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 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.
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.
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.
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.
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.
We generate the perturbative expansion of the single particle Green's function and related self-energy for a half-filled single-band Hubbard model on a square lattice. We invoke algorithmic Matsubara integration to evaluate single-particle quantities for real and Matsubara frequencies and verify results through comparison to existing data on the Matsubara axis. With low-order expansions at weak coupling we observe a number of outcomes expected at higher orders: the opening of a gap, pseudogap behavior, and Fermi-surface reconstruction. Based on low-order perturbations, we consider the phase diagram that arises from truncated expansions of the self-energy and Green's function and their relation via the Dyson equation. From Matsubara axis data, we observe insulating behavior in direct expansions of the Green's function, whereas the same order of truncation of the self-energy produces metallic behavior. This observation is supported by additional calculations for real frequencies. We attribute this difference to the order in which diagrams are implicitly summed in the Dyson series. By separating the reducible and irreducible contributions at each order we show that the reducible diagrams implicitly summed in the Dyson equation lead to incorrect physics in the half-filled Hubbard model. Our observations for this particular case lead us to question the utility of the Dyson equation for any problem that shows a disparity between reducible and irreducible contributions to the expansion of the Green's function.
This thesis is devoted to diagrammatic frameworks for computing the dynamic structure factor (DSF) and the static susceptibility of spin systems at finite temperature in the thermodynamic limit. We consider two complementary diagrammatic frameworks. First, we study functional renormalization group (FRG) approaches based on pseudo-fermion (pf-FRG) and pseudo-Majorana (pm-FRG) representations. Second, we develop high-order strong-coupling and high-temperature expansions (HTE) using free and restricted-graph techniques. To enable finite-temperature calculations in pf-FRG, we eliminate unphysical states by incorporating the Popov–Fedotov trick. Within pm-FRG, we introduce a temperature-flow scheme that significantly improves the efficiency and accuracy of finite-size scaling analyses of thermal phase transitions. We further extend pm-FRG to include external magnetic fields, allowing treatment of XXZ models with a field along the z direction. While the pseudo-Majorana representation maps local spin-1/2 operators to free Majorana fermions, higher spin values S>1/2 can instead be represented as strongly interacting real bosons, an idea underlying the recently proposed spin-FRG. Rather than implementing the spin-FRG itself, we analyze its perturbation series using a strong-coupling expansion. To evaluate the high-dimensional imaginary-time integrals arising in this expansion analytically with efficient bookkeeping, we introduce the kernel trick, which rewrites any Matsubara diagram in terms of the partial spectral functions of its constituent Green’s functions. The resulting fourth-order strong-coupling expansion for general spin S provides systematic beyond-mean-field corrections to critical temperatures and fields in both 1/d and large-interaction-range expansions. More importantly, it provides an analytic explanation of the long-standing quantum-to-classical correspondence observed in diagrammatic Monte Carlo simulations: the normalized static susceptibilities of quantum and classical Heisenberg models coincide after a simple temperature rescaling. Our analysis shows that this arises from the fact that the static susceptibility is well approximated by a simple functional form akin to a mean-field susceptibility, which we call the renormalized-mean-field form. To reach higher orders, we generalize the HTE to Matsubara-frequency-resolved Green’s functions (Dyn-HTE). This includes the static susceptibility (at zero Matsubara frequency) as a special case and validates the renormalized-mean-field form up to 12th order in perturbation theory. By analytically continuing the frequency dependence obtained from Dyn-HTE using a continued-fraction expansion, we compute DSFs for one-parameter Heisenberg models on arbitrary lattices, which can be directly compared with experimental inelastic neutron-scattering spectra. All Dyn-HTE calculations are implemented in a publicly available, well-documented open-source code. Finally, we clarify the relation between pseudo-Majorana and spin-based diagrammatic approaches by deriving the exact pseudo-Majorana action directly from the spin Hamiltonian, without invoking any parton representation. This derivation paves the way for extending Majorana-based methods to higher-spin systems.