Fragile Topology and Wannier Obstructions

A Factor-Graph Approach to Algebraic Topology, With Applications to Kramers--Wannier Duality

We consider the classical duality result of Kramers and Wannier, which expresses the partition function of the twodimensional Ising model at low temperature in terms of the partition function of the two-dimensional Ising model at high temperature. Our main contribution is to gain new insights into this result by reproving it with the help of a simple normalfactor-graph duality theorem (with ori...

Enumerative Algebraic Geometry via Techniques of Symplectic Topology and Analysis of Local Obstructions

Partly occupied Wannier functions.

We introduce a scheme for constructing partly occupied, maximally localized Wannier functions (WFs) for both molecular and periodic systems. Compared to the traditional occupied WFs the partly occupied WFs possess improved symmetry and localization properties achieved through a bonding-antibonding closing procedure. We demonstrate the equivalence between bonding-antibonding closure and the mini...

Wannier-bloch Oscillators

We consider a Wannier-Stark problem for small eld f in the one-ladder case. We prove that a generical rst band state is a metastable state (Wannier-Bloch oscillator) with the lifetime determined by the imaginary part of the Wannier-Stark ladder. The innnite resonances of the ladder cause Bloch oscillations as a global beating eeect. For an adiabatic time = ft large enough, but much smaller than...

Generalized Wannier Functions

We consider single particle Schrodinger operators with a gap in the energy spectrum. We construct a countable set of exponentially decaying functions, which form a complete, orthonormal basis set for the states below the spectral gap. Each such function is localized near a closed surface. Estimates on the exponential decay rate and a discussion of the geometry of these surfaces is included.