Quantum Hall Effects in Layered Disordered Superconductors

Layered singlet paired superconductors with disorder and broken time reversal symmetry are studied, demonstrating a phase diagram with charge-spin separation in transport. In terms of the average intergrain transmission and the interlayer tunneling we find quantum Hall phases with spin Hall coefficients of ${\sigma }_{xy}^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=0,2$ separated by a spin metal phase. We identify a spin metal-insulator localization exponent as well as a spin conductivity exponent of $\approx 0.96$. In the presence of a Zeeman term an additional ${\sigma }_{xy}^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=1$ phase appears.

Figure 1

Phase diagram of the 3D network model with spin-rotational symmetry. The solid lines describe the fit $t\sim |ϵ{|}^{1.2}$.

Figure 2

Scaling of renormalized localization length for the 3D network model with spin-rotational symmetry at $t=1/\sqrt{2}$ showing an exponent ${\nu }_{3\mathrm{D}}=0.96$. The lower and upper branches correspond to the insulating $ϵ>{ϵ}_{\mathrm{c}\mathrm{r}}$ and metallic $ϵ<{ϵ}_{\mathrm{c}\mathrm{r}}$ phases, respectively.

Figure 3

Phase diagram of the 3D network model with $\Delta =2$ showing an additional phase with ${\sigma }_{xy}^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=1$ . The outer solid lines describe the fit $t\sim [|ϵ|-0.64{]}^{2.5}$, while the inner solid line is a guide to the eye.

Figure 4

Scaling of renormalized localization length for $ϵ=\Delta =0$ as function of interlayer coupling.