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.
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 \ensuremathμ, 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\ensuremath-U\ensuremath-\ensuremathμ phase space of Hubbard-like models even in the T=0 limit at minimal computational expense.
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 a symbolic algorithm for the fully analytic treatment of perturbative expansions of Hamiltonians with general two-body interactions. The method merges well-known analytics with the recently developed symbolic integration tool, algorithmic Matsubara integration, that allows for the evaluation of the imaginary frequency/time integrals. By symbolically constructing Wick contractions at each order of the perturbative expansion we order by order construct the fully analytic solution of the Green's function and self-energy expansions. A key component of this process is the assignment of momentum/frequency conserving labels for each contraction that motivates us to present a fully symbolic Fourier transform procedure which accomplishes this feat. These solutions can be applied to a broad class of quantum chemistry problems and are valid at arbitrary temperatures and on both the real- and Matsubara-frequency axis. To demonstrate the utility of this approach, we present results for simple molecular systems as well as model lattice Hamiltonians. We highlight the case of molecular problems where our results at each order are numerically exact with no stochastic uncertainty.
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 implement a highly efficient strong-coupling expansion for the Green's function of the Hubbard model. In the limit of extreme correlations, where the onsite interaction is infinite, the evaluation of diagrams simplifies dramatically enabling us to carry out the expansion to the eighth order in powers of the hopping amplitude. We compute the finite-temperature Green's function analytically in the momentum and Matsubara frequency space as a function of the electron density. Employing Padé approximations, we study the equation of state, Kelvin thermopower, momentum distribution function, quasiparticle fraction, and quasiparticle lifetime of the system at temperatures lower than, or of the order of, the hopping amplitude. We also discuss several different approaches for obtaining the spectral functions through analytic continuation of the imaginary frequency Green's function, and show results for the system near half filling. We benchmark our results for the equation of state against those obtained from a numerical linked-cluster expansion carried out to the eleventh order.