Half-integer quantum Hall effect of disordered Dirac fermions at a topological insulator surface

The unconventional (half-integer) quantum Hall effect for a single species of Dirac fermions is analyzed. We discuss possible experimental measurements of the half-integer Hall conductance $g_{xy}$ of topological insulator surface states and explain how to reconcile Laughlin's flux insertion argument with half-integer $g_{xy}$. Using a vortex state representation of Landau level wave functions, we calculate current density beyond linear response, which is in particular relevant to the topological image monopole effect. As a major result, the field theory describing the localization physics of the quantum Hall effect of a single species of Dirac fermions is derived. In this connection, the issue of (absent) parity anomaly is revisited. The renormalization group (RG) flow and the resulting phase diagram are extensively discussed. Starting values of the RG flow are given by the semiclassical conductivity tensor which is obtained from the Boltzmann transport theory of the anomalous Hall effect.

Figure 1

Failure of transport measurement of half-integer Hall response. A thin 3D TI slab in a QH state ${\sigma }_{xy}^{\mathrm{top}}={\sigma }_{xy}^{\mathrm{bottom}}=\frac{e^2}{2h}$ (one shared green edge channel) is probed by a local four-contact measurement consisting of two opposite bias gates (orange) and, perpendicularly, two probing gates connected by an amperemeter (blue). For further explanation, see main text.

Figure 2

Since the surface of a 3D TI is itself boundaryless, the modified setup for the flux insertion argument involves a torus of 3D TI surface states. (Here, $\vec{j}$ is shown for the exemplary case of ${\sigma }_{xy}^{\mathrm{top}}>0$.)

Figure 3

Cross section of the 3D TI torus depicted in Fig. 2 in a plane perpendicular to the azimuthal unit vector. Here $r=\sqrt{x^2+y^2}$. For the discussion of length scales and boundary conditions, see main text.

Figure 4

Schematic representation of the spectrum at the inner boundary of the TI torus shown in Figs. 2 and 3.

Figure 5

RG flow diagram for Dirac fermions, based on the scaling proposed by Khmelnitskii. As compared to the case of fermions with parabolic dispersion, the diagram is shifted by half a conductance quantum.

Figure 6

RG flow diagram for noninteracting Dirac fermions for the case when the magnetic field is the only source of time-reversal-symmetry breaking (i.e., assuming no magnetic impurities and no Zeeman coupling). The QH transition at $B=0$ (${\sigma }_{xy}=0$) is qualitatively different from all others; see main text. The corresponding flow is depicted by black bold arrows. It follows the behavior of the symplectic class for $L<l_B$ and crosses over to the unitary class at $L\sim l_B$.

Figure 7

RG flow describing the $B=0$ transition near the “supermetallic” fixed point $(1/g_{xx}^{*},g_{xy}^{*})=(0,0)$; see Eqs. (48).

Figure 8

Phase diagram for the QHE of Dirac fermions in the plane of longitudinal and Hall resistivities.

Figure 9

Dependence of the classical conductivities on chemical potential. The transverse conductivity was split into intrinsic (red, ${\sigma }_{xy}^{(\mathrm{intr}.)}$) and Fermi-surface (violet, $\delta {\sigma }_{xy}$) contributions. The dashed curves are obtained by reflection with respect to the origin and visualize the magnitude of the AHE contributions.

Figure 10

Dependence of the classical conductivity tensor on magnetic field. The transverse conductivity was split into intrinsic (red, ${\sigma }_{xy}^{(\mathrm{intr}.)}$) and Fermi-surface (violet, $\delta {\sigma }_{xy}$) contributions. The dot-dashed vertical lines denote the position where ${\Omega }_c^{\mathrm{cl}}{\tau }_{tr}=1$. In the inset, the contributions to the transverse conductance are plotted in the limit of vanishing $B$ field. The ticks on the abscissa denote ${\Omega }_c^{\mathrm{cl}}{\tau }_{tr}=1/20$. The dashed curves are obtained by reflection $B\to -B$ and visualize the magnitude of the AHE contributions. In this plot, the Zeeman energy is assumed to be $B$-field independent (which could result from an exchange coupling to a nearby ferromagnetic layer).

Figure 11

Levitation of delocalized states of gapless Dirac fermions for Coulomb impurities ($\mu /{\tau }_{tr}$ and ${\Omega }_c^{\mathrm{cl}}{\tau }_{tr}\propto B$ are $\mu$-independent).

Figure 12

Phase diagram of QH phases in the $B-\mu$ plane (“levitation scenario”) for the case of short-range impurities. Here, as in Fig. 10, $E_Z<0$ is assumed to be $B$-field independent. The floating up of delocalized states with odd (even) number is depicted by solid blue (red) curves. The dashed lines correspond to ${\Omega }_c^{\mathrm{cl}}{\tau }_{tr}=1$. The insets magnify the region of weak magnetic fields. The asymmetry under $B\to -B$ (dotted blue/red curves) is a consequence of the AHE. Disorder strength is determined by $\left|\mu \right|{\tau }_{sj}=100$.

Figure 13

Sketch of the setup discussed in the main text. An electric test charge (solid dot) is placed above a double QH structure (e.g., a thin 3D TI slab) and creates a series of magnetic and electric mirror charges (circles).

Figure 14

The magnetic field configuration for 3D TI in the QH state in a setup as depicted in Fig. 13. Left: slab of thickness $10\mu \mathrm{m}$. Right: slab of thickness $20\mathrm{nm}$. In both figures $z_0=2\mu \mathrm{m}$, ${\sigma }_{xy}^{\mathrm{top}}=e^2/2h$, ${\sigma }_{xy}^{\mathrm{bottom}}=-7e^2/2h$, and, for simplicity, ${\epsilon }_1={\epsilon }_2={\epsilon }_3$ and ${\mu }_1={\mu }_2={\mu }_3$.

Figure 15

The broadened delta function entering the general solution of the kinetic equation. In the inset, the same function is shown in the vicinity of the chemical potential, where it takes the value $|{\tau }_{tr}\langle W^{\left(B\right)}\rangle /{\tau }^{\left(W\right)}{|}^{-1}$.

Figure 16

Sketch of three different scenarios for the setup discussed in Appendix 4. The left and right scenarios correspond to case I, the middle one to case II.