Topological quantum number and critical exponent from conductance fluctuations at the quantum Hall plateau transition

The conductance of a two-dimensional electron gas at the transition from one quantum Hall plateau to the next has sample-specific fluctuations as a function of magnetic field and Fermi energy. Here we identify a universal feature of these mesoscopic fluctuations in a Corbino geometry: The amplitude of the magnetoconductance oscillations has an $e^2/h$ resonance in the transition region, signaling a change in the topological quantum number of the insulating bulk. This resonance provides a signed scaling variable for the critical exponent of the phase transition (distinct from existing positive definite scaling variables).

Figure 1

Corbino geometry consisting of a ring-shaped two-dimensional electron gas (2DEG, gray) in a perpendicular magnetic field $B$. A pair of electrodes (black), at a voltage difference $V$, is attached to the inner and outer edges of the ring.

Figure 2

Example of a path taken in the complex plane $z=e^{i\phi }$ by a zero (blue dot) of $\det r$, as the system is driven through a phase transition in which $\mathcal{Q}\to \mathcal{Q}-1$. At the transition, the zero crosses the unit circle and a reflection eigenvalue vanishes for some flux ${\phi }_0$. The speed at which the zero crosses the unit circle determines the critical exponent of the quantum Hall phase transition (see Sec. 4).

Figure 3

Topological quantum number (top panels) and conductance (bottom panels) of a disordered 2DEG in the Corbino disk geometry. The left panels give a broad range of flux $f$ and Fermi energy $\mu$, while the right panels show a close-up of the $\mathcal{Q}=0\to 1$ transition. (Note the different color scales for the left and right panels.) The horizontal dashed line indicates the three reentrant phase transitions (see Fig. 7).

Figure 4

Dependence of the amplitude of the magnetoconductance oscillations on the Fermi energy for different magnetic fields. The $n$th trace from the bottom shows the $\mu$ dependence of $\Delta G=G_{\max }-G_{\min }$ in the flux interval $(f_n,f_n+\Delta f)$, with $f_n=490\times {10}^{-5}+2(n-1)\Delta f$ and $\Delta f=1.1\times {10}^5$. Successive traces are displaced vertically, with the $G_0=e^2/h$ peak value indicated by a horizontal black line.

Figure 5

Corbino disk in a uniform magnetic field $B$, containing additionally a flux tube $\Phi$ inside the inner perimeter. We choose a gauge such that the phase increment $\phi =e\Phi /\hslash$ due to the flux tube is accumulated upon crossing a cut indicated by the dashed line. Virtual leads $\gamma ,\delta$ are attached at the two sides of the cut. Together with the two physical leads $\alpha ,\beta$ they define the four-terminal scattering matrix (A1). The reflection matrix $r$ of the original two-terminal system is obtained from Eq. (A2), in a formulation which can be analytically continued to arbitrary complex $z=e^{i\phi }$.

Figure 6

Construction of a virtual lead, connecting two sides of a tight-binding lattice. The virtual lead consists of parallel 1D chains with on-site energy $\mu$ and hopping amplitudes alternating between $+1$ and $-1$. The tight-binding equations enforce that every other site in the chain (colored red or blue) has the same wave amplitude at energy $\mu$. The wave amplitudes ${\psi }_n$, ${\psi }_n^{\prime }$ at the left and right end of the $n$th chain are therefore unaffected by the insertion of the virtual lead. We use this device in Fig. 5 to convert a two-terminal geometry into a four-terminal geometry, by cutting the lead at the center and connecting the ends to terminals $\gamma ,\delta$.

Figure 7

Positions of the zeros (red) and poles (green, clustered near the origin; see inset) of $\det r$, along the path indicated by the dashed line in Fig. 3. Arrows point in the direction of increasing Fermi energy. The numbers $1,2,3$ indicate the three phase transitions in which the topological quantum number $\mathcal{Q}$ changes first from 0 to 1, then from 1 back to 0, and once again from 0 to 1.

Figure 8

Geometry of the network model used to calculate the critical exponent. This rolled-up strip is topologically equivalent to the Corbino geometry used in the main text.

Figure 9

Scaling variable as obtained numerically (data points) and as fitted to Eq. (B1) (black lines). Not all calculated data points are included, to avoid clutter.