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arXiv:0804.3043 · Physical Review B

Determination of the lattice susceptibility within the dual fermion method

Gang Li, Hunpyo Lee, H. Monien

Published Nov 7, 2008• 20 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.78.195105

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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 (D\ensuremathΓA)(\text{D}\ensuremath{\Gamma}\text{A})(D\ensuremathΓA). 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 D\ensuremathΓA\text{D}\ensuremath{\Gamma}\text{A}D\ensuremathΓA methods are satisfactory.
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.
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.
In this thesis, we use the Dual Fermion approximation (DF) to study the Hubbard model on the 2D square lattice by using two-particle quantities such as susceptibilities and full vertex functions. In the first chapter we give an introduction to the strongly correlated systems including their definitions and features. Then, in the next chapter, to solve our model we present some numerical methods. The numerical methods that we used for our study are the dynamical mean-field theory (DMFT), the continuous-time auxiliary field impurity solver (CTAUX), Fluctuation diagnostics and Dual Fermion approximation (DF). The DF method provides high-resolution results for single-particle quantities such as the density of states (DOS) and Self Energy as well two particles spin susceptibility and vertex functions. With these results, we can present high-resolution results of the fluctuation diagnostics method with minimal computational expense. We examine the full Q-vector dependence of the extended self-energy at the Nodal and Antinodal k-points and provide interpretation for the claim that spin fluctuations are responsible for the observed metal-to-insulator behaviour.
We develop a codebase for performing dynamical mean field theory (DMFT) and dual fermion (DF) calculations on the Hubbard model, by modifying, extending, and interfacing several pre-existing open-source codes. This is used to examine the Hubbard model on a face-centered cubic (fcc) lattice, to obtain numerically-exact results at the level of DMFT and investigate the effect of non-local electron correlations within the dual fermion approach. Our code centres around a modified implementation of the CTHYB quantum Monte Carlo technique used as an impurity solver. We discuss the modifications made, and validate our code by comparing its results to known results in the literature. The DF calculations are obtained by using the CTHYB results as inputs to an extension of the OpenDF code, which we likewise validate by comparison of our results to those of previous authors. We study the magnetic phase diagram, magnetic susceptibilities, and single-particle dynamics of the Hubbard model on the fcc lattice. Within DMFT a large FM region is found above half-filling, the phase boundary of which is located precisely by susceptibility and magnetisation calculations. A large AFM phase is found around and below half-filling with contributions from instabilities towards the X and W points. Single-particle spectra are obtained using the Maximum Entropy method for a wide range of temperatures and fillings demonstrating how the spectra change as the FM phase boundary is crossed. The non-local correlations taken into account by the DF approach reduce the critical temperatures of both the FM and AFM regions. Corrections in the FM region are small and strongly suggest that DMFT is able to capture the relevant physics in this region of the phase diagram. The code developed as part of this project opens the door for further investigations of the Hubbard model on the fcc, and other lattices, across wide ranges of parameter space.
We investigate the Hubbard model on the triangular lattice at half filling using the dynamical cluster approximation (DCA) and dual fermion (DF) methods in combination with continuous-time quantum Monte Carlo (CT QMC) and semiclassical approximation (SCA) methods. We study the one-particle properties and nearest-neighbor spin correlations using the DCA method. We calculate the spectral functions using the CT QMC and SCA methods. The spectral function in the SCA and obtained by analytic continuation of the Pad\'e approximation in CT QMC are in good agreement. We determine the metal-insulator transition and the hysteresis associated with a first-order transition in the double-occupancy and nearest-neighbor spin-correlation functions as functions of temperature. As a further check, we employ the DF method and discuss the advantages and limitation of the dynamical mean-field theory, DCA, and recently developed DF methods by comparing the Green's functions. We find an enhancement of antiferromagnetic correlations and provide evidence for magnetically ordered phases by calculating the spin susceptibility.
The dynamical susceptibility of strongly correlated electronic systems can be calculated within the framework of dynamical mean-field theory (DMFT). The required measurement of the four-point vertex of the auxiliary impurity model is however costly and restricted to a finite grid of Matsubara frequencies, leading to a cutoff error. It is shown that the propagation of this error to the lattice response function can be minimized by virtue of an exact decomposition of the DMFT polarization function into local and nonlocal parts. The former is measured directly by the impurity solver, while the latter is given in terms of a ladder equation for the Hedin vertex that features an unprecedentedly fast decay of frequency summations compared to previous calculation schemes, such as the one of the dual boson approach. At strong coupling, the local approximation of the TRILEX approach is viable, but vertex corrections to the polarization should be dropped on equal footing to recover the correct prefactor of the effective exchange. In finite dimensions, the DMFT susceptibility exhibits spurious mean-field criticality, therefore, a two-particle self-consistent and frequency-dependent correction term is introduced, similar to the Moriya $\ensuremath{\lambda}$ correction of the dynamical vertex approximation. Applications to the two- and three-dimensional Hubbard models on the square and cubic lattices show that the expected critical behavior near an antiferromagnetic instability is recovered.