Topological Order Parameters for Interacting Topological Insulators

We propose a topological order parameter for interacting topological insulators, expressed in terms of the full Green’s functions of the interacting system. We show that it is exactly quantized for a time-reversal invariant topological insulator, and it can be experimentally measured through the topological magneto-electric effect. This topological order parameter can be applied to both interacting and disordered systems, and used for determining their phase diagrams.

Figure 1

Extension of $G(k)$ to higher dimensional manifolds ($k$, $u$) and ($k$, $u$, $v$). (a) Extension of $G(k)$ to $G(k,u)$, $u\in [0,1]$ in the definition of $P_3$, with $G(k,u=0)=G(k)$. The $(3+1)\mathrm{D}$ momenta space is illustrated as horizontal $1\mathrm{D}$ interval. (b) Extension of $G(k)$ to $G(k,u,v)$ with $G(k,u=v=0)=G(k)$ in the definition of $P_2$.

Figure 2

Phase diagram on the ($\lambda$, $g$) plane. The dark curve $ab$ is the phase boundary separating the normal insulators (NI) and topologically nontrivial insulators (TI). The ground states are gapped everywhere except on $ab$. The true parameter space is in fact infinite dimensional, but this 2D diagram can illustrate the main features. This diagram is applicable in both $(4+1)\mathrm{D}$ and $(3+1)\mathrm{D}$.