Wannier functions are localized basis functions in condensed matter physics, offering an alternative to extended Bloch states. They bridge real-space and reciprocal-space descriptions of electronic states in periodic systems, providing insights into chemical bonding and local properties.

These functions are obtained by Fourier transforming Bloch states over the Brillouin zone. Their mathematical formulation involves complex transformations, considering symmetry and periodicity. Wannier functions possess unique properties like spatial localization and orthogonality, making them valuable in various applications.

Definition and concept

• Wannier functions serve as localized basis functions in condensed matter physics, providing an alternative representation to extended Bloch states
• These functions play a crucial role in understanding electronic structure and properties of crystalline solids
• Wannier functions bridge the gap between real-space and reciprocal-space descriptions of electronic states in periodic systems

Localized basis functions

• Wannier functions represent localized electronic states in crystalline solids
• Provide a real-space description of electronic wavefunctions in periodic systems
• Enable intuitive understanding of chemical bonding and local electronic properties
• Resemble atomic orbitals but account for the periodic potential of the crystal lattice

Relation to Bloch functions

• Wannier functions are linear combinations of Bloch functions
• Transform between extended Bloch states and localized Wannier functions through a unitary transformation
• Preserve the complete information contained in the Bloch functions
• Allow for a more intuitive interpretation of electronic structure in real space

Fourier transform of Bloch states

• Wannier functions obtained by Fourier transforming Bloch states over the Brillouin zone
• Mathematical expression: $w_n(\mathbf{r} - \mathbf{R}) = \frac{V}{(2\pi)^3} \int_{\text{BZ}} e^{-i\mathbf{k}\cdot\mathbf{R}} \psi_{n\mathbf{k}}(\mathbf{r}) d\mathbf{k}$
• Inverse transformation reconstructs Bloch functions from Wannier functions
• Provides a connection between reciprocal space (k-space) and real space representations

Mathematical formulation

• Mathematical framework of Wannier functions involves complex transformations and considerations of symmetry and periodicity
• Understanding the mathematical formulation is crucial for accurately constructing and utilizing Wannier functions in condensed matter physics
• Mathematical properties of Wannier functions determine their usefulness in various applications and numerical methods

Wannier function construction

• General form of Wannier functions: $w_n(\mathbf{r} - \mathbf{R}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{k}} e^{-i\mathbf{k}\cdot\mathbf{R}} \psi_{n\mathbf{k}}(\mathbf{r})$
• Involves a sum over all k-points in the Brillouin zone
• Requires careful choice of phase factors for Bloch functions
• Results in a set of localized functions centered at lattice sites R

Gauge freedom

• Wannier functions not uniquely defined due to gauge freedom in Bloch functions
• Bloch functions can be multiplied by a k-dependent phase factor without changing physical observables
• Gauge transformation: $\psi_{n\mathbf{k}}(\mathbf{r}) \rightarrow e^{i\phi_n(\mathbf{k})} \psi_{n\mathbf{k}}(\mathbf{r})$
• Different gauge choices lead to different sets of Wannier functions with varying degrees of localization

Periodic boundary conditions

• Wannier functions must satisfy periodic boundary conditions in the crystal lattice
• Periodicity requirement: $w_n(\mathbf{r} + \mathbf{R}) = w_n(\mathbf{r})$ for any lattice vector R
• Ensures consistency with the underlying periodic structure of the crystal
• Imposes constraints on the allowed forms of Wannier functions

Properties of Wannier functions

• Wannier functions possess unique properties that make them valuable tools in condensed matter physics
• These properties arise from their construction and relationship to Bloch functions
• Understanding these properties is essential for effectively utilizing Wannier functions in various applications

Spatial localization

• Wannier functions are spatially localized around specific lattice sites
• Decay exponentially with distance from their center in many cases
• Localization extent depends on the band structure and choice of gauge
• Enables efficient description of local electronic properties and interactions

Orthogonality

• Wannier functions form an orthonormal set of basis functions
• Orthogonality condition: $\langle w_n(\mathbf{r} - \mathbf{R}) | w_m(\mathbf{r} - \mathbf{R}') \rangle = \delta_{nm} \delta_{\mathbf{R}\mathbf{R}'}$
• Ensures linear independence and completeness of the Wannier basis
• Simplifies calculations and matrix elements in the Wannier representation

Symmetry considerations

• Wannier functions can be constructed to respect the symmetry of the crystal lattice
• Symmetry-adapted Wannier functions transform according to irreducible representations of the crystal's space group
• Symmetry properties aid in understanding and classifying electronic states
• Can be used to study topological properties and symmetry-protected phases of matter

Applications in condensed matter

• Wannier functions find extensive applications in various areas of condensed matter physics
• Their localized nature and connection to Bloch states make them powerful tools for analyzing electronic structure and properties
• Applications range from simple tight-binding models to complex topological materials

Tight-binding models

• Wannier functions serve as a natural basis for constructing tight-binding Hamiltonians
• Enable efficient description of electronic band structures in solids
• Hopping integrals between Wannier functions determine band dispersions
• Facilitate the study of electron correlation effects in strongly interacting systems

Chemical bonding analysis

• Wannier functions provide insights into the nature of chemical bonding in solids
• Center positions and spreads of Wannier functions reveal bonding characteristics
• Allow for identification of covalent, ionic, and metallic bonding patterns
• Aid in understanding the electronic origins of material properties

Topological insulators

• Wannier functions play a crucial role in characterizing topological insulators
• Obstruction to constructing exponentially localized Wannier functions indicates non-trivial topology
• Hybrid Wannier functions used to study edge states and bulk-boundary correspondence
• Enable calculation of topological invariants and classification of topological phases