# On the construction of Wannier functions in topological insulators: the 3D case

## TL;DR

This paper explores the construction of localized Wannier functions in 3D topological insulators with time-reversal symmetry, identifying topological invariants and providing an algorithm for basis construction when invariants vanish.

## Contribution

It introduces a method to construct Wannier bases in 3D topological insulators by analyzing homotopy classes and invariants, offering a constructive algorithm for the case when invariants are trivial.

## Key findings

- Identified three $\
- Provided an algorithm for constructing Wannier bases when topological invariants vanish.
- Established a link between homotopy classes and Wannier function construction.

## Abstract

We investigate the possibility of constructing exponentially localized composite Wannier bases, or equivalently smooth periodic Bloch frames, for 3-dimensional time-reversal symmetric topological insulators, both of bosonic and of fermionic type, so that the bases in question are also compatible with time-reversal symmetry. This problem is translated in the study, of independent interest, of homotopy classes of continuous, periodic, and time-reversal symmetric families of unitary matrices. We identify three $\mathbb{Z}_2$-valued complete invariants for these homotopy classes. When these invariants vanish, we provide an algorithm which constructs a "multi-step" logarithm that is employed to continuously deform the given family into a constant one, identically equal to the identity matrix. This algorithm leads to a constructive procedure to produce the composite Wannier bases mentioned above.