The Iterated Projected Position Algorithm for Constructing Exponentially Localized Generalized Wannier Functions for Periodic and Non-Periodic Insulators in Two Dimensions and Higher

Abstract

Localized bases play an important role in understanding electronic structure. In periodic insulators, a natural choice of localized basis is given by the Wannier functions which depend a choice of unitary transform known as a gauge transformation. Over the past few decades, there have been many works which have focused on optimizing the choice of gauge so that the corresponding Wannier functions are maximally localized or reflect some symmetry of the underlying system. In this work, we consider fully non-periodic materials where the usual Wannier functions are not well defined and gauge optimization is impossible. To tackle the problem of calculating exponentially localized generalized Wannier functions in both periodic and non-periodic system we discuss the "Iterated Projected Position (IPP)" algorithm. The IPP algorithm is based on matrix diagonalization and therefore unlike optimization based approaches it does not require initialization and cannot get stuck at a local minimum. Furthermore, the IPP algorithm is guaranteed by a rigorous analysis to produce exponentially localized functions under certain mild assumptions. We numerically demonstrate that the IPP algorithm can be used to calculate exponentially localized bases for the Haldane model, the Kane-Mele model (in both ${\mathbb{Z}}_2$ invariant even and ${\mathbb{Z}}_2$ invariant odd phases), and the $p_x+i{p}_y$ model on a quasi-crystal lattice.

arXiv comment19 pages, 26 figures

References in corpus (23)

• Quantum Spin Hall Effect in Graphene
• Maximally-localized generalized Wannier functions for composite energy bands
• Wannier90: A Tool for Obtaining Maximally-Localised Wannier Functions
• Maximally localized Wannier functions: Theory and applications
• Wannier90 as a community code: new features and applications
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• The Quantum-Mechanical Position Operator in Extended Systems
• Computing topological invariants without inversion symmetry
• Wannier representation of Z_2 topological insulators
• Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials
• Exponential localization of Wannier functions in insulators
• Maximally-localized Wannier functions for disordered systems: application to amorphous silicon
• Band Structure and Quantum Conductance of Nanostructures from Maximally-Localized Wannier Functions: The Case of Functionalized Carbon Nanotubes
• Triviality of Bloch and Bloch-Dirac bundles
• Aperiodic Weak Topological Superconductors
• Accurate polarization within a unified Wannier function formalism
• Optimal decay of Wannier functions in Chern and Quantum Hall insulators
• Automated construction of maximally localized Wannier functions: Optimized projection functions method
• ${\mathbb{Z}}_2$ invariants of topological insulators as geometric obstructions
• Robust determination of maximally-localized Wannier functions
• Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators
• On the construction of composite Wannier functions
• Generalized Wannier Functions

Cited by in corpus (9)

• The Wannier Function Software Ecosystem for Materials Simulations
• Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher
• Localization of generalized Wannier bases implies Chern triviality in non-periodic insulators
• Localised module frames and Wannier bases from groupoid Morita equivalences
• Algebraic localization implies exponential localization in non-periodic insulators
• Ultra-generalized Wannier bases: Are they relevant to topological transport?
• Approximating Maximally Localized Wannier Functions with Position Scaling-Eigenfunction
• Structural Dynamics and Strong Correlations in Dynamical Quantum Optical Lattices
• Shaping Maximally Localized Wannier Functions via Discrete Adiabatic Transport