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arXiv:2202.08071 · Computation

Local Potential Functional Embedding Theory: A Self-Consistent Flavor of Density Functional Theory for Lattices without Density Functionals

Sajanthan Sekaran, Matthieu Saubanère, Emmanuel Fromager

Published Mar 18, 2022• 13 citations
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Abstract

Quantum embedding is a divide and conquer strategy that aims at solving the electronic Schrödinger equation of sizeable molecules or extended systems. We establish in the present work a clearer and in-principle-exact connection between density matrix embedding theory (DMET) and density-functional theory (DFT) within the simple but nontrivial one-dimensional Hubbard model. For that purpose, we use our recent reformulation of single-impurity DMET as a Householder transformed density-matrix functional embedding theory (Ht-DMFET). On the basis of well-identified density-functional approximations, a self-consistent local potential functional embedding theory (LPFET) is formulated and implemented. Combining both LPFET and DMET numerical results with our formally exact density-functional embedding theory reveals that a single statically embedded impurity can in principle describe the density-driven Mott–Hubbard transition, provided that a complementary density-functional correlation potential (which is neglected in both DMET and LPFET) exhibits a derivative discontinuity (DD) at half filling. The extension of LPFET to multiple impurities (which would enable to circumvent the modeling of DDs) and its generalization to quantum chemical Hamiltonians are left for future work.

DOI
10.3390/computation10030045

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Quantum embedding is a divide and conquer strategy that aims at solving the electronic Schrödinger equation of sizeable molecules or extended systems. We establish in the present work a clearer and in-principle-exact connection between density matrix embedding theory (DMET) and density-functional theory (DFT) within the simple but nontrivial one-dimensional Hubbard model. For that purpose, we use our recent reformulation of single-impurity DMET as a Householder transformed density-matrix functional embedding theory (Ht-DMFET). On the basis of well-identified density-functional approximations, a self-consistent local potential functional embedding theory (LPFET) is formulated and implemented. Combining both LPFET and DMET numerical results with our formally exact density-functional embedding theory reveals that a single statically embedded impurity can in principle describe the density-driven Mott–Hubbard transition, provided that a complementary density-functional correlation potential (which is neglected in both DMET and LPFET) exhibits a derivative discontinuity (DD) at half filling. The extension of LPFET to multiple impurities (which would enable to circumvent the modeling of DDs) and its generalization to quantum chemical Hamiltonians are left for future work.
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 noninteracting 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\mathrm{impurity}+\mathrm{bath}impurity+bath'' cluster's 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 noninteracting 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 neither described 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.
We introduce density matrix embedding theory (DMET), a quantum embedding theory for computing frequency-independent quantities, such as ground-state properties, of infinite systems. Like dynamical mean-field theory, DMET maps the bulk interacting system to a simpler impurity model and is exact in the noninteracting and atomic limits. Unlike dynamical mean-field theory, DMET is formulated in terms of the frequency-independent local density matrix, rather than the local Green's function. In addition, it features a finite, algebraically constructible bath of only one bath site per impurity site, with no bath discretization error. Frequency independence and the minimal bath make DMET a computationally simple and efficient method. We test the theory in the one-dimensional and two-dimensional Hubbard models at and away from half filling, and we find that compared to benchmark data, total energies, correlation functions, and metal-insulator transitions are well reproduced, at a tiny computational cost.
The equivalence in one-electron quantum baths between the practical implementation of density matrix embedding theory (DMET) and the more recent Householder-transformed density matrix functional embedding theory has been shown previously in the standard but special case where the reference full-size (one-electron reduced) density matrix, from which the bath is constructed, is idempotent [S. Yalouz et al., J. Chem. Phys. 157, 214112 (2022)]. We prove mathematically that the equivalence remains valid when the density matrix is not idempotent anymore, thus allowing for the construction of correlated (one-electron) quantum baths. A density-matrix functional exactification of DMET is derived within the present unified quantum embedding formalism. Numerical examples reveal that the embedding cluster can be quite sensitive to the level of density-matrix functional approximation used for computing the reference density matrix.
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 noninteracting) one. It provides a rigorous framework for combining wave-function (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 wave function 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.
Site-occupation embedding theory (SOET) is a density functional theory (DFT)-based method which aims at modelling strongly correlated electrons. It is in principle exact and applicable to model and quantum chemical Hamiltonians. The theory is presented here for the Hubbard Hamiltonian. In contrast to conventional DFT approaches, the site (or orbital) occupations are deduced in SOET from a partially interacting system consisting of one (or more) impurity site(s) and non-interacting bath sites. The correlation energy of the bath is then treated implicitly by means of a site-occupation functional. In this work, we propose a simple impurity-occupation functional approximation based on the two-level (2L) Hubbard model which is referred to as two-level impurity local density approximation (2L-ILDA). Results obtained on a prototypical uniform eight-site Hubbard ring are promising. The extension of the method to larger systems and more sophisticated model Hamiltonians is currently in progress.
Density matrix embedding theory (DMET) is a quantum embedding theory for strongly correlated systems. From a computational perspective, one bottleneck in DMET is the optimization of the correlation potential to achieve self-consistency, especially for heterogeneous systems of large size. We propose a new method, called projected DMET (p-DMET), which achieves self-consistency without needing to optimize the correlation potential. We demonstrate the performance of p-DMET on the two-dimensional Hubbard model.