CECAM - Maximally Localized Wannier Functions: Concepts, Applications, and BeyondMaximally Localized Wannier Functions: Concepts, Applications, and Beyond

The electronic structure of periodic systems is usually described in terms of the energy eigenfunctions, the Bloch functions, which are also eigenstates of the translation operator. However, these functions are spatially delocalized and offer a description which is not very intuitive. An alternative description of the electronic structure is provided by Maximally Localized Wannier Functions (MLWFs), which favor a representation of the electronic structure in terms of functions localized in real space [1,2]. These functions are obtained by unitary transformation from the energy eigenfunctions and therefore provide a fully equivalent description. Through their center, their extension and their shape, MLWFs provide a description of the electronic structure which is physically insightful. The electron orbitals can be visualized and located with respect to the positions of the atoms. Early investigations on complex crystals [2] and on disordered solids [3] and liquids [4] immediately illustrated the strength of this new concept. Essentially, a MLWF provides information on the local electronic structure in a straightforward and compact form. Maximally localized Wannier functions are experiencing an increasing success as a monitoring tool of the electronic structure within both the electronic structure [5-9] and ab-initio molecular dynamics communities [10-12]. However, more recently, the particular properties of MLWFs have also attracted interest as a primary methodological tool within more complex theoretical developments. The most interesting aspect of MLWFs is that they allow a decomposition of the electronic structure in local components [1,2]. This property naturally allows one to locally monitor the response of the electronic structure when the investigated system is perturbed, either by the application of external (electric or magnetic) fields or by structural modification. In the study of the polarization, the usefulness of MLWFs is further enhanced by the fact that the displacement of their centers naturally connects with the modern definition of the polarization [1,2]. Hence MLWFs have become the method of choice to investigate the dielectric response in insulator-insulator [13-16] or insulator-metal interfaces [17]. Similarly, MLWFs have shown their usefulness in the study of the response of the electronic structure to a magnetic field [18-20]. These functions have been used in the calculation of NMR shifts [18], but have also turned out to be valuable in establishing the correct definition of the orbital magnetization in periodic systems [19,20]. Recent work on the interpretation of photoemission shifts at semiconductor surfaces and interfaces relies on MLWFs to highlight the role of strain fields [21]. Another exploited property of a set of MLWFs is that they constitute a complete basis set for the occupied states in which hierarchy can be established on the basis of distance to a particular point in space. This property is particularly useful for the introduction of approximations in which role of the local electronic structure is prioritized. Such basis functions have found a natural application in the development of methodologies for large-scale electronic calculations [22,23]. Recent developments also concern the research area of strongly correlated electrons, where MLWFs have recently been introduced as preferred functions for describing the local correlation [24-31]. Maximally localized Wannier functions have also found application in the construction of simplified Hamiltonians for multi-scale approaches. In this case, one uses the property that MLWFs can represent a particular energy window. This is straightforward when the relevant electronic bands are well separated from the other ones, but recent developments have extended such applications also to coupled bands introducing purposely designed disentanglement procedures [32-35]. In particular, these methodological developments are at the basis of several successful applications in the context of transport in nano-sized structures [36-38]. Although most of the early applications of MLWFs were to insulating systems, very recently the usefulness of partially-occupied MLWFs as fast and accurate interpolators for describing Fermi-surface properties of metals has been explored. For example, this has been used to reduce dramatically the computational load needed to evaluate properties such as the anomalous Hall effect of ferromagnets [39] and the electron-phonon coupling [40]. A further recent extension has shown that generalized Wannier functions can be constructed which show small spreads both in space and in energy [41]. The raise of interest in MLWFs has also prompted theoretical work in order to complete the characterization of these functions. Very recent work has been able to demonstrate the exponential nature of the localization [42], extending to three dimensions previous work for one dimension [43]. Furthermore, the real nature of MLWFs, empirically observed in many applications, could be formally demonstrated [42]. The wide-spread use of MLWFs in the ab initio simulation community has also led to the development of molecular dynamics integrators which preserve the localization properties during the course of the evolution [44-46]. A computer module for the generation of MLWFs is freely available [47] and facilitates their wide-spread use across the electronic-structure community. The success of MLWFs has brought several major software packages, such as DACAPO, CPMD and QUANTUM-ESPRESSO, to offer to their users a module for their generation. From the technical point of view, there have been important evolutions in the algorithms that are used to generate MLWFs [48-51]. Finally, it should be mentioned that the concept of MLWF has acquired generality giving rise to applications which extend beyond the description of the electronic structure. For instance, the concept has been used in the context of photonic crystals [52] and recent work shows how generalized Wannier functions can be used to study phonon Hamiltonians [41,40]. As clearly illustrated by this evolution, the use of MLWFs is rapidly expanding across diverse research areas. One of the primary aims of the present workshop is to bring together the major players in these different disciplines in order to share their experiences. These will include researchers in the electronic structure and ab initio simulation community, in physics and in chemistry. Researchers that are primarily interested in new developments and researchers that are more concerned with practical applications will be both represented. While the research areas are diverse, the optimal integration of MLWFs in broader schemes necessarily will face difficulties which are to a large extent common to all concerned research areas. The workshop represents a forum where researchers from various fields meet to expose their views, discuss their ideas, and confront their solutions. Furthermore the interaction between development and application will be beneficial to both parties leading to more focused developments on the one side, and to richer applications on the other. The planned workshop is the very first one of this nature. The impact of the workshop is expected to be twofold. First, at this time it is not clear whether all the major researchers that have been identified in this proposal are fully aware of all the developments that are occurring in parallel in other research areas. The workshop will therefore contribute in removing potential communication barriers. This will be beneficial to several research fields. Second, the workshop will promote the interaction between development and application. Eventually this will lead to the optimal exploitation of the concept of MLWFs in various contexts.

Maximally Localized Wannier Functions: Concepts, Applications, and Beyond

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