the superconducting phase is indeed stable near the magnetic phase where the pairing interaction mediated by spin fluctuations is dominantly enhanced. The non-local correlation effect to superconductivity is also discussed using the dual fermion approach and the dynamical vertex approximation. We finally apply the vertex function approach within DFT+DMFT to a Fe-based superconductor, BaFe2As2, and compute the dynamical magnetic susceptibility in this material. Our calculation results show a good agreement with the magnetic excitation spectra observed in a neutron scattering experiment. The response function calculation method derived in this thesis can capture both a localized and an itinerant nature of collective excitations in strongly correlated electron systems.
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A general cluster dual fermion approach to nonlocal correlations in crystals is formulated. The scheme allows the treatment of long-range correlations beyond the cluster dynamical mean field theory and nonlocal effects in realistic calculations of multiorbital systems. The simplest approximation is shown to exactly correspond to the free-cluster dynamical mean field theory. This approach is applied to the one-dimensional Hubbard model. Already the first dual-fermion correction to the free cluster leads to a drastic improvement of the calculated Green’s function.
We study the finite-temperature Fermi-liquid to non-Fermi-liquid crossover in the two-dimensional Hubbard model for a range of dopings using the self-consistent ladder dual fermion method. We consider relatively high temperatures where by comparing the single-particle self-energy and the two-particle spin susceptibility we identify a suppression of the density of states near the Fermi level caused by a quasi-antiferromagnetic behavior that is itself characterized by a long, but finite, correlation length scale which we extract. We compute the single-particle self-energy, and the charge/spin susceptibilities for the entire doping range at relatively high temperature, and also for the half-filled weak-coupling case at relatively low temperatures. We perform fluctuation diagnostics to decompose the single-particle self-energy into scattering qqq-vector and bosonic frequency contributions in the spin channel. Within this framework we find that the key contributions to the single-particle self-energy that give non-Fermi-liquid character show complicated dependence on zero, positive, and negative frequency bosonic modes. We find that for a nominal choice of ${t}^{\ensuremath{'}}$ that electron- and hole-doped systems with similar Fermi-liquid character have distinct spin and charge susceptibility and distinct spin-fluctuation diagnostics spectra.