Group theory of Wannier functions providing the basis for a deeper understanding of magnetism and superconductivity

TL;DR

This paper develops a group theory framework for symmetry-adapted Wannier functions in crystals, enabling analysis of their localization and symmetry properties to predict magnetic and superconducting states without relying on specific physical models.

Contribution

It provides a comprehensive group theoretical method for analyzing Wannier functions' symmetry and localization, applicable to any crystal symmetry group, and links these properties to magnetic and superconducting phenomena.

Findings

Determines when Bloch functions can be transformed into localized Wannier functions

Connects Wannier function symmetry to magnetic and superconducting states

Offers a model-independent analysis based on symmetry considerations

Abstract

The paper presents the group theory of best localized and symmetry-adapted Wannier functions in a crystal of any given space group G or magnetic group M. Provided that the calculated band structure of the considered material is given and that the symmetry of the Bloch functions at all the points of symmetry in the Brillouin zone is known, the paper details whether or not the Bloch functions of particular energy bands can be unitarily transformed into best localized Wannier functions symmetry-adapted to the space group G, to the magnetic group M, or to a subgroup of G or M. In this context, the paper considers usual as well as spin-dependent Wannier functions, the latter representing the most general definition of Wannier functions. The presented group theory is a review of the theory published by one of the authors in several former papers and is independent of any physical model of magnetism or superconductivity. However, it is suggested to interpret the special symmetry of the best localized Wannier functions in the framework of a nonadiabatic extension of the Heisenberg model, the nonadiabatic Heisenberg model. On the basis of the symmetry of the Wannier functions, this model of strongly correlated localized electrons makes clear predictions whether or not the system can possess superconducting or magnetic eigenstates.