We present a multi-reference configuration mixing scheme for describing ground and excited states, with well defined spin and space group symmetry quantum numbers, of the one-dimensional Hubbard model with nearest-neighbor hopping and periodic boundary conditions. Within this scheme, each state is expanded in terms of non-orthogonal and variationally determined symmetry-projected configurations. The results for lattices up to 30 and 50 sites compare well with the exact LiebWu solutions as well as with results from other state-of-the-art approximations. In addition to spin-spin correlation functions in real space and magnetic structure factors, we present results for spectral functions and density of states computed with an ansatz whose quality can be well-controlled by the number of symmetry-projected configurations used to approximate the systems with Ne and Ne ± 1 electrons. The intrinsic symmetry-broken determinants resulting from the variational calculations have rich structures in terms of defects that can be regarded as basic units of quantum fluctuations. Given the quality of the results here reported, as well as the parallelization properties of the considered scheme, we believe that symmetry-projection techniques, which have found ample applications in nuclear structure physics, deserve further attention in the study of low-dimensional correlated many-electron systems.

We derive and implement a new way of solving coupled cluster equations with lower computational scaling. Our method is based on the decomposition of both amplitudes and two electron integrals, using a combination of tensor hypercontraction and canonical polyadic decomposition. While the original theory scales as O(N) with respect to the number of basis functions, we demonstrate numerically that we achieve sub-millihartree difference from the original theory with O(N) scaling. This is accomplished by solving directly for the factors that decompose the cluster operator. The proposed scheme is quite general and can be easily extended to other many-body methods.

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Background:The computational resources needed to generate the ab initio solution of the nuclear many-body problem for increasing mass number and/or accuracy necessitates innovative developments to improve upon (1) the storage of many-body operators and (2) the scaling of many-body methods used to evaluate nuclear observables. The storing and efficient handling of many-body operators with high particle ranks is currently one of the major bottlenecks limiting the applicability range of ab initio studies with respect to mass number and accuracy. Recently, the application of tensor decomposition techniques to many-body tensors has proven highly beneficial to reduce the computational cost of ab initio calculations in quantum chemistry and solid-state physics.Purpose: The impact of applying state-of-the-art tensor factorization techniques to modern nuclear Hamiltonians derived from chiral effective field theory is investigated. Subsequently, the error induced by the tensor decomposition of the input Hamiltonian on ground-state energies of closed-shell nuclei calculated via second-order many-body perturbation theory is benchmarked. Methods:The first proof-of-principle application of tensor-decomposition techniques to the nuclear Hamiltonian is performed. Two different tensor formats are investigated by systematically benchmarking the approximation error on matrix elements stored in various bases of interest. The analysis is achieved while including normalordered three-nucleon interactions that are nowadays used as input to the most advanced ab initio calculations in medium-mass nuclei. With the aid of the factorized Hamiltonian, the second-order perturbative correction to ground-state energies is decomposed and the scaling properties of the underlying tensor network are discussed. Results:The employed tensor formats are found to lead to an efficient data compression of two-body matrix elements of the nuclear Hamiltonian. In particular, the sophisticated tensor hypercontraction (THC) scheme yields low tensor ranks with respect to both harmonic-oscillator and Hartree-Fock single-particle bases. It is found that the tensor rank depends on the two-body total angular momentum J for which one performs the decomposition, which is itself directly related to the sparsity the corresponding tensor. Furthermore, including normal-ordered two-body contributions originating from three-body interactions does not compromise the efficient data compression. Ultimately, the use of factorized matrix elements authorizes controlled approximations of the exact second-order ground-state energy corrections. In particular, a small enough error is obtained from low-rank factorizations in 4 He, 16 O and 40 Ca. Conclusions:It is presently demonstrated that tensor-decomposition techniques can be efficiently applied to systematically approximate the nuclear many-body Hamiltonian in terms of lower-rank tensors. Beyond the input Hamiltonian, tensor-decomposition techniques can be envisioned to scale down the cost of state-of-the-art non-perturbative man...