We investigate the charge- and spin-dynamical structure factors for the two-dimensional one-band Hubbard model in the strong coupling regime within the dynamical cluster approximation (DCA) that is explicitly extended 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 excitation. In the spirit of the DCA, the effective vertex is calculated by using the quantum Monte Carlo methods on a finite cluster. On the basis of a comparison to high temperature auxiliary field quantum Monte Carlo data, we show that near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin, and single-particle excitations.
We investigate the charge- and spin-dynamical structure factors for the two-dimensional one-band Hubbard model in the strong coupling regime within the dynamical cluster approximation (DCA) that is explicitly extended 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 excitation. In the spirit of the DCA, the effective vertex is calculated by using the quantum Monte Carlo methods on a finite cluster. On the basis of a comparison to high temperature auxiliary field quantum Monte Carlo data, we show that near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin, and single-particle excitations.
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.
In this thesis, we use the dynamical cluster approximation to study strongly correlated electron systems, especially from the angle of two-particle quantities, such as dynamical susceptibilities and vertex functions. The thesis starts with an introduction to the strongly correlated systems, including their definitions, prominent features, applications, difficulties in explaining them theoretically and some numerical approaches developed. The following section is an introduction to one family of strongly correlated systems we focus on in this thesis, the high temperature cuprates. The salient features in their general phase diagrams are described and discussed. Then an overview of the model we used to study high Tc cuprates is provided, with its limitations and extensions. To solve this model, the numerical method we employ for our study is the dynamical mean-field theory, the dynamical cluster approximation and the continuous time auxiliary field impurity solver. the last part of Chap.1 contains brief derivations for these algorithms. In Chap. 2, we apply dynamical cluster approximation to solve the one-band 2D Hub- bard model. The physical quantities of interest are two-particle quantitites, such as the dynamical susceptibility, irreducible vertex functions and full vertex functions. In this chapter, we describe how to obtain these susceptibilities via linear response theory and write down a detailed example for superconducting susceptibility. Finally we show how to calculate two particle quantities within DCA and obtain phase boundary with them. Chap. 3 is based on one of our publications. In this work, we specifically address the problem of optimizing the superconducting transition temperature in the 2D Hubbard model by analyzing wide regions of parameter space in density, interaction, and second- nearest-neighbor hopping strength.We mainly focus on d-wave superconductivity but show results of other symmetries in the last section. Chap. 4 follows another publication of ours. We study the temperature and doping evolution of NMR response in the normal state of the 2D Hubbard model using cluster dynamical mean-field theory. We simulate the Knight shift, the spin-echo decay rate and the spin-lattice relaxation time, and compare them to the cuprates experimental results. The last chapter extends the calculation of the NMR response to the superconducting state with the Nambu formalism. We show the detailed formulas and diagrams to calculate two-particle quantities, including the Dyson-Schwinger equation of motion.
The DCA+algorithm was recently introduced by Stear, Maier, and Schulthess [Phys. Rev. B 88, 115101 (2013)] to extend the dynamic cluster approximation (DCA) with a continuous lattice self-energy in order to achieve better convergence with cluster size. Here we extend the DCA+algorithm to the calculation of two-particle correlation functions by introducing irreducible vertex functions with continuous momentum dependence consistent with the DCA+self-energy. This enables a significantly more controlled and reliable study of phase transitions than with the DCA. We test the new method by calculating the superconducting transition temperature Tc in the attractive Hubbard model and show that it reproduces previous determinantal quantum Monte Carlo results. We then calculate Tc in the doped repulsive Hubbard model, for which previous DCA calculations could only access the weak-coupling (U=4t) regime for large clusters. We show that the new algorithm provides access to much larger clusters and delivers asymptotically converged results for Tc for both the weak (U=4t) and intermediate (U=7t) coupling regimes, and thereby enables the accurate determination of the exact infinite cluster size result.
Dynamic cluster Monte Carlo calculations for the doped two-dimensional Hubbard model are used to study the irreducible particle-particle vertex responsible for dx2-y2 pairing in this model. This vertex increases with increasing momentum transfer and decreases when the energy transfer exceeds a scale associated with the Q=(pi, pi) spin susceptibility. Using an exact decomposition of this vertex into a fully irreducible two-fermion vertex and charge and magnetic exchange channels, the dominant part of the effective pairing interaction is found to come from the magnetic, spin S=1 exchange channel.
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.
The high-temperature superconducting cuprates are governed by intertwined spin, charge, and superconducting orders. While various state-of-the-art numerical methods have demonstrated that these phases also manifest themselves in doped Hubbard models, they differ on which is the actual ground state. Finite-cluster methods typically indicate that stripe order dominates, while embedded quantum-cluster methods, which access the thermodynamic limit by treating long-range correlations with a dynamical mean field, conclude that superconductivity does. Here, we report the observation of fluctuating spin and charge stripes in the doped single-band Hubbard model using a quantum Monte Carlo dynamical cluster approximation (DCA) method. By resolving both the fluctuating spin and charge orders using DCA, we demonstrate that they survive in the doped Hubbard model in the thermodynamic limit. This discovery also provides an opportunity to study the influence of fluctuating stripe correlations on the model's pairing correlations within a unified numerical framework. Using this approach, we also find evidence for pair-density-wave correlations whose strength is correlated with that of the stripes.