What about U? -- Strong correlations from first principles
CECAM-HQ-EPFL, Lausanne, Switzerland
The first principles description of correlated materials (typically materials containing partially filled d- or f-shells) is currently regarded as one of the great challenges in condensed matter physics. Correlated materials, such as complex oxides, offer a rich variety of physical phenomena (e.g. high temperature superconductivity, metal-insulator transitions, and phenomena resulting from a complex entanglement of spin-, charge, orbital- and structural degrees of freedom) that continue to challenge our current understanding of quantum mechanics. Current and future applications are plentiful, ranging from oxide electronics, fuel cells, battery materials, homogeneous and heterogeneous catalysts, high Tc superconductors, spintronics devices to colossal magnetoresistant materials. Different approaches to tackle the theoretical description of strong correlations are currently being pursued in the different subfields of the electronic structure community. We envisage that a workshop dedicated to the first principles treatment of strong correlations (possibly evolving into a series of workshops) would provide the synergy necessary to shape future developments in this important field and would provide a unique opportunity to promote this emerging field in the European research landscape. The proposed workshop builds on three successful previous events:
• the "Strong Correlations from First Principles" workshop at Kloster Seeon in 2011
• the "Strong Correlations from First Principles" symposium at the March Meeting of the German Physical Society in Berlin in 2012
• the "What about U?" workshop in Lausanne in 2012
These workshops gathered the key players in the sub genres of (first principles) correlated physics and can be viewed as stock-taking. Many new and exciting developments have flourished from these events. Building on this ground work, we feel that it is time now to consolidate the efforts and to use the momentum of the previous events to channel progress into new, interdisciplinary directions.
The term “correlated” refers to systems or phenomena in which the single-particle picture fails to describe the electronic properties. This happens typically in materials with open d- or f-shells, but also occurs in sp systems such as, for instance, low-dimensional organic materials, or during molecular dissociations. The simultaneous presence of localized electrons and itinerant band states, structural, orbital and spin degrees of freedom gives rise to a rich variety of physics and chemistry that makes these materials attractive for a wide range of applications. This richness, combined with the considerable challenges posed to our (theoretical) understanding of their physics, has made strongly correlated systems a hot and fascinating topic.
Although, by definition, a one-particle picture is not able to capture correlation effects (due to many-body interactions), an explicit reference to the one-particle (band) picture is needed in practice. The failure of density functional theory (DFT) in the local-density (LDA) or generalized gradient (GGA) approximations, that are based on an effective one-particle picture (with charge- and charge-gradient-dependent interactions), is generally considered as an indication for correlated physics. It is known, however, that the assessment of the importance of electronic correlations depends on the quantities considered. If only ground state properties are of interest, DFT teaches us that those are accessible within an (effective) one-particle picture, and that LDA or GGA are often remarkably good, e.g. for determining lattice constants, bond lengths, phase stability etc. . Extensions like the addition of a partially screened Coulomb interaction ("Hubbard U") in DFT+U or hybrid functionals often further improve the description of the ground state of correlated systems. In reality, however, the interesting regions in the phase diagram of strongly correlated systems are usually at finite temperature (and possibly finite doping), which immediately illustrates some of the challenges for first principles theories. If the observable of interest is the spectral function of the system (as e.g. measured by direct or inverse photoemission) or other excited state properties, one clearly has to go beyond the effective Kohn-Sham picture of DFT.
A major deficiency of LDA and GGA is the delocalization (or self-interaction) error , which is particularly severe for systems with partially occupied d- or f-states. Hybrid functionals, on the other hand, partly correct the self-interaction error by incorporating a certain portion of exact exchange, which significantly improves the descriptions of d- or f-electron systems [3,4]. The dependence on adjustable parameters, however, remains a concern. A parameter free ground state approach beyond LDA/GGA is to combine exact exchange with correlation in the random-phase approximation (RPA) [5,6]. This has the additional advantage that many-body perturbation theory in the GW approach (see below) gives a spectral function that is consistent with the RPA energy . The application of RPA to paradigmatic strong correlation problems like the alpha-gamma transition in Cerium  or bond dissociation  are emerging and RPA is found to perform well.
Conversely, correlation effects that govern e.g. the spectroscopy of partially filled d- or f-shells in transition metal oxides or f-electron materials can in principle be treated systematically by dynamical mean field theory (DMFT) . The metal-insulator transitions in V2O3 , the alpha-gamma transition in Cerium [11, 12], or the paramagnetic (Mott-) insulating phase of Ce2O3  are classical examples of successful DMFT applications. Recent years have also seen applications to new materials classes, such as to the celebrated iron pnictides , or to spin-orbit materials , where the interplay of correlations and spin-orbit coupling leads to new exotic states of matter. On the methodological side, several issues have been resolved recently, the most important ones probably being the accurate determination of many-body (Hubbard U) corrections from first principles, the treatment of dynamical screening effects (“dynamical Hubbard U”) [16,17], and the simultaneous correction of strongly correlated (d- or f-) states by DMFT and itinerant (eg. s- or p-) states by weak-coupling corrections beyond DFT-LDA. A great advantage of DMFT is that it is intrinsically a finite-temperature formalism.
Many-body perturbation theory (MBPT) provides an alternative approach for systems with moderate electronic correlations. Hedin's GW approximation to the many-body self-energy  is a step towards a systematic ab initio understanding of such systems. The GW approach corresponds to the first order term of a systematic expansion in MBPT and has become the method of choice for the description of quasiparticle band structures in moderately correlated solids . Through the screened Coulomb interaction W it captures the screening among itinerant electrons while treating exchange at the exact exchange level at the same time. The latter should account for a large part of the interactions among localized d- or f-electrons, with the additional advantage that localized and itinerant states are treated on the same footing. The first promising calculations to open-shell transition metal oxides or lanthanide compounds [20,21,22], actinides  and even correlated phenomena like the Kondo resonance in quantum transport  are emerging, but are -- at least in the strong coupling limit of f-electron compounds -- limited to the magnetic phases. The description of paramagnetic phases and finite temperatures remains beyond the scope of perturbative approaches.
A very promising direction is to combine DMFT and GW, such as to retain the best of both worlds . This area has seen tremendous progress recently [26, 27], after the bottleneck of how to handle dynamical screening effects within the DMFT equations has been solved (both, by the development of accurate approximate impurity solvers , as well as efficient Monte Carlo techniques ), and the first fully dynamical GW+DMFT calculations for real materials have recently been published [26,27].
A further recent, but very active field is currently emerging from the interface between dynamical mean field theory and quantum chemistry approaches , both by using quantum chemistry techniques as solvers for the DMFT equations and by importing dynamical mean field ideas into the field of quantum chemistry. Quantum chemistry approaches, based on solving a simplified Schrodinger equation, within some approximations/assumptions on the structure of the many-body wave function of the system and on the resulting electronic interactions, represent in themselves a very active field of research in the study of correlated systems. Prominent examples are complete active space self-consistent field (CASSCF) and complete active space perturbation theory to second order (CASPT2) [30,31]. Although the numerical cost of these methods makes their use prohibitive for all but the smallest (atomic or molecular) systems, developing a more rigorous theoretical connection between their wave-function-based formulation and DFT would be inspirational and beneficial for both classes of numerical approaches. In particular it would facilitate the definition of more flexible and accurate exchange-correlation functionals, able to render the many body features of certain electronic states more accurately. Very recently, a stochastic sampling of the configuration interaction (CI) space has become available also for solids . The full CI space would give the exact many-body wave function, but grows exponentially with the number of electrons and basis functions. The stochastic description makes CI computationally tractable and provides benchmarks for many-body methods .
Other methods, being developed to alleviate or eliminate the deficiencies of typical approximate DFT functionals in the description of correlated systems, are based on using more complex (yet tractable) quantities than density (as, e.g., Greens functions or the one-body density matrix ) as variables of quantum mechanics equations to be solved in the computer. Due to the larger amount of information these quantities carry, these methods naturally offer a more precise and flexible way to represent the quantum many body features of the electronic ground state of correlated materials, and disclose a built-in capability to treat both ground state and excited state properties on the same theoretical ground.
Last but not least, let us mention Monte Carlo techniques addressing electronic structure problems either directly, or after construction of appropriate lattice models. Developing (or refining) a precise interface of these methods with DFT is very important. On the one hand, the possible calculation of the effective electronic interactions in the model Hamiltonians and the initial guess of the electronic wave functions from DFT can certainly contribute to making these methods more quantitative. On the other hand, the use of these models in parallel with DFT would be very useful, not only for the interpretation of DFT results, but also to design more flexible and accurate functionals able to capture many-body features of the electronic structure of correlated systems. DFT+DMFT and its current developments are a clear example of how successful this strategy can be.
Patrick Rinke ( Aalto University, Helsinki ) - Organiser
Silke Biermann ( Ecole Polytechnique ) - Organiser
Stefano de Gironcoli ( Scuola Internazionale Superiore di Studi Avanzati - International School for Advanced Studies ) - Organiser
Matteo Cococcioni ( Swiss Federal Institute of Technology, Lausanne ) - Organiser