Löwdin's symmetry dilemma within Green functions theory for the one‐dimensional Hubbard model

Abstract

The energy gap of correlated Hubbard clusters is well studied for one-dimensional systems using analytical methods and density-matrixrenormalization-group (DMRG) simulations. Beyond 1D, however, exact results are available only for small systems by quantum Monte Carlo. For this reason and, due to the problems of DMRG in simulating 2D and 3D systems, alternative methods such as Green functions combined with many-body approximations (GFMBA), that do not have this restriction, are highly important. However, it has remained open whether the approximate character of GFMBA simulations prevents the computation of the Hubbard gap. Here we present new GFMBA results that demonstrate that GFMBA simulations are capable of producing reliable data for the gap which agrees well with the DMRG benchmarks in 1D.An interesting observation is that the accuracy of the gap can be significantly increased when the simulations give up certain symmetry restriction of the exact system, such as spin symmetry and spatial homogeneity. This is seen as manifestation and generalization of the "symmetry dilemma" introduced by Löwdin for Hartree-Fock wave function calculations.

Exact and many-body perturbation solutions of the Hubbard model applied to linear chains

“…For even numbers of atoms, we obtain results in agreement with those reported in Ref. [19] for both MF and ED approaches (Ref [19] does not report results for odd number of atoms). Random initial occupations have been used in the case of the MF approximation such that it has to be compared with unrestricted-spin Hartree-Fock ("usHF") method employed in Ref.…”

“…11 As a key result of this study, we show that the GW correction extends the parameter range for which the ground states display the correct spin symmetry compare to the MF approximation. We also show that the results from SOA for linear chains 19 remain valid within the GW approximation, leading to a Löwdin's symmetry dilemma.…”

“…It is thus important to compare in details the MF results to ED or other advanced techniques (e.g. quantum Monte Carlo or configuration interaction) 10,17,19 to determine the validity of the MF approximation for different systems as a function of the model's parameter.…”

“…A recent study 19 showed that the so-called Löwdin's symmetry dilemma 31 is not satisfactorily resolved using GFMBA, and more specifically in second-order Born approximation (SOA). Löwdin's symmetry dilemma was first described within the MF approximation where, in some cases, solutions with unphysical spatial and spin symmetries are more stable.…”

“…More precisely, the authors of Ref. [19] highlight the case of one-dimensional (1D) single-orbital Hubbard chains with an even number of atoms for which the ED predicts a paramagnetic (PM) ground state for all values of the Hubbard parameters whereas MF predicts an antiferromagnetic (AFM) ground state above a critical Hubbard parameters value. If the PM symmetry is imposed in the MF algorithm, the total energy increases and deviates from the ED total energy.…”