This thesis presents the development of new numerical methods for the treatment of strongly correlated electron systems based on self-consistent approaches at both the one and the two-particle level such as the parquet formalism. The parquet formalism was solved for the first time on a two-dimensional cluster. When the fully irreducible vertex is approximated by the bare vertex, we obtain the parquet approximation. Its validity was investigated by comparing results that it produces to those of other conserving approximations such as the FLuctuation EXchange (FLEX) approximation or the Second Order Perturbation Theory (SOPT). We found that it provides a significant improvement of FLEX or SOPT and a satisfactory agreement with Quantum Monte Carlo results despite instabilities in the self-consistency at low temperatures and for strong Coulomb interaction. We use the parquet formalism to study the Quantum Critical Point at finite doping in the Hubbard model by decomposing the vertex into its contributions from different channels. We apply this decomposition to the pairing channel and we find that the dominant contribution to the vertex originates in the spin channel even at the quantum critical doping. Furthermore, we explore the divergence of the two parts of the pairing matrix at optimal doping and observe that the irreducible vertex decreases monotonically as the doping is increased while the bare susceptibility exhibits an algebraic divergence at the quantum critical doping supporting the Quantum Critical BCS scenario proposed by She and Zaanen. To circumvent the instabilities in the iteration of the parquet formalism, we explored the dual fermion approach introduced by Rubtsov et al. Here, we extended the formalism to the Dynamical Cluster Approximation, in the process introducing a small parameter in the dual fermions perturbation theory. We demonstrate the quality of the resulting Dual Fermion DCA through a systematic study of the cluster size dependence and of the different perturbative approximations. These efforts represent the initial steps in the development of the Multi-Scale Many Body approach that appropriately treats correlations at different length scales.
This thesis presents the development of new numerical methods for the treatment of strongly correlated electron systems based on self-consistent approaches at both the one and the two-particle level such as the parquet formalism. The parquet formalism was solved for the first time on a two-dimensional cluster. When the fully irreducible vertex is approximated by the bare vertex, we obtain the parquet approximation. Its validity was investigated by comparing results that it produces to those of other conserving approximations such as the FLuctuation EXchange (FLEX) approximation or the Second Order Perturbation Theory (SOPT). We found that it provides a significant improvement of FLEX or SOPT and a satisfactory agreement with Quantum Monte Carlo results despite instabilities in the self-consistency at low temperatures and for strong Coulomb interaction. We use the parquet formalism to study the Quantum Critical Point at finite doping in the Hubbard model by decomposing the vertex into its contributions from different channels. We apply this decomposition to the pairing channel and we find that the dominant contribution to the vertex originates in the spin channel even at the quantum critical doping. Furthermore, we explore the divergence of the two parts of the pairing matrix at optimal doping and observe that the irreducible vertex decreases monotonically as the doping is increased while the bare susceptibility exhibits an algebraic divergence at the quantum critical doping supporting the Quantum Critical BCS scenario proposed by She and Zaanen. To circumvent the instabilities in the iteration of the parquet formalism, we explored the dual fermion approach introduced by Rubtsov et al. Here, we extended the formalism to the Dynamical Cluster Approximation, in the process introducing a small parameter in the dual fermions perturbation theory. We demonstrate the quality of the resulting Dual Fermion DCA through a systematic study of the cluster size dependence and of the different perturbative approximations. These efforts represent the initial steps in the development of the Multi-Scale Many Body approach that appropriately treats correlations at different length scales.
This thesis represents our effort to develop the next generation multi-scale quantum simulation methods suitable for strongly-correlated materials, where complicated phase-diagrams prevail, suggesting complicated underlying physics. We first give a detailed description of the parquet formalism. With its help, different approximate methods can be unified and a hierarchy of approximate methods with different accuracies and computational complexity can thus be designed. Next, we present a numerical solution of the parquet approximation. Results on the Hubbard model are compared to those obtained from Determinant Quantum Monte Carlo (DQMC), FLuctuation EXchange (FLEX), and self-consistent second-order approximation methods. The comparison shows a satisfactory agreement with DQMC and a significant improvement over the FLEX or the self-consistent second-order approximation. The parquet formalism can also be used to analyze the superconducting mechanism of the high-temperature superconductors. The dynamical cluster approximation (DCA) method is used to understand the proximity of the superconducting dome to the quantum critical point in the 2-D Hubbard model. At optimal doping, where Vd is revealed to be featureless, we find a power-law behavior of chi_0d(w=0), replacing the BCS logarithm behavior, and strongly enhanced T_c. After that we propose another multi-scale approach by combining the DCA and the recently introduced dual-fermion formalism. Within this approach, short and long length scale physics is addressed by the DCA cluster calculation, while intermediate length scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as O(1/Lc), so perturbation theory on the dual lattice converges very quickly. Lastly, we study the responses to dynamical modulation of the optical lattice potential by analyzing properties of the repulsive fermionic Hubbard model in an optical lattice. We provide numerical evidence showing the modulations by on-site local interaction cannot be ignored, and instead can even strongly contribute to the dynamical behaviors of the system in highly-doped cases.
We present a numerical solution of the parquet approximation, a conserving diagrammatic approach which is self-consistent at both the single-particle and the two-particle levels. The fully irreducible vertex is approximated by the bare interaction thus producing the simplest approximation that one can perform with the set of equations involved in the formalism. The method is applied to the Hubbard model on a half-filled 4x4 cluster. Results are compared to those obtained from determinant quantum Monte Carlo (DQMC), FLuctuation EXchange (FLEX), and self-consistent second-order approximation methods. This comparison shows a satisfactory agreement with DQMC and a significant improvement over the FLEX or the self-consistent second-order approximation.
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.
Disorder is ubiquitous in real materials and influences the physical properties like the conductivity to varying degrees. If electron-electron interactions are strong, theoretical and numerical treatment of these systems becomes challenging. In this thesis a numerical approach is developed to address these systems, treating both interactions and disorder on equal footing. The approach is based on the dual fermion approach for interacting systems developed by Rubtsov et al. Terletska et al. applied the ideas of the dual fermion approach to disordered non-interacting systems. In this approach, the replica trick is used to integrate out the disorder in favor of an effective electron-electron interaction. We extended the approach from Terletska et al. to treat disordered interacting systems. Dual Fermions allow to take into account non-local fluctuations by means of a perturbative expansion around an impurity problem. The impurity reference system is determined self-consistently, analogously to the dynamical mean-field theory. The perturbative expansion is expected to yield good results for small and large values of interaction strength and disorder. A priori, it is not clear what to expect for intermediate values, but experience shows that oftentimes good results are obtained for this region. An advantage of the dual fermion approach is that there is no sign-problem for a single orbital model if quantum Monte Carlo is used to solve the interacting reference system. Additionally, perturbation theory is usually numerically much cheaper than fully solving an interacting lattice or cluster problem. Thus, the dual fermion approach allows to address regions of parameter space that are not accessible to lattice quantum Monte Carlo calculations or cluster extension of dynamical mean-field theory. Cluster extensions of the dynamical mean-field theory are for example the dynamical cluster approximation or the cellular dynamical mean-field theory. The new approach is benchmarked with respect to the Anderson-Falicov-Kimball model. The Anderson-Hubbard model is the main application. For both models, the algorithm is tested in one dimension for the single particle Green function and compared to the dynamical cluster approximation to estimate the quality of the results. The Anderson-Falicov-Kimball model is studied further in two dimensions and the phase diagram on the disorder-interaction plane is calculated from both the single particle Green function and the dc conductivity. For the Anderson-Hubbard model, the antiferromagnetic phase transition, the Mott transition and the Anderson transition are investigated in three dimensions. The dynamical mean-field theory results are qualitatively confirmed for the most part. Quantitative corrections are in agreement with dynamical cluster approximation results, thus confirming the validity of the results. We found a qualitative difference between the dual fermion approach and dynamical mean-field theory for the temperature dependence of the hysteresis of the double occupancy. The dual fermion result shows a decreasing lower critical interaction strength for decreasing temperature, which is in contrast with the dynamical mean-field result that shows an increase of the lower critical interaction strength. This behavior happens for small temperatures that are not accessible using for example the dynamical cluster approximation, thus highlighting the usefulness of the dual fermion approach. We found this behavior for all values of the disorder that we considered.
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.
Disorder is ubiquitous in real materials and influences the physical properties like the conductivity to varying degrees. \nIf electron-electron interactions are strong, theoretical and numerical treatment of these systems becomes challenging. In this thesis a numerical approach is developed to address these systems, \ntreating both interactions and disorder on equal footing. \nThe approach is based on the dual fermion approach for interacting systems developed by Rubtsov et al. \nTerletska et al. applied the ideas of the dual fermion approach to disordered non-interacting systems. In this approach, the replica trick is used to integrate out the disorder in favor of an effective electron-electron interaction. \nWe extended the approach from Terletska et al. to treat disordered interacting systems. \n \nDual Fermions allow to take into account non-local fluctuations by means of a perturbative expansion around an impurity problem. The impurity reference system is determined self-consistently, analogously to the dynamical mean-field \ntheory. \nThe perturbative expansion is expected to yield good results for small and large values of interaction strength and disorder. A priori, it is not clear what to expect for intermediate values, \nbut experience shows that oftentimes good results are obtained for this region. \n \nAn advantage of the dual fermion approach is that there is no sign-problem for a single orbital model if quantum Monte Carlo is used to solve the interacting reference system. \nAdditionally, perturbation theory is usually numerically much cheaper than fully solving an interacting lattice or cluster problem. \nThus, the dual fermion approach allows to address regions of parameter space that are not accessible to lattice quantum Monte Carlo calculations or cluster extension of dynamical mean-field theory. \nCluster extensions of the dynamical mean-field theory are for example the dynamical cluster approximation or the cellular dynamical mean-field theory. \n \nThe new approach is benchmarked with respect to the Anderson-Falicov-Kimball model. The Anderson-Hubbard model is the main application. \nFor both models, the algorithm is tested in one dimension for the single particle Green function and compared to the dynamical cluster approximation to estimate the quality of the results. \nThe Anderson-Falicov-Kimball model is studied further in two dimensions and the phase diagram on the disorder-interaction plane is calculated from both the single particle Green function and the dc conductivity. \n \nFor the Anderson-Hubbard model, the antiferromagnetic phase transition, the Mott transition and the Anderson transition are investigated in three dimensions. The dynamical mean-field theory results are \nqualitatively confirmed for the most part. Quantitative corrections are in agreement with dynamical cluster approximation results, thus confirming the validity of the results. \nWe found a qualitative difference between the dual fermion approach and dynamical mean-field theory for the temperature dependence of the hysteresis of the double occupancy. The dual fermion result \nshows a decreasing lower critical interaction strength for decreasing temperature, which is in contrast with the dynamical mean-field result that shows an increase of the lower critical interaction strength. \nThis behavior happens for small temperatures that are not accessible using for example the dynamical cluster approximation, thus highlighting the usefulness of the dual fermion approach. \nWe found this behavior for all values of the disorder that we considered.