Quantum Hall Network Models as Floquet Topological Insulators

Network models for equilibrium integer quantum Hall (IQH) transitions are described by unitary scattering matrices that can also be viewed as representing nonequilibrium Floquet systems. The resulting Floquet bands have zero Chern number, and are instead characterized by a chiral Floquet winding number. This begs the question, How can a model without Chern number describe IQH systems? We resolve this puzzle by showing that nonzero Chern number is recovered from the network model via the energy dependence of network model scattering parameters. This relationship shows that, despite their topologically distinct origins, IQH and chiral Floquet topology-changing transitions share identical universal scaling properties.

Figure 1

Network model and Floquet band structure. (a) Schematic of Chalker-Coddington network model. (b) Floquet bands $ϵ(\mathit{k})$ of the clean network model evolution operator at criticality, $\theta =\pi /4$.

Figure 2

Chern bands from the scattering network. (a) Semiclassical orbits for an electron in a periodic potential and uniform magnetic field and (b) corresponding energy dependence of scattering parameters (schematic). (c) Partial set of energy bands from the scattering network with $\theta (E)=(\pi /4)(\tanh \{[(E+\pi /4)/4\pi ]\}+1)$ and $\phi (E)=E$. The horizontal axis is placed at the critical energy $E=E_c$ for which $\theta (E_c)=\pi /4$. (d) Berry curvature for two bands near $E=0$ for which $\theta$ respectively crosses (lower) and does not cross (upper) $\pi /4$, giving total Chern number 1 or 0. All other bands have Chern number zero.