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arXiv:1208.4703 · Physical Review A

Reduced-density-matrix-functional theory at finite temperature: Theoretical foundations

Tim Baldsiefen, Attila Cangi, Eberhard K. U. Gross

Published Nov 20, 2015• 36 citations
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Abstract

We present an ab initio approach for grand-canonical ensembles in thermal equilibrium (eq) with local or nonlocal external potentials based on the one-reduced density matrix (1RDM). We show that equilibrium properties of a grand-canonical ensemble are determined uniquely by the eq-1RDM and establish a variational principle for the grand potential with respect to its 1RDM. We further prove the existence of a Kohn-Sham system capable of reproducing the 1RDM of an interacting system at finite temperature. Utilizing this Kohn-Sham system as an unperturbed system, we deduce a many-body approach to iteratively construct approximations to the correlation contribution of the grand potential.

DOI
10.1103/physreva.92.052514

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We present an ab initio approach for grand-canonical ensembles in thermal equilibrium (eq) with local or nonlocal external potentials based on the one-reduced density matrix (1RDM). We show that equilibrium properties of a grand-canonical ensemble are determined uniquely by the eq-1RDM and establish a variational principle for the grand potential with respect to its 1RDM. We further prove the existence of a Kohn-Sham system capable of reproducing the 1RDM of an interacting system at finite temperature. Utilizing this Kohn-Sham system as an unperturbed system, we deduce a many-body approach to iteratively construct approximations to the correlation contribution of the grand potential.
We develop a systematic approach to construct energy functionals of the one-particle reduced density matrix (1RDM) for equilibrium systems at finite temperature. The starting point of our formulation is the grand potential Ω [ G ] regarded as variational functional of the Green’s function G based on diagrammatic many-body perturbation theory and for which we consider either the Klein or Luttinger–Ward form. By restricting the input Green’s function to be one-to-one related to a set on one-particle reduced density matrices (1RDM) this functional becomes a functional of the 1RDM. To establish the one-to-one mapping we use that, at any finite temperature and for a given 1RDM γ in a finite basis, there exists a non-interacting system with a spatially non-local potential v [ γ ] which reproduces the given 1RDM. The corresponding set of non-interacting Green’s functions defines the variational domain of the functional Ω . In the zero temperature limit we obtain an energy functional E [ γ ] which by minimisation yields an approximate ground state 1RDM and energy. As an application of the formalism we use the Klein and Luttinger–Ward functionals in the GW-approximation to compute the binding curve of a model hydrogen molecule using an extended Hubbard Hamiltonian. We compare further to the case in which we evaluate the functionals on a Hartree–Fock and a Kohn–Sham Green’s function. We find that the Luttinger–Ward version of the functionals performs the best and is able to reproduce energies close to the GW energy which corresponds to the stationary point.
We establish one-body reduced density matrix (1RDM) functional theory for the canonical ensemble in a finite basis set at an elevated temperature. Including temperature guarantees the differentiability of the universal functional by occupying all states and additionally not fully occupying the states in a fermionic system. We use the convexity of the universal functional and invertibility of the potential-to-1RDM map to show that the subgradient contains only one element which is equivalent to differentiability. This allows us to show that all 1RDMs with a purely fractional occupation number spectrum ($0<{n}_{i}<1\phantom{\rule{0.28em}{0ex}}{\ensuremath{\forall}}_{i}$) are uniquely vvv-representable up to a constant.
We present an ab initio theory for superconductors, based on a unique mapping between the statistical density operator at equilibrium, on the one hand, and the corresponding one-body reduced density matrix $\ensuremath{\gamma}$ and the anomalous density $\ensuremath{\chi}$, on the other. This formalism for superconductivity yields the existence of a universal functional ${\mathfrak{F}}_{\ensuremath{\beta}}[\ensuremath{\gamma},\ensuremath{\chi}]$ for the superconductor ground state, whose unique properties we derive. We then prove the existence of a Kohn-Sham system at finite temperature and derive the corresponding Bogoliubov--de Gennes--type single-particle equations. By adapting the decoupling approximation from density functional theory for superconductors we bring these equations into a computationally feasible form. Finally, we use the existence of the Kohn-Sham system to extend the Sham-Schl\"uter connection and derive a first exchange-correlation functional for our theory. This reduced density matrix functional theory for superconductors has the potential of overcoming some of the shortcomings and fundamental limitations of density functional theory of superconductivity.
A system of electrons in a local or nonlocal external potential can be described with one-matrix functional theory (1MFT), which is similar to density-functional theory (DFT) but takes the one-particle reduced density matrix (one-matrix) instead of the density as its basic variable. Within 1MFT, Gilbert derived [Phys. Rev. B 12, 2111 (1975)] effective single-particle equations analogous to the Kohn-Sham (KS) equations in DFT. The self-consistent solution of these 1MFT-KS equations reproduces not only the density of the original electron system but also its one-matrix. While in DFT it is usually possible to reproduce the density using KS orbitals with integer (0 or 1) occupancy, in 1MFT reproducing the one-matrix requires in general fractional occupancies. The variational principle implies that the KS eigenvalues of all fractionally occupied orbitals must collapse at self-consistency to a single level. We show that as a consequence of the degeneracy, the iteration of the KS equations is intrinsically divergent. Fortunately, the level-shifting method, commonly introduced in Hartree-Fock calculations, is always able to force convergence. We introduce an alternative derivation of the 1MFT-KS equations that allows control of the eigenvalue collapse by constraining the occupancies. As an explicit example, we apply the 1MFT-KS scheme to calculate the ground state one-matrix of an exactly solvable two-site Hubbard model.
In parallel with standard density-functional theory, we study the energy of the ground state of a finite many-body system as a functional of the one-particle density matrix. We show that the formulation of a variational principle that is valid beyond the Hartree-Fock limit requires that two-body correlations be included not only in the ground-state energy but also in the constraints. As an illustration, we apply a linear-response argument to derive formulas for first-order corrections to the Hartree-Fock density matrix. Further analysis suggests an approach in terms of the density matrix of an independent-particle system, which can be introduced by the application of an alternative variational principle. This approach is reminiscent of Kohn-Sham theory, but the effective external potential is not required to be local. This variational method can be implemented in a systematic fashion by means of the linked-cluster expansion. In an appendix we study a variant of the Hohenberg-Kohn theorem for nonlocal potentials.
Accurately describing chemical bond breaking requires capturing strong/static electron correlation, a major challenge for single-reference self-consistent-field methods such as Hartree--Fock and Kohn--Sham density-functional theory (DFT). Fractional occupations substantially improve this situation, as demonstrated by information-density-matrix-functional theory (i-DMFT) and thermally-assisted-occupation DFT (TAO-DFT). Although useful, they are formulated within rather different theoretical frameworks, are somewhat semi-empirical, and their relationship remains unclear. Here we introduce finite-temperature adiabatic-connection reduced density-matrix functional theory (FT-AC-RDMFT), in which i-DMFT and TAO-DFT emerge as different reductions of a unified framework. The effective eigenvalue equation for natural orbitals and the Fermi--Dirac occupation numbers follow directly from the variational principle, while a thermal adiabatic connection relates the empirical entropy coefficient of i-DMFT to a characteristic temperature of the auxiliary non-interacting system. Within this framework, i-DMFT is the exact-exchange-based reduction, whereas TAO-DFT results when exact exchange is replaced by a density-based exchange-correlation functional. Numerical calculations confirm that i-DMFT, apart from a constant, coincides with the exact-exchange realization of the framework, and show that the improved dissociation behavior from finite-temperature occupations extends beyond exact exchange to hybrid and semilocal density functionals. For strongly correlated V2O3, finite-temperature occupations substantially reduce the severely overestimated non-magnetic LDA bulk modulus, bringing it close to experiment. These results establish FT-AC-RDMFT as a common theoretical foundation for finite-temperature occupation-based electronic-structure methods in molecular and solid-state systems.