We use tensor network techniques to obtain high-order perturbative diagrammatic expansions for the quantum many-body problem at very high precision. The approach is based on a tensor train parsimonious representation of the sum of all Feynman diagrams, obtained in a controlled and accurate way with the tensor cross interpolation algorithm. It yields the full time evolution of physical quantities in the presence of any arbitrary time-dependent interaction. Our benchmarks on the Anderson quantum impurity problem, within the real-time nonequilibrium Schwinger-Keldysh formalism, demonstrate that this technique supersedes diagrammatic quantum Monte Carlo by orders of magnitude in precision and speed, with convergence rates $1/N^2$ or faster, where $N$ is the number of function evaluations. The method also works in parameter regimes characterized by strongly oscillatory integrals in high dimension, which suffer from a catastrophic sign problem in quantum Monte Carlo calculations. Finally, we also present two exploratory studies showing that the technique generalizes to more complex situations: a double quantum dot and a single impurity embedded in a two-dimensional lattice.

We present an efficient basis for imaginary time Green's functions based on a low-rank decomposition of the spectral Lehmann representation. The basis functions are simply a set of well-chosen exponentials, so the corresponding expansion may be thought of as a discrete form of the Lehmann representation using an effective spectral density which is a sum of $\delta$ functions. The basis is determined only by an upper bound on the product $\beta {\omega }_{max}$, with $\beta$ the inverse temperature and ${\omega }_{max}$ an energy cutoff, and a user-defined error tolerance $\epsilon$. The number $r$ of basis functions scales as $O(log\left(\beta {\omega }_{max}\right)log(1/\epsilon ))$. The discrete Lehmann representation of a particular imaginary time Green's function can be recovered by interpolation at a set of $r$ imaginary time nodes. Both the basis functions and the interpolation nodes can be obtained rapidly using standard numerical linear algebra routines. Due to the simple form of the basis, the discrete Lehmann representation of a Green's function can be explicitly transformed to the Matsubara frequency domain, or obtained directly by interpolation on a Matsubara frequency grid. We benchmark the efficiency of the representation on simple cases, and with a high-precision solution of the Sachdev-Ye-Kitaev equation at low temperature. We compare our approach with the related intermediate representation method, and introduce an improved algorithm to build the intermediate representation basis and a corresponding sampling grid.

Cover crops have the potential to be agricultural nitrogen (N) regulators that reduce leaching through soils and then deliver N to subsequent cash crops. Yet, regulating N in this way has proven difficult because the few cover crop species that are well-studied excel at either reducing N leaching or increasing N supply to cash crops, but they fail to excel at both simultaneously. We hypothesized that mixed species cover crop stands might balance the N fixing and N scavenging capabilities of individual species. We tested six cover crop monocultures and four mixtures for their effects on N cycling in an organically managed maize-soybean-wheat feed grain rotation in Pennsylvania, USA. For three years, we used a suite of integrated approaches to quantify N dynamics, including extractable soil inorganic N, buried anion exchange resins, bucket lysimeters, and plant N uptake. All cover crop species, including legume monocultures, reduced N leaching compared to fallow plots. Cereal rye monocultures reduced N leaching to buried resins by 90% relative to fallow; notably, mixtures with just a low seeding rate of rye did almost as well. Austrian winter pea monocultures increased N uptake in maize silage by 40 kg N ha^-1 relative to fallow, and conversely rye monocultures decreased N uptake into maize silage by 40 kg N ha^-1 relative to fallow. Importantly, cover crop mixtures had larger impacts on leaching reduction than on maize N uptake, when compared to fallow plots. For example, a three-species mixture of pea, red clover, and rye had similar maize N uptake to fallow plots, but leaching rates were 80% lower in this mixture than fallow plots. Our results show clearly that cover crop species selection and mixture design can substantially mitigate tradeoffs between N retention and N supply to cash crops, providing a powerful tool for managing N in temperate cropping systems.

We propose a method to improve the computational and memory efficiency of numerical solvers for the nonequilibrium Dyson equation in the Keldysh formalism. It is based on the empirical observation that the nonequilibrium Green's functions and self energies arising in many problems of physical interest, discretized as matrices, have low rank off-diagonal blocks, and can therefore be compressed using a hierarchical low rank data structure. We describe an efficient algorithm to build this compressed representation on the fly during the course of time stepping, and use the representation to reduce the cost of computing history integrals, which is the main computational bottleneck. For systems with the hierarchical low rank property, our method reduces the computational complexity of solving the nonequilibrium Dyson equation from cubic to near quadratic, and the memory complexity from quadratic to near linear. We demonstrate the full solver for the Falicov-Kimball model exposed to a rapid ramp and Floquet driving of system parameters, and are able to increase feasible propagation times substantially. We present examples with 262 144 time steps, which would require approximately five months of computing time and 2.2 TB of memory using the direct time stepping method, but can be completed in just over a day on a laptop with less than 4 GB of memory using our method. We also confirm the hierarchical low rank property for the driven Hubbard model in the weak coupling regime within the GW approximation, and in the strong coupling regime within dynamical mean-field theory.