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Precise understanding of strongly interacting fermions, from electrons in modern materials to nuclear matter, presents a major goal in modern physics. However, the theoretical description of interacting Fermi systems is usually plagued by the intricate quantum statistics at play. Here we present a cross-validation between a new theoretical approach, Bold Diagrammatic Monte Carlo (BDMC), and precision experiments on ultra-cold atoms. Specifically, we compute and measure with unprecedented accuracy the normal-state equation of state of the unitary gas, a prototypical example of a strongly correlated fermionic system. Excellent agreement demonstrates that a series of Feynman diagrams can be controllably resummed in a non-perturbative regime using BDMC. This opens the door to the solution of some of the most challenging problems across many areas of physics.
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We develop a new representation for the integrals associated with Feynman diagrams. This leads directly to a novel method for the numerical evaluation of these integrals, which avoids the use of Monte Carlo techniques. Our approach is based on based on the theory of generalized sinc ($sin(x)/x$) functions, from which we derive an approximation to the propagator that is expressed as an infinite sum. When the propagators in the Feynman integrals are replaced with the approximate form all integrals over internal momenta and vertices are converted into Gaussians, which can be evaluated analytically. Performing the Gaussians yields a multi-dimensional infinite sum which approximates the corresponding Feynman integral. The difference between the exact result and this approximation is set by an adjustable parameter, and can be made arbitrarily small. We discuss the extraction of regularization independent quantities and demonstrate, both in theory and practice, that these sums can be evaluated quickly, even for third or fourth order diagrams. Lastly, we survey strategies for numerically evaluating the multi-dimensional sums. We illustrate the method with specific examples, including the the second order sunset diagram from quartic scalar field theory, and several higher-order diagrams. In this initial paper we focus upon scalar field theories in Euclidean spacetime, but expect that this approach can be generalized to fields with spin.
We present a detailed study of the population imbalanced three-component Hubbard chain with attractive interactions. Such a system can be realized experimentally with three different hyperfine states of ultra cold $^6$Li atoms in an optical lattice. We find that there are different phases that compete with each other in this system: A molecular superfluid phase in which the three fermion species pair up to form molecules (trions), a usual pairing phase involving two species with exactly opposite momenta, and a more exotic generalized Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase consisting of three competing pairing tendencies with different non-zero center-of-mass momenta. At large attractive interactions the system exhibits strong tendencies towards collapse and phase separation. Employing the density-matrix-renormalization-group-method (DMRG) to determine the decay exponents of the various correlators we establish the phase diagram of this model for different fillings and interactions. We also discuss the experimentally relevant situation in a trap and report the existence of an additional region where two species are dynamically balanced.
We present an iterative algorithm to count Feynman diagrams via many-body relations. The algorithm allows us to count the number of diagrams of the exact solution for the general fermionic many-body problem at each order in the interaction. Further, we apply it to different parquet-type approximations and consider spin-resolved diagrams in the Hubbard model. Low-order results and asymptotics are explicitly discussed for various vertex functions and different two-particle channels. The algorithm can easily be implemented and generalized to many-body relations of different forms and levels of approximation.
We review recent progress that we have achieved in evaluating the class of fully massive vacuum integrals at five loops. After discussing topics that arise in classification, evaluation and algorithmic codification of this specific set of Feynman integrals, we present some selected new results for their expansions around $4-2varepsilon$ dimensions.
Few analytical methods exist for quantitative studies of large fluctuations in stochastic systems. In this article, we develop a simple diagrammatic approach to the Chemical Master Equation that allows us to calculate multi-time correlation functions which are accurate to a any desired order in van Kampens system size expansion. Specifically, we present a set of Feynman rules from which this diagrammatic perturbation expansion can be constructed algorithmically. We then apply the methodology to derive in closed form the leading order corrections to the linear noise approximation of the intrinsic noise power spectrum for general biochemical reaction networks. Finally, we illustrate our results by describing noise-induced oscillations in the Brusselator reaction scheme which are not captured by the common linear noise approximation.
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