Topological phases, contrarily to many other phases of matter, cannot be understood in terms of local order parameters. Depending on the symmetries and the dimension of the system under consideration, an appropriate topological invariant describes the topological phase. For instance, the (first) Chern number characterizes the quantum anomalous Hall (QAH) phase associated to the Haldane model. With “two copies” of the Haldane model, we can restore time-reversal symmetry; the system is then characterized by a Z2 topological invariant. In this talk, we explore different topological phases, associated to non zero Chern number or Z2 invariant, in different physical systems on the kagome and the honeycomb lattices.
First, we are interested by the kagome magnet Co3Sn2S2. It shows an impressive behavior of the QAH conductivity driven by the interplay between ferromagnetism in the z direction and antiferromagnetism in the xy plane. Motivated by these facts, we show how such a tuning of the QAH conductivity via the external tuning of the magnetic order can be described.
Then, we investigate the topological phases of a spin-orbit coupled tight-binding model with flux on the kagome lattice. This model is time-reversal invariant and shows Z2 topological insulating phases. We show the stability of the topological phase towards spin-flip processes and different types of on-site potentials. To describe the topological properties of the system we develop an analytical approach related to smooth fields in the Brillouin Zone. We also discuss the effects of Hubbard interactions for a specific type of on-site potential at 2/3 filling.
Finally we present a protocol for the realization of a topological circuit quantum electrodynamics system and the measurement of its topological properties. This protocol is related to the dynamical susceptibility of the system, which is measurable via micro-wave light. The topological system which we have been interested in is a Haldane model (honeycomb lattice). The results may be expanded to a kagome lattice system, although more tedious.”