In these lecture notes, partly based on a course taught at the Karpacz Winter School in
March 2014, we explore the close connections between non-adiabatic response of a system …

Topological insulators and their intriguing edge states can be understood in a single-particle
picture and can as such be exhaustively classified. Interactions significantly complicate this …

The geometry of quantum states is well established as a basis for understanding the
response of electronic systems to static electromagnetic fields, as exemplified by the theory …

We find exact analytic expressions for the energies and wave functions of the charged and
neutral excitations above the exact ground states (at rational filling per unit cell) of projected …

Topological invariants built from the periodic Bloch functions characterize new phases of
matter, such as topological insulators and topological superconductors. The most important …

We study superconductivity of twisted bilayer graphene with local and nonlocal attractive
interactions. We obtain the superfluid weight and Berezinskii-Kosterlitz-Thouless (BKT) …

The ground state and transport properties of the Lieb lattice flat band in the presence of an
attractive Hubbard interaction are considered. It is shown that the superfluid weight can be …

In flat bands, superconductivity can lead to surprising transport effects. The superfluid
“mobility”, in the form of the superfluid weight D s, does not draw from the curvature of the …

We review recent progresses in the study of flat band systems, especially focusing on the
fundamental physics related to the singularity of the flat band's Bloch wave functions. We first …

Semiclassical quantization of electronic states under a magnetic field, as proposed by
Onsager, describes not only the Landau level spectrum but also the geometric responses of …