Semicircle Law (Quantum Hall Effect)

The semicircle law, in condensed matter physics, is a mathematical relationship that occurs between quantities measured in the quantum Hall effect. It describes a relationship between the anisotropic and isotropic components of the macroscopic conductivity tensor $σ$ , and, when plotted, appears as a semicircle. The semicircle law was first described theoretically in Dykhne and Ruzin's analysis of the quantum Hall effect as a mixture of 2 phases: a free electron gas, and a free hole gas.^[1]^[2] Mathematically, it states that [math]\displaystyle{ \sigma_{xx}^2+(\sigma_{xy}-\sigma_{xy}^0)^2=(\sigma_{xx}^0)^2 }[/math]where $σ$ is the mean-field Hall conductivity, and $σ 0$ is a parameter that encodes the classical conductivity of each phase. A similar law also holds for the resistivity.^[1]

A convenient reformulation of the law mixes conductivity and resistivity: [math]\displaystyle{ \sigma_{xy}^0=\frac{\rho_{xy}^0}{(\rho_{xy}^0)^2+(\rho_{xx}^0)^2}=\frac{e^2}{h}\left(n+\frac{1}{2}\right) }[/math]where $n$ is an integer, the Hall divisor.^[3]

Although Dykhne and Ruzin's original analysis assumed little scattering, an assumption that proved empirically unsound, the law holds in the coherent-transport limits commonly observed in experiment.^[2]^[4]

Theoretically, the semicircle law originates from a representation of the modular group $Γ 0 (2)$ , which describes a symmetry between different Hall phases. (Note that this is not a symmetry in the conventional sense; there is no conserved current.)^[5]^[6] That group's strong connections to number theory also appear: Hall phase transitions (in a single layer)^[5] exhibit a selection rule[math]\displaystyle{ |pq'-p'q|=1 }[/math]that also governs the Farey sequence.^[5]^[6] Indeed, plots of the semicircle law are also Farey diagrams.

In striped quantum Hall phases, the relationship is slightly more complex, because of the broken symmetry:[math]\displaystyle{ \begin{cases} \sigma_1 \sigma_2 +(\sigma_h - \sigma^0_h)^2=(e^2/(2h))^2 \\ \sigma^0_h = (N + 1/2) e^2 / h \end{cases} }[/math]Here $σ 1$ and $σ 2$ describe the macroscopic conductivity in directions aligned with and perpendicular to the stripes.^[7]