The class C quantum network model with random tunneling and its nonlinear sigma model representation

The spin quantum Hall effect is a relative of the integer quantum Hall effect, characterized by integer quantized spin Hall conductance. In this work, we formulate and investigate a quantum network model consisting of $\textsf{N}$ channels per chiral link, preserving the fundamental symmetries of the spin quantum Hall effect. We demonstrate that, in the general case, the triplet sector of the theory remains coupled to the singlet sector. In the large-$\textsf{N}$ limit, we systematically derive the effective long-distance, low-energy field theory, identified as a nonlinear sigma model. Our analysis reveals that while triplet modes are typically massive and do not influence the large-$\textsf{N}$ nonlinear sigma model, specific conditions exist where these modes become `soft', thereby increasing the ultraviolet cutoff length of the effective theory. Furthermore, by calculating the bare longitudinal and spin Hall conductances, we show that the standard saddle-point approximation fails in regimes with significant tunneling asymmetry between even and odd links. Finally, we establish that the introduction of a Zeeman field not only breaks the SU(2) symmetry of the nonlinear sigma model action but also generates a term that explicitly violates inversion symmetry.