Turning reduced density matrix theory into a practical tool for studying the Mott transition

Properties of atoms, molecules, and solids arise as consequences of the physical behavior of electrons in the electrical fields of nuclei. Solving exactly a fundamental many-electron Schrödinger equation becomes a very complex and computationally intractable problem for situations involving more than two electrons. Luckily, for many systems an independent electron picture, within which a single electron is considered in an effective field, serves as a good approximation. A much better description of electron systems is provided by density functional theory-based approaches, since they account for electron correlation. Approximate density functionals have been the workhorse of computational materials science and computational chemistry for many years. Unfortunately, there is no systematic way of improving density functionals, and most of the available functionals are likely to yield incorrect predictions for systems whose electron structure is far from the independent particle picture. Such systems are called strongly correlated. Although there is no standard definition of strong correlation a common understanding is that it occurs when valence electrons tend to 'feel' each other, Coulomb electron repulsion takes over the kinetic energy, and, as a result, electrons tend to localize in space instead of being delocalized over the entire molecule or crystal lattice. Thus, in the limit of strong correlation a classical picture of electrons is recovered. Thus, it is not surprising that approximate many-electron methods, having at their base a model of a delocalized electron, are likely to fail when strong correlation effects come into play.

Systems with strongly correlated electrons are neither rare nor exotic in chemistry and physics. For example, in chemistry the problem of a covalent bond breaking—a fundamental process in chemical reactions—is governed by strong correlation of electrons, and the one-electron picture breaks down in the dissociation limit [2]. In solids, materials with open d and f shells, for example transition metals and their oxides, have properties, that cannot be explained within a standard band theory. The (strong) correlation of d,f electrons gives rise to a rich set of fascinating physical phenomena [3]. To investigate strongly correlated systems one has to go beyond an independent electron picture. Borrowing from quantum chemistry and using as a main quantity a correlated many-electron wave function for the whole solid is an enormously complex task, although some initial successful examples of such efforts have been reported [4]. On the other hand, density functional methods employing local or semilocal functionals are able to capture only a short-range dynamical electron correlation; they miss a long-range static correlation which is crucial for proper description of strong correlations. Between wave function-based methods and density functional theory there is another approach—reduced density matrix functional theory (RDMFT) [5, 6]—which seems to be particularly well suited for treating strongly correlated materials. RDMFT employs a one-electron reduced density matrix (1-RDM) as its main variable, and the properties of systems are derived from a ground state 1-RDM functional. Given access to an interacting 1-RDM, strong correlation can be immediately detected by the appearance of eigenvalues of 1-RDM (natural occupation numbers) significantly deviating from the integer values 0 and 1, which characterize uncorrelated electrons. RDMFT relies on approximation to the electron repulsion functional, and recently a number of approximations have been proposed [7–10].

It has already been shown by Sharma et al [11] that one of the 1-RDM functionals,the power functional [8], correctly predicts that transition metal oxides are Mott insulators. The proposed approach to obtaining the density of states within RDMFT has led to correct photoelectron spectra of MnO, a known test case Mott insulator. Moreover, the predicted spectra displayed an insulator-to-metal phase transition under volume reduction of the system. This is the first evidence that the RDMFT method can successfully compete with more computationally demanding many-body methods in describing strongly correlated systems undergoing phase transitions.

In a recent paper of Shinohara et al [1] RDMFT is put on even firmer footing as a practical tool for studying Mott insulators. A transition from insulator to metal phase of NiO subject to doping is studied by an RDMFT method that employs a power functional. Its performance is compared with that of density functional approaches: local spin density approximation (LSDA) [12] and a local functional corrected with the on-site Coulomb repulsion term (LSDA+U) [13]. It is shown that neither LSDA nor LSDA+U is able to capture the physics of the studied transition. Density of states spectra predicted by both methods show a mere shift of the chemical potential without reducing the gap between Hubbard bands. For RDMFT, on the other hand, encouraging performance reported in [11] is confirmed: the method yields accurate quantitative predictions together with the correct physical picture for the Mott transition. Namely, it has been found not only that RDMFT spectra for NiO are in good agreement with the experimental data but also that upon doping the spectra undergo qualitative changes—spectral weights are redistributed towards lower bands—in agreement with more computationally demanding many-body theories. Another interesting finding, confirming that the good performance of RDMFT is not incidental but the method also yields physical insight into the studied system, is confirmation of the importance of the charge transfer effects accompanying the localization of orbitals in strongly correlated NiO.

Strongly correlated materials are not just physical curiosities or test fields for new theories anymore. They are becoming a realm of materials science, and in the near future they may find applications as high-temperature superconductors, or photovoltaic [14] or thermochromic materials [15]. So there is a growing demand for computational methods useful in designing such systems and studying their properties. The paper of Shinohara et al [1] is an important step toward providing a new tool for this purpose. This is still an initial work and more applications of the proposed reduced density matrix functional method are eagerly awaited.