The single particle excitation spectrum of the periodic Anderson model in two dimensions is calculated by using the dynamical cluster approximation (DCA). The DCA method is an extention of the dynamical mean field approximation, that is, the short range inter-site correlations are restored by considering the cluster. For the half filling model there is no sharp coherence peak at the Fermi level due to the anti-ferromagnetic Ruderman–Kittel–Kasuya–Yosida (RKKY)-like interaction. When the filling of the conduction electron band is small, there is a sharp coherence peak. It can be explaind that the effect of the RKKY-like interaction is weaker than that for the half filling model or the RKKY-like interaction changes to the ferromagnetic one. For the intermediate filling model the intensity of the coherence peak becomes very small.
The single particle excitation spectrum of the periodic Anderson model in two dimensions is calculated by using the dynamical cluster approximation (DCA). The DCA method is an extention of the dynamical mean field approximation, that is, the short range inter-site correlations are restored by considering the cluster. For the half filling model there is no sharp coherence peak at the Fermi level due to the anti-ferromagnetic Ruderman–Kittel–Kasuya–Yosida (RKKY)-like interaction. When the filling of the conduction electron band is small, there is a sharp coherence peak. It can be explaind that the effect of the RKKY-like interaction is weaker than that for the half filling model or the RKKY-like interaction changes to the ferromagnetic one. For the intermediate filling model the intensity of the coherence peak becomes very small.
We present the algorithmic details of the dynamical cluster approximation (DCA), with a quantum Monte Carlo (QMC) method used to solve the effective cluster problem. The DCA is a fully causal approach which systematically restores nonlocal correlations to the dynamical mean field approximation (DMFA) while preserving the lattice symmetries. The DCA becomes exact for an infinite cluster size, while reducing to the DMFA for a cluster size of unity. We present a generalization of the Hirsch-Fye QMC algorithm for the solution of the embedded cluster problem. We use the two-dimensional Hubbard model to illustrate the performance of the DCA technique. At half filling, we show that the DCA drives the spurious finite-temperature antiferromagnetic transition found in the DMFA slowly towards zero temperature as the cluster size increases, in conformity with the Mermin-Wagner theorem. Moreover, we find that there is a finite-temperature metal to insulator transition which persists into the weak-coupling regime. This suggests that the magnetism of the model is Heisenberg-like for all nonzero interactions. Away from half filling, we find that the sign problem that arises in QMC simulations is significantly less severe in the context of DCA. Hence, we were able to obtain good statistics for small clusters. For these clusters, the DCA results show evidence of non-Fermi-liquid behavior and superconductivity near half filling.
The dynamical cluster approximation (DCA) is a quantum cluster extension to the single-site dynamical mean-field theory that incorporates spatially nonlocal dynamic correlations systematically and nonperturbatively. The DCA+ algorithm addresses the cluster shape dependence of the DCA and improves the convergence with cluster size by introducing a lattice self-energy with continuous momentum dependence. However, we show that the DCA+ algorithm is plagued by a fundamental problem when its self-consistency equations are formulated using the bare Green's function of the cluster. This problem is most severe in the strongly correlated regime at low doping, where the DCA+ self-energy becomes overly metallic and local, and persists to cluster sizes where the standard DCA has long converged. In view of the failure of the DCA+ algorithm, we propose to complement DCA simulations with a post-interpolation procedure for single-particle and two-particle correlation functions to preserve continuous momentum dependence and the associated benefits in the DCA. We demonstrate the effectiveness of this practical approach with results for the half-filled and hole-doped two-dimensional Hubbard model.
We recently introduced the dynamical cluster approximation (DCA), a technique that includes short-ranged dynamical correlations in addition to the local dynamics of the dynamical mean-field approximation while preserving causality. The technique is based on an iterative self-consistency scheme on a finite-size periodic cluster. The dynamical mean-field approximation (exact result) is obtained by taking the cluster to a single site (the thermodynamic limit). Here, we provide details of our method, explicitly show that it is causal, systematic, $\ensuremath{\Phi}$ derivable, and that it becomes conserving as the cluster size increases. We demonstrate the DCA by applying it to a quantum Monte Carlo and exact enumeration study of the two-dimensional Falicov-Kimball model. The resulting spectral functions preserve causality, and the spectra and the charge-density-wave transition temperature converge quickly and systematically to the thermodynamic limit as the cluster size increases.
We investigate the inter-play of f-electron band dispersion and inter-f electron interactions on the gap in the one electron density of states (DOS) of the periodic Anderson model. The calculations are based on dynamical mean field theory and the numerical renormalization group method for the model in infinite dimensions. The local f-electron DOS at low energies for the electron-hole symmetric model with interactions is found to have a complete-gap, a pseudo-gap or to be gap-less, whenever these structures occur for the non-interacting model but the parameters that characterize these features are strongly renormalized. As the degree of f-electron dispersion is increased, when there is a complete gap, the many-body renormalization remains almost constant, but in the pseudo-gap case the renormalization increases. We calculate the sizes of gaps both in the local DOS and in the dynamic local susceptibilities. When the gap width is defined by the edges in the spectrum for the complete-gap model, the widths of the indirect gap and the spin gap are equal, independent of the degree of electron correlation. However, if the width is defined as the excitation energy between the peaks in the spectrum, then the ratio of the indirect gap to the spin gap increases and eventually saturates as the correlation increases. The value of this ratio depends on the f-electron band dispersion. We have also investigated the dependence of the shape of the interacting DOS on the f-band filling for different forms of f-band dispersion. When the f-band dispersion is small, the gap structure in the symmetric model vanishes as the filling factor decreases. When the model, including the dispersion, has no gap structure, the filling dependence on the DOS in low energies is similar to that for a single impurity Anderson model. The energy dispersion of the quasi-particle in the range, ε f < ε < ε f + U , is approximately explained by the renormalized band model in the cases of both large and small f-band dispersion.
This thesis is dedicated to a theoretical study of the 1-band Hubbard model in the strong coupling limit. The investigation is based on the Dynamical Cluster Approximation (DCA) which systematically restores non-local corrections to the Dynamical Mean Field approximation (DMFA). The DCA is formulated in momentum space and is characterised by a patching of the Brillouin zone where momentum conservation is only recovered between two patches. The approximation works well if k-space correlation functions show a weak momentum dependence. In order to study the temperature and doping dependence of the spin- and charge excitation spectra, we explicitly extend the Dynamical Cluster Approximation to two-particle response functions. The full irreducible two-particle vertex with three momenta and frequencies is approximated by an effective vertex dependent on the momentum and frequency of the spin and/or charge excitations. The effective vertex is calculated by using the Quantum Monte Carlo method on the finite cluster whereas the analytical continuation of dynamical quantities is performed by a stochastic version of the maximum entropy method. A comparison with high temperature auxiliary field quantum Monte Carlo data serves as a benchmark for our approach to two-particle correlation functions. Our method can reproduce basic characteristics of the spin- and charge excitation spectrum. Near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin and single-particle excitations: a collective spin mode emerges at optimal doping and sufficiently low temperatures in the spin response spectrum and exhibits the energy scale of the magnetic exchange interaction J. Simultaneously, the low energy single-particle excitations are characterised by a coherent quasiparticle with bandwidth J. The origin of the quasiparticle can be quite well understood in a picture of a more or less antiferromagnetic ordered background in which holes are dressed by spin-excitations to allow for a coherent motion. By increasing doping, all features which are linked to the spin-polaron vanish in the single-particle as well as two-particle spin response spectrum. In the second part of the thesis an analysis of superconductivity in the Hubbard model is presented. The superconducting instability is implemented within the Dynamical Cluster Approximation by essentially allowing U(1) symmetry breaking baths in the QMC calculations for the cluster. The superconducting transition temperature T_c is derived from the d-wave order parameter which is directly estimated on the Monte Carlo cluster. The critical temperature T_c is in astonishing agreement with the temperature scale estimated by the divergence of the pair-field susceptibility in the paramagnetic phase. A detailed study of the pseudo and superconducting gap is continued by the investigation of the local and angle-resolved spectral function.
We have studied the ground-state properties of the symmetric and asymmetric one-dimensional periodic Anderson model as a function of band filling and the model parameters (V,U,and Ef) using the local mean-field (LMF) approach. We have constructed the phase diagram for both cases and have calculated the energy, local moment, and occupation number of the f level. The LMF approach can reveal the basic properties of the system throughout the entire parameter space. For the symmetric case ${(E}_{f}=\ensuremath{-}U/2),$ close to quarter and half filling the antiferromagnetic phase is the ground state. For intermediate band-filling there is a second-order phase transition to a ferromagnetic state. At quarter filling there is a transition from a metallic paramagnetic state to an insulating antiferromagnetic state as the on-site Coulomb interaction increases. The asymmetric case is treated in the mixed-valence regime (Ef=0). At half filling the ground state is antiferromagnetic while at or close to quarter filling it is paramagnetic. Below half filling the ground state becomes ferromagnetic, suggesting that doping destroys rapidly the antiferromagnetic $f\ensuremath{-}f$ spin correlations. For small values of U we find a paramagnetic state at all band fillings. Results for the average occupation number of the f level compare well with density matrix renormalization group calculations.