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 σ0 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 Γ0(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 σ1 and σ2 describe the macroscopic conductivity in directions aligned with and perpendicular to the stripes.[7]

## References

1. ↑ 1.0 1.1 Dykhne, A. M.; Ruzin, I. M. (1994-07-15). "Theory of the fractional quantum Hall effect: The two-phase model". Physical Review B 50 (4): 2369–2379. doi:10.1103/physrevb.50.2369. ISSN 0163-1829. PMID 9976455. Bibcode: 1994PhRvB..50.2369D. https://journals.aps.org/prb/pdf/10.1103/PhysRevB.50.2369.
2. ↑ 2.0 2.1 Hilke, M.; Shahar, D.; Song, S. H.; Tsui, D. C.; Xie, Y. H.; Shayegan, M. (1999-06-15). "Semicircle: An exact relation in the integer and fractional quantum Hall effect" (in en). Europhysics Letters 46 (6): 775. doi:10.1209/epl/i1999-00331-2. ISSN 0295-5075. Bibcode: 1999EL.....46..775H. https://iopscience.iop.org/article/10.1209/epl/i1999-00331-2/meta.
3. ↑ Kivelson, Steven; Lee, Dung-Hai; Zhang, Shou-Cheng (July 1992). "Global phase diagram in the quantum Hall effect". Physical Review B 46 (4): 2223–2238. doi:10.1103/physrevb.46.2223. PMID 10003898. Bibcode: 1992PhRvB..46.2223K. http://prb.aps.org/abstract/PRB/v46/i4/p2223_1.
4. ↑ Ruzin, Igor; Feng, Shechao (1995-01-02). "Universal Relation between Longitudinal and Transverse Conductivities in Quantum Hall Effect". Physical Review Letters 74 (1): 154–157. doi:10.1103/physrevlett.74.154. ISSN 0031-9007. PMID 10057722. Bibcode: 1995PhRvL..74..154R. https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.74.154.
5. ↑ 5.0 5.1 5.2 Burgess, C. P.; Dolan, B. P. (2007-10-11). "Modular symmetry, the semicircle law, and quantum Hall bilayers". Physical Review B 76 (15): 155310. doi:10.1103/physrevb.76.155310. ISSN 1098-0121. Bibcode: 2007PhRvB..76o5310B. https://journals.aps.org/prb/pdf/10.1103/PhysRevB.76.155310.
6. ↑ 6.0 6.1 Burgess, C. P.; Dib, Rim; Dolan, Brian P. (2000-12-15). "Derivation of the semicircle law from the law of corresponding states". Physical Review B 62 (23): 15359–15362. doi:10.1103/physrevb.62.15359. ISSN 0163-1829. Bibcode: 2000PhRvB..6215359B. https://journals.aps.org/prb/pdf/10.1103/PhysRevB.62.15359.
7. ↑ Felix von Oppen; Halperin, Bertrand I; Stern, Ady (1999). "Conductivity tensor of striped quantum Hall phases". Physical Review Letters 84 (13): 2937–40. doi:10.1103/PhysRevLett.84.2937. PMID 11018980. Bibcode: 2000PhRvL..84.2937V.

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