Disordered topological metals

Topological behavior can be masked when disorder is present. A topological insulator, either intrinsic or interaction induced, may turn gapless when sufficiently disordered. Nevertheless, the metallic phase that emerges once a topological gap closes retains several topological characteristics. By considering the self-consistent disorder-averaged Green function of a topological insulator, we derive the condition for gaplessness. We show that the edge states survive in the gapless phase as edge resonances and that, similar to a doped topological insulator, the disordered topological metal also has a finite, but nonquantized topological index. We then consider the disordered Mott topological insulator. We show that within mean-field theory, the disordered Mott topological insulator admits a phase where the symmetry-breaking order parameter remains nonzero but the gap is closed, in complete analogy to “gapless superconductivity” due to magnetic disorder.

Figure 1

The topological index for $s_z=-1$ as a function of the scattering rate $1/\tau$. As soon as $1/\tau$ exceeds $1/{\tau }_{\mathrm{gap}}=\Delta /2$, the topological index plunges from a quantized value to a nonquantized and monotonically decreasing, disorder-dependent value.

Figure 2

Penetration of the edge state into the bulk: The transverse localization length of the edge state, $L_{\perp }\left(\tau \right)=v_F/\text{Re}\left[\tilde{\Delta }\right]$, in units of the bare decay length, $L_{\perp }\left(0\right)=v_F/\Delta$, is plotted as a function of disorder. It varies by a factor of 2.

Figure 3

Dispersion of the edge states: $k_{\parallel }=\text{Re}\left[\tilde{\omega }\right]/v_F$ (full line) and inverse decay length $L_{\parallel }^{-1}=\text{Im}\left[\tilde{\omega }\right]/v_F$ (dashed line) versus $\omega$. (a) Gapped phase ($\zeta =0.8{\zeta }_{\mathrm{gap}}$). (b) Gapless phase ($\zeta =1.2{\zeta }_{\mathrm{gap}}$).

Figure 4

Inverse decay length along the edge, $L_{\parallel }^{-1}$, in units of the inverse bulk mean free path, $l^{-1}=1/\left(v_F\tau \right)$, as a function of disorder. $L_{\parallel }^{-1}$ is zero in the gapped phase and approaches $l^{-1}$ as ${\tau }^{-1}\to \infty$.

Figure 5

Properties of the edge states as a function of the scattering rate ${\tau }^{-1}$. (a) Effective velocity $v$ of the edge states in units of the Fermi velocity. (b) Decay rate $\Gamma$ of the edge states in units of the scattering rate ${\tau }^{-1}$. As the velocity vanishes at ${\zeta }_{\mathrm{gap}}$, $\Gamma$ increases at a much slower rate than $L_{\parallel }^{-1}$ in the gapless phase, namely $\Gamma \propto L_{\parallel }^{-3}$.

Figure 6

Order parameter in the absence of disorder, ${\Delta }_0/D$, and critical scattering rates, $1/\left(D{\tau }_c\right)$ (vanishing of the order parameter) and $1/\left(D{\tau }_{\mathrm{gap}}\right)$ (vanishing of the spectral gap), as a function of interaction strength $c$.

Figure 7

Region of gaplessness at $T=0$: $({\tau }_c^{-1}-{\tau }_{\mathrm{gap}}^{-1})/{\tau }_c^{-1}$ as a function of $1/\left(D{\tau }_c\right)$.