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.
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.
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.
By merging algorithmic Matsubara integration with discrete pole representations we present a procedure to generate fully analytic closed form results for impurity problems at fixed perturbation order. To demonstrate the utility of this approach we study the Bethe lattice and evaluate the second-order self-energy for which reliable benchmarks exist. We show that, when evaluating diagrams on the Matsubara axis, the analytic sums of pole representations are extremely precise. We point out the absence of a numerical sign problem in the evaluation, and explore the application of the same procedure for real-frequency evaluation of diagrams. We find that real-frequency results are subject to noise that is controlled at low temperatures and can be mitigated at additional computational expense. We further demonstrate the utility of this approach by evaluating dynamical mean field and bold diagrammatic self-consistency schemes at both second and fourth order and compare to benchmarks where available.
We systematically generate the perturbative expansion for the two-particle spin susceptibility in the Feynman diagrammatic formalism and apply this expansion to a model system---the single-band Hubbard model on a square lattice. We make use of algorithmic Matsubara integration (AMI) [A. Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 99, 035120 (2019)] to analytically evaluate Matsubara frequency summations, allowing us to symbolically impose analytic continuation to the real-frequency axis. We minimize our computational expense by applying graph invariant transformations [A. Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 101, 125109 (2020)]. We highlight extensions of the random-phase approximation and T-matrix methods that, due to AMI, become tractable. We present results for weak interaction strength where the direct perturbative expansion is convergent, and verify our results on the Matsubara axis by comparison to other numerical methods. By examining the spin susceptibility as a function of real frequency via an order-by-order expansion, we can identify precisely what role higher-order corrections play on spin susceptibility and demonstrate the utility and limitations of our approach.
We present an algorithm to evaluate Matsubara sums for Feynman diagrams composed of bare Green's functions with single-band dispersions and local U Hubbard interaction vertices. The algorithm provides an exact construction of the analytic result for the frequency integrals of a diagram that can then be evaluated for all parameters U, temperature T, chemical potential $\ensuremath{\mu}$, external frequencies, and internal/external momenta. This method allows for symbolic analytic continuation of results to the real frequency axis, avoiding any ill-posed numerical procedure. This method can also be used to simultaneously evaluate diagrams throughout the entire $T\text{\ensuremath{-}}U\text{\ensuremath{-}}\ensuremath{\mu}$ phase space of Hubbard-like models even in the T=0 limit at minimal computational expense.
Diagrammatic expansions are a central tool for treating correlated electron systems. At thermal equilibrium, they are most naturally defined within the Matsubara formalism. However, extracting any dynamic response function from a Matsubara calculation ultimately requires the ill-defined analytical continuation from the imaginary- to the real-frequency domain. It was recently proposed [A. Taheridehkordi et al., Phys. Rev. B 99, 035120 (2019)] that the internal Matsubara summations of any interaction-expansion diagram can be performed analytically by using symbolic algebra algorithms. The result of the summations is then an analytical function of the complex frequency rather than Matsubara frequency. Here we apply this principle and develop a diagrammatic Monte Carlo technique which yields results directly on the real-frequency axis. We present results for the self-energy $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})$ of the doped $32\ifmmode\times\else\texttimes\fi{}32$ cyclic square-lattice Hubbard model in a nontrivial parameter regime, where signatures of the pseudogap appear close to the antinode. We discuss the behavior of the perturbation series on the real-frequency axis and in particular show that one must be very careful when using the maximum entropy method on truncated perturbation series. Our approach holds great promise for future application in cases when analytical continuation is difficult and moderate-order perturbation theory may be sufficient to converge the result.
In quantum many-body physics, Quantum Field Theory (QFT) provides powerful tools to describe collective particle behavior. This framework employs three crucial theoretical tools: Green's functions for characterizing particle propagation and response to perturbations, Feynman diagrams for organizing perturbative expansions of quantum processes, and renormalization for improving the convergence of diagrammatic series. A key challenge in QFT is the efficient representation of finite-temperature Green's functions, particularly at low temperatures. We develop a universal framework linking the effective degrees of freedom in these functions to the rank of Lehmann eigenspace, explaining the observed compressibility in Discrete Lehmann representation (DLR). To address numerical instability of DLR in self-consistent solvers, we introduce the symmetrized discrete Lehmann representation (SDLR), which incorporates physical symmetries while maintaining computational efficiency. Applying SDLR, we study phonon-mediated superconductivity in the uniform electron gas (UEG), focusing on the Coulomb interaction effects. Our analysis reveals that the conventional Coulomb pseudopotential μ∗ stems from two distinct effects: modifications to the Fermi Liquid factor and alterations of effective couplings. To extend our first-principle approach to lower electron densities and real materials, vertex corrections in higher-order Feynman diagrams must be systematically evaluated. This presents a significant challenge due to the factorial growth in diagram numbers at higher orders. We address this by transforming Feynman diagrams into computational graphs, enabling efficient computation reuse across shared sub-diagrams. Furthermore, we incorporate Taylor-mode automatic differentiation to streamline the computationally intensive diagrammatic re-expansion process. This approach reduces the computational complexity from exponential to sub-exponential scale relative to the differential order, as demonstrated through calculations of the UEG effective mass ratio. This research advances computational approaches within QFT by combining theoretical physics with advanced computational methods, establishing a foundation for future interdisciplinary studies of quantum many-body systems.