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arXiv:cond-mat/0203492 · Journal of Physical Studies

Dynamical susceptibilities in the strong coupling approach: A general scheme and the Falicov–Kimball model

Andrij M. Shvaika

Published Jan 1, 2001• 15 citations
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Abstract

A general scheme to calculate dynamical susceptibilities of strongly correlated electron systems within the dynamical mean eld theory is developed.The approach is based on an expansion over electron hopping around the atomic limit (within the diagrammatic technique for site operators: projection and Hubbard ones) in in nite dimensions.As an example, the Falicov{Kimball and simpli ed pseudospin-electron models are considered for which an analytical expressions for dynamical susceptibilities are obtained.

DOI
10.30970/jps.05.349

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In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2.\n\nIn the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT \nalgorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy.\nThe main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined\nFalicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase.\n\nIn the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory.\nThis model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The \nrenormalization factor q can be evaluated as a function of the filling n for 0<n<1. In the limit n->0, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.

•Jan 1, 2006•1 citation
A general scheme to calculate dynamical susceptibilities of strongly correlated electron systems within the dynamical mean eld theory is developed.The approach is based on an expansion over electron hopping around the atomic limit (within the diagrammatic technique for site operators: projection and Hubbard ones) in in nite dimensions.As an example, the Falicov{Kimball and simpli ed pseudospin-electron models are considered for which an analytical expressions for dynamical susceptibilities are obtained.
The Falicov-Kimball model with correlated hopping is studied in the limit of infinite spatial dimensions.Dynamical susceptibilities are calculated using the generalization of the Dynamical Mean-Field Theory (DMFT), which is based on expansion over electron hopping around the single-site limit.A special case of semi-elliptic density of states and diagonal correlated hopping is considered numerically and the absence of phase transition for all temperatures except T = 0 is demonstrated, which corresponds to the known results.
The Falicov-Kimball model with a correlated-hopping interaction is solved using an extended dynamical mean-field theory that becomes exact in the limit of large dimensions. The effect of correlated hopping is to introduce nonlocal self-energy components that retain full dynamics as $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty},$ thus introducing an explicit k dependence to the single-particle self-energy. An explicit solution for the homogeneous phase at D=2D=2D=2 reveals significant nonlocal dynamical contributions in the physically relevant regime of a moderately large correlated-hopping amplitude, indicating that important nonlocal correlations are omitted in Hubbard-like models upon neglecting the correlated-hopping interaction.
A perturbation theory scheme in terms of electron hopping, which is based on the Wick theorem for Hubbard operators, is developed. Diagrammatic series contain single-site vertices connected by hopping lines and it is shown that for each vertex the problem splits into the subspaces with ``vacuum states'' determined by the diagonal Hubbard operators and only excitations around these vacuum states are allowed. The rules to construct diagrams are proposed. In the limit of infinite spatial dimensions the total auxiliary single-site problem exactly splits into subspaces that allows to build an analytical thermodynamically consistent approach for a Hubbard model. Some analytical results are given for the simple approximations when the two-pole (alloy-analogy solution) and four-pole (Hartree-Fock approximation) structure for Green's function is obtained. Two poles describe contribution from the Fermi-liquid component, which is dominant for small electron and hole concentrations (``overdoped case'' of high-Tc′s),{T}_{c}'\mathrm{s}),Tc′​s), whereas other two describe contribution from the non-Fermi liquid and are dominant close to half-filling (``underdoped case'').
Strong electronic correlations generally require non-perturbative treatment. Local correlations are captured by dynamical mean-field theory while nonlocal correlations can be treated with diagrammatic extensions such as the Dual Fermion approach. Dual Fermion is built on physically motivated, but in principle uncontrolled approximations, so careful benchmarking is needed to understand the strengths and limitations of the method. In this work, we benchmark ladder Dual Fermion and dynamical mean-field theory for the Falicov-Kimball model with the exact classical Monte Carlo solution. We focus on the thermodynamics, electronic structure and susceptibility, especially at the combined frequency and momentum structure, and find that Dual Fermion clearly outperforms dynamical mean-field theory. Somewhat surprisingly, Dual Fermion is not as accurate for the relation between orbital density versus chemical potential in the doped system. These results demonstrate the need for rigorous benchmarking of diagrammatic extensions of dynamical mean-field theory for models with inequivalent orbitals, which is essential for modelling materials.
In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2.\n\nIn the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT \nalgorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy.\nThe main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined\nFalicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase.\n\nIn the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory.\nThis model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The \nrenormalization factor q can be evaluated as a function of the filling n for 0<n<1. In the limit n->0, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.