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تسجيل مستخدم جديدMultiple impurities and combined local density approximations in Site-Occupation Embedding Theory
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Site-occupation embedding theory (SOET) is an in-principle-exact multi-determinantal extension of density-functional theory for model Hamiltonians. Various extensions of recent developments in SOET [Senjean et al., Phys. Rev. B 97, 235105 (2018)] are explored in this work. An important step forward is the generalization of the theory to multiple impurity sites. We also propose a new single-impurity density-functional approximation (DFA) where the density-functional impurity correlation energy of the two-level (2L) Hubbard system is combined with the Bethe ansatz local density approximation (BALDA) to the full correlation energy of the (infinite) Hubbard model. In order to test the new DFAs, the impurity-interacting wavefunction has been computed self-consistently with the density matrix renormalization group method (DMRG). Double occupation and per-site energy expressions have been derived and implemented in the one-dimensional case. A detailed analysis of the results is presented, with a particular focus on the errors induced either by the energy functionals solely or by the self-consistently converged densities. Among all the DFAs (including those previously proposed), the combined 2L-BALDA is the one that performs the best in all correlation and density regimes. Finally, extensions in new directions, like a partition-DFT-type reformulation of SOET, a projection-based SOET approach, or the combination of SOET with Green functions, are briefly discussed as a perspective.
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Site-occupation embedding theory (SOET) is an alternative formulation of density-functional theory (DFT) for model Hamiltonians where the fully-interacting Hubbard problem is mapped, in principle exactly, onto an impurity-interacting (rather than a n
on-interacting) one. It provides a rigorous framework for combining wavefunction (or Green function) based methods with DFT. In this work, exact expressions for the per-site energy and double occupation of the uniform Hubbard model are derived in the context of SOET. As readily seen from these derivations, the so-called bath contribution to the per-site correlation energy is, in addition to the latter, the key density functional quantity to model in SOET. Various approximations based on Bethe ansatz and perturbative solutions to the Hubbard and single impurity Anderson models are constructed and tested on a one-dimensional ring. The self-consistent calculation of the embedded impurity wavefunction has been performed with the density matrix renormalization group method. It has been shown that promising results are obtained in specific regimes of correlation and density. Possible further developments have been proposed in order to provide reliable embedding functionals and potentials.
Quantum embedding based on the (one-electron reduced) density matrix is revisited by means of the unitary Householder transformation. While being exact and equivalent to (but formally simpler than) density matrix embedding theory (DMET) in the non-in
teracting case, the resulting Householder transformed density matrix functional embedding theory (Ht-DMFET) preserves, by construction, the single-particle character of the bath when electron correlation is introduced. In Ht-DMFET, the projected impurity+bath clusters Hamiltonian (from which approximate local properties of the interacting lattice can be extracted) becomes an explicit functional of the density matrix. In the spirit of single-impurity DMET, we consider in this work a closed (two-electron) cluster constructed from the full-size non-interacting density matrix. When the (Householder transformed) interaction on the bath site is taken into account, per-site energies obtained for the half-filled one-dimensional Hubbard lattice match almost perfectly the exact Bethe Ansatz results in all correlation regimes. In the strongly correlated regime, the results deteriorate away from half-filling. This can be related to the electron number fluctuations in the (two-site) cluster which are not described neither in Ht-DMFET nor in regular DMET. As expected, the per-site energies dramatically improve when increasing the number of embedded impurities. Formal connections with density/density matrix functional theories have been briefly discussed and should be explored further. Work is currently in progress in this direction.
Recently a novel approach to find approximate exchange-correlation functionals in density-functional theory (DFT) was presented (U. Mordovina et. al., JCTC 15, 5209 (2019)), which relies on approximations to the interacting wave function using densit
y-matrix embedding theory (DMET). This approximate interacting wave function is constructed by using a projection determined by an iterative procedure that makes parts of the reduced density matrix of an auxiliary system the same as the approximate interacting density matrix. If only the diagonal of both systems are connected this leads to an approximation of the interacting-to-non-interacting mapping of the Kohn-Sham approach to DFT. Yet other choices are possible and allow to connect DMET with other DFTs such as kinetic-energy DFT or reduced density-matrix functional theory. In this work we give a detailed review of the basics of the DMET procedure from a DFT perspective and show how both approaches can be used to supplement each other. We do so explicitly for the case of a one-dimensional lattice system, as this is the simplest setting where we can apply DMET and the one that was originally presented. Among others we highlight how the mappings of DFTs can be used to identify uniquely defined auxiliary systems and auxiliary projections in DMET and how to construct approximations for different DFTs using DMET inspired projections. Such alternative approximation strategies become especially important for DFTs that are based on non-linearly coupled observables such as kinetic-energy DFT, where the Kohn-Sham fields are no longer simply obtainable by functional differentiation of an energy expression, or for reduced density-matrix functional theories, where a straightforward Kohn-Sham construction is not feasible.
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