This thesis is devoted to the development of numerical methods for correlated electrons in realistic systems based on perturbation theories using Feynman diagrams. Electron correlations in realistic systems give rise to a wide range of interesting physical properties. Diagrammatic expansions and its low-order approximations are key building blocks of theoretical and numerical methods for studying correlated electrons. Recent developments of the diagrammatic quantum Monte Carlo (DiagMC) methods enable efficient stochastic evaluations of diagrams and does not suffer from numerical sign problem which worsens with increasing system size. It is therefore a promising direction to apply DiagMC techniques to realistic electron systems in the pursuit of ab initio calculations of correlated materials. This thesis starts by reviewing basic concepts and definitions of realistic electron systems, the perturbation theory based on Feynman diagrams, Monte Carlo evaluations of the series, as well as self-consistent diagrammatic methods. Low-order diagrammatic methods are applied to realistic calculations and benchmarked against large collections of state-of-the-art many-body methods. The thesis then proposes a diagrammatic Monte Carlo solver for realistic molecules and impurities based on a recursive evaluation of diagrams, which is tested on realistic molecules and impurities with varying temperature, system size, and interaction strength. We observe that this method is ideal for problems with many orbitals with moderate correlations, but may suffer from series divergence at strong correlations. Next, we propose the inchworm Monte Carlo algorithm for interaction expansion, which overcomes limitations of divergent diagrammatic series by replacing the standard perturbation series with a sequence of incremental series expansions. A prototype implementation is presented for system with on-site Hubbard interactions. Finally, we discuss technical aspects of numerical representations of the Green's function, which has significant impact on the efficiency of diagrammatic methods. We review several common numerical representations, and introduce our development of the sparse sampling method, which provides compact representations of the Green's function in both imaginary time and frequency as well as fast transformation formula between the two.
This thesis is devoted to the development of numerical methods for correlated electrons in realistic systems based on perturbation theories using Feynman diagrams. Electron correlations in realistic systems give rise to a wide range of interesting physical properties. Diagrammatic expansions and its low-order approximations are key building blocks of theoretical and numerical methods for studying correlated electrons. Recent developments of the diagrammatic quantum Monte Carlo (DiagMC) methods enable efficient stochastic evaluations of diagrams and does not suffer from numerical sign problem which worsens with increasing system size. It is therefore a promising direction to apply DiagMC techniques to realistic electron systems in the pursuit of ab initio calculations of correlated materials. This thesis starts by reviewing basic concepts and definitions of realistic electron systems, the perturbation theory based on Feynman diagrams, Monte Carlo evaluations of the series, as well as self-consistent diagrammatic methods. Low-order diagrammatic methods are applied to realistic calculations and benchmarked against large collections of state-of-the-art many-body methods. The thesis then proposes a diagrammatic Monte Carlo solver for realistic molecules and impurities based on a recursive evaluation of diagrams, which is tested on realistic molecules and impurities with varying temperature, system size, and interaction strength. We observe that this method is ideal for problems with many orbitals with moderate correlations, but may suffer from series divergence at strong correlations. Next, we propose the inchworm Monte Carlo algorithm for interaction expansion, which overcomes limitations of divergent diagrammatic series by replacing the standard perturbation series with a sequence of incremental series expansions. A prototype implementation is presented for system with on-site Hubbard interactions. Finally, we discuss technical aspects of numerical representations of the Green's function, which has significant impact on the efficiency of diagrammatic methods. We review several common numerical representations, and introduce our development of the sparse sampling method, which provides compact representations of the Green's function in both imaginary time and frequency as well as fast transformation formula between the two.
This thesis contains a series of numerical studies of strongly correlated electron systems. In these systems, interesting emergent properties are brought about by strong electron correlations, leading to unusual fluctuations and phase transitions. We start with a review of basic concepts in many-body physics from a field theory point of view, including second quantization and path integral formalisms. We then introduce extensions of the single-particle and two-particle Green’s function formalisms in the singlet superconducting state, which enable us to extract physical information of the systems in the symmetry-broken phase. Next, we proceed to the models and numerical methods we use to study the strongly correlated systems. We introduce a low energy effective model -- the Hubbard model, which contains a subset of the electron degrees of freedom and can be solved numerically using advanced many-body methods. With the dynamical cluster approximation and continuous-time quantum Monte Carlo impurity solver, we are able to study the competing fluctuations in the paramagnetic state and analyze the fluctuations behind superconductivity in the singlet superconducting state. We introduce self-consistent diagrammatic methods used in realistic material calculations in the last part of this thesis, with an outline of group theory concepts that can be used to optimize the simulations. Numerical representations and methods for effectively solving equations in realistic material calculations are discussed afterwards. We briefly review some of the developments in this field, and then introduce spectral methods that are based on mathematical properties of Legendre polynomials for solving both the imaginary- and real-time Dyson equations.
Accurately describing many-body effects in multi-orbital systems remains a major challenge in theoretical condensed matter physics. At present, there is a significant methodological gap between the numerical tools used in ab initio computational materials science and those developed to study strong electronic correlations. The former can treat realistic, large-scale systems but typically neglect many-body effects, while the latter focus on simplified models with only a few degrees of freedom, as only such models can be solved accurately in the presence of strong interactions. The purpose of this thesis is to bridge these two approaches and establish a systematic theoretical framework for realistic correlated electronic materials. This involves a full-cycle methodology that begins with constructing ab initio interacting models from density-functional theory, solving them using dynamical mean-field theory to capture local correlations, and extending beyond to incorporate non-local collective electronic fluctuations. To this end, we introduce the "dual" approach to strong correlations, which includes the dual fermion, dual boson, and dual triply irreducible local expansion methods. The central idea of the dual theories is to shift the reference point of the conventional Feynman diagrammatic expansion from a non-interacting electronic system to an interacting but exactly solvable one. Integrating out this reference system recasts the expansion in an effective dual space, where all diagrammatic building blocks are renormalized by the corresponding impurity quantities. This procedure transforms a non-perturbative expansion in the original variables into a perturbative expansion in terms of dual fermionic and bosonic fields, exact in both the weak- and strong-coupling limits. In this thesis, we collect and systematize the major developments of dual techniques achieved to date.
This thesis develops and applies numerical tools for studying strongly correlated models at finite temperature. Strongly correlated materials exhibit rich emergent behavior, including high temperature superconductivity, exotic magnetism, and metal–insulator transitions. To study these phenomena, Feynman diagrammatics based on perturbation theory provides the foundation for both perturbative and non-perturbative numerical techniques. The thesis focuses on the diagrammatic evaluation of singleand two-particle response functions, with the truncation order acting as a control parameter. Recent advances in symbolic computation have led to the Algorithmic Matsubara Integration (AMI) technique, a central computational technique utilized in this work. AMI generates a symbolic representation of arbitrary Feynman diagrams and resolves the nested Matsubara sums via the residue theorem, yielding closedform semi-analytical expressions, with the remaining momentum integrals evaluated stochastically. The first part of the thesis adopts a pedagogical approach, introducing the generic many-body Hamiltonian, motivating the Hubbard model as a prototype for correlated systems, and reviewing Green’s function formalism in many-body theory. In the first application, we employ AMI to evaluate the pairing susceptibility to measure superconducting fluctuations in the single-band Hubbard model. We identify two dominant 𝑞 = [0,0] and 𝑞 = [π,π] 𝑑-wave pairing modes and map the correlation length and amplitude across the phase space 𝑇–𝑡'– 𝜇, defined by temperature 𝑇, nearest-neighbor hopping 𝑡' and chemical potential 𝜇. Next, we present an algorithm that assigns band indices to arbitrary Feynman topologies symbolically and iteratively using the non-zero elements of interaction tensor Uabcd. Applied to all connected topologies, it generates the full set of contributing diagrams for multiorbital problems. Combined with AMI, this framework termed MB-AMI enables systematic diagrammatic expansion for any low-energy multi-band system with arbitrary longrange interactions. We benchmark the method on molecular H₂ and on the Hubbard ii model under a Zeeman field. Using MB-AMI, we study the multilayer Hubbard model and the layer-resolved features of the pairing susceptibility. We report a monotonic but sublinear increase in the pairing response with layer stacking, assisted by pair-hopping interactions, with a well defined three-dimensional limit. Finally, motivated by the spin-dependent renormalization of screened interactions 𝑊ᵤᵤ and 𝑊ud, we propose a phenomenological extended Hubbard model where nearest-neighbor interactions 𝑉 are spin-dependent and independently tunable. We explore all four quadrants of the 𝑉ᵤᵤ–𝑉ud phase space, focusing particularly on the ferromagnetic-like regime 𝑉ud > 𝑉ᵤᵤ. In this regime, we find that competing ferromagnetic and antiferromagnetic spin fluctuations give rise to an exotic spin-density wave (SDW). We study the resulting fermi surface reconstruction and the emergence of triplet 𝑝-wave pairing from the induced SDW. In conclusion, we propose that merging MB-AMI with the Discrete Lehmann Representation (DLR) into a unified framework opens a pathway to performing high order, fully self-consistent diagrammatic calculations for accurate ab initio simulations of realistic materials.
The description of electron correlation has been a critical problem among theoretical and computational chemistry researchers. To describe the physics of many new materials, this interaction is crucial. Typically, chemists and physicists have constructed approximations to classic electronic structure methods - wavefunction methods and density functional theory - to attempt to solve this problem. Both have had many successes in the computational chemistry field, but are hindered by either their computational cost or ability to rigorously describe the electron correlation. More recently, researchers have come back to investigating how Green's functions can be of use to study this correlation. This thesis focuses on the second-order Green's function which has shown successes in its moderate computational cost and ability to describe electron correlation. However, in solid-state chemistry where the systems to be studied are quite large, the method still must be implemented by other means. Instead of exploring this method deterministically, this thesis moves towards investigating the method via a stochastic method. While stochastic methods such as diagrammatic Monte Carlo are quite common in the physics community, they are far lesser explored among theoretical chemists. This work attempts to develop a computationally feasible way to implement diagrammatic Monte Carlo within the second-order Green's function scheme. This method is preferred to be implemented in a fully self-consistent matter. Thus, chapter 4 of this work will explore the algorithmics required to sample for the second-order self-energy component of the method. Further in chapter 5, the correct statistical exploration required to obtain important expectation values such as the one-body and two-body energies are discussed. It is found that due to the non-linearity of the data, non-parametric statistical resampling methods are required. In this work, it is shown that via a jackknife algorithm built into the second-order Green's function scheme, the stochastic error from a Monte Carlo evaluation of the self-energy can be controlled. To show the power of this analysis, in chapter 6 self-consistency is shown to be possible via calculations of a few model systems as well as larger more realistic chemical systems. Finally, chapter 7 of this thesis segues into a problem within the chemistry community. Coding skills are at the core of developing the methods described in this thesis; however, coding is not required in the chemistry curriculum in the United States. This chapter describes a curriculum that encouraged chemistry students to develop coding skills in a low stakes environment. While the overall hypothesis going into the study was that students who use coding to study a quantitative problem in chemistry will increase their understanding of that problem, the study left with further results. Via surveys, it was discovered that many chemistry students desire to learn coding and feel that developing the skill positively impacts their studies and future goals. Overall, this thesis has made great progress in developing a diagrammatic Monte Carlo technique that could be of interest to the chemistry community. The method has proven to be computationally feasibly and quantitatively correct in comparison to the analogous deterministic method. This work has formulated a scheme that can provide an accessible approach to solving other Green's function methods of interest and hopefully bring us closer to finding a method that is computationally approachable to quantitatively describing electronic correlation.
We show that Monte Carlo sampling of the Feynman diagrammatic series (DiagMC) can be used for tackling hard fermionic quantum many-body problems in the thermodynamic limit by presenting accurate results for the repulsive Hubbard model in the correlated Fermi liquid regime. Sampling Feynman's diagrammatic series for the single-particle self-energy, we can study moderate values of the on-site repulsion ( U / t ∼4) and temperatures down to T / t =1/40. We compare our results with high-temperature series expansions and with single-site and cluster dynamical mean-field theory.
This thesis develops and applies numerical methods for computing correlated electrons at finite temperatures within perturbation theory, formulated in terms of Feynman diagrams. First we compute the fully renormalized effective interaction W in the 2D Hubbard model beyond the conventional low-order random phase approximation (RPA) and additionally resolve W into same-spin and opposite-spin channels. This analysis shows that starting from a purely local repulsive bare interaction U we get attractive effective nearest-neighbor interaction. We identify regimes of inverse temperature t and interaction U=t, where t denotes the nearest-neighbor hopping amplitude, in which this attraction appears. Such an emergent of attractive interaction is known to be able to promote superconductivity in the repulsive Hubbard model. Next, we introduce a method for incorporating higher-order diagrams within selfconsistent perturbative schemes. We merge algorithmic Matsubara integration (AMI) with discrete pole representations, including the discrete Lehmann representation and Prony (minimal pole) representation. AMI automates nested internal Matsubara frequency summations for arbitrary diagrams topologies through symbolic contour integration, which requires explicit access to the analytic pole structure of the input Green’s functions. By compressing the Green’s function into a small set of poles and corresponding weights, we regain access to its analytic pole structure, which allows us to iterate the calculation within self-consistent schemes. Benchmarks on the Bethe lattice demonstrate accurate single-shot and self-consistent self-energies on both Matsubara and real axes. For self-consistent calculations, we use two standard schemes, where AMI combined with the discrete pole representation works as a selfenergy and impurity solver. We further extend calculations to third and fourth order of interaction in self-energy diagrams. This approach yields the ability to perform self-consistent calculations at a fraction of the cost compared to other Matsubara-axis methods without moving to the real axis at every step of the iteration. Finally we demonstrate the universality of the AMI+DLR framework, by decomposing any diagram evaluation into a computational kernel that depends only on the diagram topology and DLR parameters, and a set of coefficients determined by the physical system. This separation enables precomputation and reuse of the computational kernel for a certain diagram topology and any physical problem, both in single-shot and iterative calculations. We illustrate the universality by reusing the same computational kernel for two simple systems: the 2D Hubbard square lattice and the H₂ molecule.