AG Mathematische Physik, Tom Stoiber, (Yeshiva University): Higher-order Topological Insulators and K-theory

Tom Stoiber, (Yeshiva University)

Higher-order Topological Insulators and K-theory

Abstract:
Topological insulators are materials that are insulating in their
interior but exhibit protected surface states, a phenomenon that is
mathematically well-understood through the use of C*-algebras and
operator K-theory. In higher-order topological insulators, all faces of
a crystal are insulating, while surface states appear only at boundaries
of higher codimension, such as hinges or corners. For so-called
intrinsic topological insulators the existence of these states is
independent of boundary conditions but requires spatial symmetries like
mirror, rotational, or inversion symmetry. Since the topological
protection emerges only in the infinite volume limit, we need to
construct C*-algebras that describe symmetric infinite crystals with
various boundary configurations. These algebras naturally admit
cofiltrations based on the codimensions of the boundaries, leading to
spectral sequences in equivariant K-theory. We then demonstrate that the
classification and phenomenology of higher-order topological insulators
align seamlessly with this formalism. Specifically, the higher
differentials in the spectral sequence identify precisely the classes of
intrinsic higher-order topological insulators and their associated
surface states.