Cluster perturbation theory (CPT) is a computationally economic method commonly used to estimate the momentum- and energy-resolved single-particle Green's function. It has been used extensively in direct comparisons with experiments that effectively measure the single-particle Green's function, e.g., angle-resolved photoemission spectroscopy. However, many experimental observables are given by two-particle correlation functions. CPT can be extended to compute two-particle correlation functions by approximately solving the Bethe-Salpeter equation. We implement this method and focus on the transverse spin susceptibility, measurable via inelastic neutron scattering or with optical probes of atomic gases in optical lattices. We benchmark the method with the one-dimensional Fermi-Hubbard model by comparing with known results.
Cluster perturbation theory (CPT) is a computationally economic method commonly used to estimate the momentum- and energy-resolved single-particle Green's function. It has been used extensively in direct comparisons with experiments that effectively measure the single-particle Green's function, e.g., angle-resolved photoemission spectroscopy. However, many experimental observables are given by two-particle correlation functions. CPT can be extended to compute two-particle correlation functions by approximately solving the Bethe-Salpeter equation. We implement this method and focus on the transverse spin susceptibility, measurable via inelastic neutron scattering or with optical probes of atomic gases in optical lattices. We benchmark the method with the one-dimensional Fermi-Hubbard model by comparing with known results.
The physics of high-Tc superconducting cuprates is obscured by the effect of strong electronic correlations. One way to overcome this problem is to seek an exact solution at least within a small cluster and expand it to the whole crystal. Such an approach is at the heart of cluster perturbation theory (CPT). Here, we developed CPT for the dynamic spin and charge susceptibilities (spin-CPT and charge-CPT), with the correlation effects explicitly taken into account by the exact diagonalization. We applied spin-CPT and charge-CPT to the effective two-band Hubbard model for the cuprates obtained from the three-band Emery model and calculated one- and two-particle correlation functions, namely, a spectral function and spin and charge susceptibilities. The doping dependence of the spin susceptibility was studied within spin-CPT and CPT-RPA, that is, the CPT generalization of the random phase approximation (RPA). In the underdoped region, both our methods resulted in the signatures of the upper branch of the spin excitation dispersion with the lowest excitation energy at the (π,π) wave vector and no presence of low-energy incommensurate excitations. In the high doping region, both methods produced a low energy response at four incommensurate wave vectors in qualitative agreement with the results of the inelastic neutron scattering experiments on overdoped cuprates.
In this dissertation, we use exact diagonalization to study a few strongly correlated systems, ranging from the Fermi-Hubbard model to the fractional quantum Hall effect (FQHE). The discussion starts with an overview of strongly correlated systems and what is meant by strongly correlated. Then, we extend cluster perturbation theory (CPT), an economic method for computing the momentum and energy resolved Green's function for Hubbard models to higher order correlation functions, specifically the spin susceptibility. We benchmark our results for the one-dimensional Fermi-Hubbard model at half-filling. In addition we study the FQHE at fillings nu=5/2 for fermions and nu=1/2 for bosons. For the nu=5/2 system we investigate a two-body model that effectively captures the three-body model that generates the Moore-Read Pfaffian state. The Moore-Read Pfaffian wave function pairs composite fermions and is believed to cause the FQHE at nu=5/2. For the nu=1/2 system we estimate the entropy needed to observe Laughlin correlations with cold atoms via an ansatz partition function. We find entropies achieved with conventional cooling techniques are adequate.
The cluster perturbation theory (CPT) is one of the simplest but systematic quantum cluster approaches to lattice models of strongly correlated electrons with local interactions. By treating the interactions, in addition to the intercluster potential, as a perturbation, it is shown that the CPT can be reformulated as an all-order resummation of diagrams within standard weak-coupling perturbation theory where vertex corrections are neglected. This reformulation is shown to allow for a straightforward generalization of the CPT to the general nonequilibrium case using contour-ordered Green's functions. By solving the resulting generalized CPT equation on the discretized Keldysh-Matsubara time contour, the transient dynamics of an essentially arbitrary initial pure or mixed state can be traced. In this way, the time-dependent expectation values of one-particle observables can be obtained within an approximation that neglects spatial correlations beyond the extension of the reference cluster. The necessary computational effort is very moderate. A detailed discussion and simple test calculations are presented to demonstrate the strengths and the shortcomings of the proposed approach. The nonequilibrium CPT is systematic and is controlled in principle by the inverse cluster size. It interpolates between the noninteracting and the atomic or decoupled-cluster limit which are recovered exactly and is found to predict the correct dynamics at very short times in a general nontrivial case. The effects of initial-state correlations on the subsequent dynamics and the necessity to extend the Keldysh contour by the imaginary Matsubara branch are analyzed carefully and demonstrated numerically. It is further shown that the approach can describe the dissipation of spin and charge to an uncorrelated bath with an essentially arbitrary number of degrees of freedom.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
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.
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.