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 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: 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 rulethat 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:Here σ1 and σ2 describe the macroscopic conductivity in directions aligned with and perpendicular to the stripes.[7]

References

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  1. ^ a b 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. Bibcode:1994PhRvB..50.2369D. doi:10.1103/physrevb.50.2369. ISSN 0163-1829. PMID 9976455.
  2. ^ a b 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". Europhysics Letters. 46 (6): 775. arXiv:cond-mat/9810217. Bibcode:1999EL.....46..775H. doi:10.1209/epl/i1999-00331-2. ISSN 0295-5075. S2CID 119420170.
  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. Bibcode:1992PhRvB..46.2223K. doi:10.1103/physrevb.46.2223. PMID 10003898.
  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. arXiv:cond-mat/9408043. Bibcode:1995PhRvL..74..154R. doi:10.1103/physrevlett.74.154. ISSN 0031-9007. PMID 10057722. S2CID 11745393.
  5. ^ a b c Burgess, C. P.; Dolan, B. P. (2007-10-11). "Modular symmetry, the semicircle law, and quantum Hall bilayers". Physical Review B. 76 (15): 155310. arXiv:cond-mat/0701535. Bibcode:2007PhRvB..76o5310B. doi:10.1103/physrevb.76.155310. ISSN 1098-0121. S2CID 76651973.
  6. ^ a b 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. arXiv:cond-mat/9911476. Bibcode:2000PhRvB..6215359B. doi:10.1103/physrevb.62.15359. ISSN 0163-1829. S2CID 119500790.
  7. ^ Felix von Oppen; Halperin, Bertrand I; Stern, Ady (1999). "Conductivity tensor of striped quantum Hall phases". Physical Review Letters. 84 (13): 2937–40. arXiv:cond-mat/9910132. Bibcode:2000PhRvL..84.2937V. doi:10.1103/PhysRevLett.84.2937. PMID 11018980. S2CID 4501701.