Group Theory of Wannier Functions Providing the Basis for a Deeper Understanding of Magnetism and Superconductivity

Abstract

Abstract:The paper presents the group theory of optimally-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 of 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 optimally-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 (Ekkehard Krüger) 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 optimally-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 of whether or not the system can possess superconducting or magnetic eigenstates.

Structural Distortion Stabilizing the Antiferromagnetic and Semiconducting Ground State of BaMn2As2

“…5.2) is a complex process described in Ref. [9] (see Theorems 5 and 7 ibidem) which is also performed by a C++ computer program. The notes to Table A5 give short remarks on the (symmetry) properties of the Wannier functions.…”

k-Space Magnetism as the Origin of Superconductivity

“…In this case, superconducting bands are single bands [5]. Furthermore, we assume that this metal possesses a narrow, half-filled superconducting band in its band structure.…”

“…Furthermore, we assume that this metal possesses a narrow, half-filled superconducting band in its band structure. By definition we can unitarily transform the Bloch functions of this band into optimally localized and symmetry-adapted spin-dependent Wannier functions [5]. We do this by replacing the Bloch functions ϕ k (r) of the superconducting band by Bloch spinors…”

“…(To simplify, we ignore that in some points of symmetry the Bloch spinors may not be written in the form of Equation (7) [5].) The coefficients f ms (k) in Equation (7) form a k dependent two-dimensional matrix…”

“…Hence, it is generally not possible to find narrow, roughly half-filled closed energy bands in the band structures of the metals as they are required for the construction of optimally localized Wannier functions. However, in the band structures of those metals that experimentally prove to be superconductors, the construction of such Wannier functions becomes possible if we allow the Wannier functions to be spin dependent [5]. This observation leads to the definition of "superconducting bands" [5]: The Bloch functions of a superconducting band can be unitarily transformed into optimally localized spin-dependent Wannier functions that are symmetry-adapted to the full space group of the metal.…”