373:
685:
205:
179:
488:
386:
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.
468:
65:
368:{\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)}
680:{\displaystyle {\begin{cases}\sigma _{1}\sigma _{2}+(\sigma _{h}-\sigma _{h}^{0})^{2}=(e^{2}/(2h))^{2}\\\sigma _{h}^{0}=(N+1/2)e^{2}/h\end{cases}}}
401:, which describes a symmetry between different Hall phases. (Note that this is not a symmetry in the conventional sense; there is no
1183:
50:
The semicircle law was first described theoretically in Dykhne and Ruzin's analysis of the quantum Hall effect as a
415:
1114:
Felix von Oppen; Halperin, Bertrand I; Stern, Ady (1999). "Conductivity tensor of striped quantum Hall phases".
21:
1116:
37:
195:
is a parameter that encodes the classical conductivity of each phase. A similar law also holds for the
1188:
1135:
1072:
1004:
926:
869:
814:
746:
391:
59:
792:
497:
706:
380:
55:
25:
1159:
1125:
1096:
1062:
1028:
994:
958:
916:
838:
804:
485:
quantum Hall phases, the relationship is slightly more complex, because of the broken symmetry:
1151:
1088:
1020:
982:
950:
942:
905:"Universal Relation between Longitudinal and Transverse Conductivities in Quantum Hall Effect"
904:
885:
830:
770:
762:
402:
186:
1050:
1143:
1080:
1012:
934:
877:
822:
754:
734:
482:
174:{\displaystyle \sigma _{xx}^{2}+(\sigma _{xy}-\sigma _{xy}^{0})^{2}=(\sigma _{xx}^{0})^{2}}
1139:
1076:
1008:
930:
873:
818:
791:
Hilke, M.; Shahar, D.; Song, S. H.; Tsui, D. C.; Xie, Y. H.; Shayegan, M. (1999-06-15).
750:
471:
410:
1177:
1100:
842:
710:
475:
406:
1032:
962:
1163:
857:
793:"Semicircle: An exact relation in the integer and fractional quantum Hall effect"
24:, is a mathematical relationship that occurs between quantities measured in the
1147:
826:
196:
1016:
938:
1084:
44:
29:
1092:
1024:
946:
881:
834:
766:
758:
1155:
954:
889:
774:
390:
Theoretically, the semicircle law originates from a representation of the
202:
A convenient reformulation of the law mixes conductivity and resistivity:
1130:
1067:
999:
921:
809:
33:
1051:"Derivation of the semicircle law from the law of corresponding states"
51:
856:
Kivelson, Steven; Lee, Dung-Hai & Zhang, Shou-Cheng (July 1992).
735:"Theory of the fractional quantum Hall effect: The two-phase model"
409:
also appear: Hall phase transitions (in a single layer) exhibit a
983:"Modular symmetry, the semicircle law, and quantum Hall bilayers"
673:
1049:
Burgess, C. P.; Dib, Rim; Dolan, Brian P. (2000-12-15).
491:
418:
208:
68:
705:
describe the macroscopic conductivity in directions
679:
462:
367:
173:
858:"Global phase diagram in the quantum Hall effect"
474:. Indeed, plots of the semicircle law are also
8:
981:Burgess, C. P.; Dolan, B. P. (2007-10-11).
28:. It describes a relationship between the
733:Dykhne, A. M.; Ruzin, I. M. (1994-07-15).
1129:
1066:
998:
920:
903:Ruzin, Igor; Feng, Shechao (1995-01-02).
808:
662:
656:
641:
620:
615:
601:
580:
574:
558:
548:
543:
530:
514:
504:
492:
490:
449:
419:
417:
350:
328:
322:
310:
300:
292:
276:
266:
258:
244:
236:
230:
221:
213:
207:
165:
155:
147:
131:
121:
113:
97:
81:
73:
67:
722:
405:.) That group's strong connections to
62:gas. Mathematically, it states that
7:
1044:
1042:
976:
974:
972:
786:
784:
728:
726:
38:macroscopic conductivity tensor
43:, and, when plotted, appears as a
14:
649:
629:
598:
594:
585:
567:
555:
523:
450:
420:
307:
285:
273:
251:
162:
140:
128:
90:
1:
1148:10.1103/PhysRevLett.84.2937
463:{\displaystyle |pq'-p'q|=1}
1205:
1017:10.1103/physrevb.76.155310
939:10.1103/physrevlett.74.154
1085:10.1103/physrevb.62.15359
827:10.1209/epl/i1999-00331-2
1184:Condensed matter physics
882:10.1103/physrevb.46.2223
759:10.1103/physrevb.50.2369
22:condensed matter physics
1117:Physical Review Letters
909:Physical Review Letters
189:Hall conductivity, and
681:
470:that also governs the
464:
369:
175:
682:
465:
370:
176:
489:
416:
206:
66:
1140:2000PhRvL..84.2937V
1077:2000PhRvB..6215359B
1061:(23): 15359–15362.
1009:2007PhRvB..76o5310B
931:1995PhRvL..74..154R
874:1992PhRvB..46.2223K
819:1999EL.....46..775H
797:Europhysics Letters
751:1994PhRvB..50.2369D
625:
553:
379:is an integer, the
305:
271:
249:
226:
160:
126:
86:
26:quantum Hall effect
677:
672:
611:
539:
460:
365:
288:
254:
232:
209:
171:
143:
109:
69:
36:components of the
1055:Physical Review B
987:Physical Review B
862:Physical Review B
739:Physical Review B
403:conserved current
358:
337:
317:
56:free electron gas
1196:
1168:
1167:
1133:
1131:cond-mat/9910132
1111:
1105:
1104:
1070:
1068:cond-mat/9911476
1046:
1037:
1036:
1002:
1000:cond-mat/0701535
978:
967:
966:
924:
922:cond-mat/9408043
900:
894:
893:
868:(4): 2223–2238.
853:
847:
846:
812:
810:cond-mat/9810217
788:
779:
778:
745:(4): 2369–2379.
730:
711:perpendicular to
704:
695:
686:
684:
683:
678:
676:
675:
666:
661:
660:
645:
624:
619:
606:
605:
584:
579:
578:
563:
562:
552:
547:
535:
534:
519:
518:
509:
508:
469:
467:
466:
461:
453:
445:
434:
423:
400:
378:
374:
372:
371:
366:
364:
360:
359:
351:
338:
333:
332:
323:
318:
316:
315:
314:
304:
299:
281:
280:
270:
265:
248:
243:
231:
225:
220:
194:
184:
180:
178:
177:
172:
170:
169:
159:
154:
136:
135:
125:
120:
105:
104:
85:
80:
42:
1204:
1203:
1199:
1198:
1197:
1195:
1194:
1193:
1174:
1173:
1172:
1171:
1124:(13): 2937–40.
1113:
1112:
1108:
1048:
1047:
1040:
980:
979:
970:
902:
901:
897:
855:
854:
850:
790:
789:
782:
732:
731:
724:
719:
703:
697:
694:
688:
671:
670:
652:
608:
607:
597:
570:
554:
526:
510:
500:
493:
487:
486:
438:
427:
414:
413:
398:
394:
376:
343:
339:
324:
306:
272:
250:
204:
203:
190:
182:
161:
127:
93:
64:
63:
54:of 2 phases: a
40:
12:
11:
5:
1202:
1200:
1192:
1191:
1186:
1176:
1175:
1170:
1169:
1106:
1038:
993:(15): 155310.
968:
915:(1): 154–157.
895:
848:
780:
721:
720:
718:
715:
701:
692:
674:
669:
665:
659:
655:
651:
648:
644:
640:
637:
634:
631:
628:
623:
618:
614:
610:
609:
604:
600:
596:
593:
590:
587:
583:
577:
573:
569:
566:
561:
557:
551:
546:
542:
538:
533:
529:
525:
522:
517:
513:
507:
503:
499:
498:
496:
476:Farey diagrams
472:Farey sequence
459:
456:
452:
448:
444:
441:
437:
433:
430:
426:
422:
411:selection rule
396:
363:
357:
354:
349:
346:
342:
336:
331:
327:
321:
313:
309:
303:
298:
295:
291:
287:
284:
279:
275:
269:
264:
261:
257:
253:
247:
242:
239:
235:
229:
224:
219:
216:
212:
168:
164:
158:
153:
150:
146:
142:
139:
134:
130:
124:
119:
116:
112:
108:
103:
100:
96:
92:
89:
84:
79:
76:
72:
18:semicircle law
13:
10:
9:
6:
4:
3:
2:
1201:
1190:
1187:
1185:
1182:
1181:
1179:
1165:
1161:
1157:
1153:
1149:
1145:
1141:
1137:
1132:
1127:
1123:
1119:
1118:
1110:
1107:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1069:
1064:
1060:
1056:
1052:
1045:
1043:
1039:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1001:
996:
992:
988:
984:
977:
975:
973:
969:
964:
960:
956:
952:
948:
944:
940:
936:
932:
928:
923:
918:
914:
910:
906:
899:
896:
891:
887:
883:
879:
875:
871:
867:
863:
859:
852:
849:
844:
840:
836:
832:
828:
824:
820:
816:
811:
806:
802:
798:
794:
787:
785:
781:
776:
772:
768:
764:
760:
756:
752:
748:
744:
740:
736:
729:
727:
723:
716:
714:
713:the stripes.
712:
708:
700:
691:
667:
663:
657:
653:
646:
642:
638:
635:
632:
626:
621:
616:
612:
602:
591:
588:
581:
575:
571:
564:
559:
549:
544:
540:
536:
531:
527:
520:
515:
511:
505:
501:
494:
484:
479:
477:
473:
457:
454:
446:
442:
439:
435:
431:
428:
424:
412:
408:
407:number theory
404:
393:
392:modular group
388:
384:
382:
361:
355:
352:
347:
344:
340:
334:
329:
325:
319:
311:
301:
296:
293:
289:
282:
277:
267:
262:
259:
255:
245:
240:
237:
233:
227:
222:
217:
214:
210:
200:
198:
193:
188:
166:
156:
151:
148:
144:
137:
132:
122:
117:
114:
110:
106:
101:
98:
94:
87:
82:
77:
74:
70:
61:
58:, and a free
57:
53:
48:
46:
39:
35:
31:
27:
23:
19:
1121:
1115:
1109:
1058:
1054:
990:
986:
912:
908:
898:
865:
861:
851:
800:
796:
742:
738:
707:aligned with
698:
689:
480:
389:
385:
381:Hall divisor
201:
191:
49:
17:
15:
1189:Hall effect
197:resistivity
30:anisotropic
1178:Categories
803:(6): 775.
717:References
187:mean-field
45:semicircle
1101:119500790
1093:0163-1829
1025:1098-0121
947:0031-9007
843:119420170
835:0295-5075
767:0163-1829
613:σ
541:σ
537:−
528:σ
512:σ
502:σ
436:−
290:ρ
256:ρ
234:ρ
211:σ
145:σ
111:σ
107:−
95:σ
71:σ
34:isotropic
1156:11018980
1033:76651973
963:11745393
955:10057722
890:10003898
443:′
432:′
1164:4501701
1136:Bibcode
1073:Bibcode
1005:Bibcode
927:Bibcode
870:Bibcode
815:Bibcode
775:9976455
747:Bibcode
483:striped
185:is the
52:mixture
1162:
1154:
1099:
1091:
1031:
1023:
961:
953:
945:
888:
841:
833:
773:
765:
375:where
181:where
1160:S2CID
1126:arXiv
1097:S2CID
1063:arXiv
1029:S2CID
995:arXiv
959:S2CID
917:arXiv
839:S2CID
805:arXiv
687:Here
20:, in
1152:PMID
1089:ISSN
1021:ISSN
951:PMID
943:ISSN
886:PMID
831:ISSN
771:PMID
763:ISSN
709:and
696:and
478:.
383:.
199:.
60:hole
47:.
32:and
16:The
1144:doi
1081:doi
1013:doi
935:doi
878:doi
823:doi
755:doi
481:In
399:(2)
1180::
1158:.
1150:.
1142:.
1134:.
1122:84
1120:.
1095:.
1087:.
1079:.
1071:.
1059:62
1057:.
1053:.
1041:^
1027:.
1019:.
1011:.
1003:.
991:76
989:.
985:.
971:^
957:.
949:.
941:.
933:.
925:.
913:74
911:.
907:.
884:.
876:.
866:46
864:.
860:.
837:.
829:.
821:.
813:.
801:46
799:.
795:.
783:^
769:.
761:.
753:.
743:50
741:.
737:.
725:^
1166:.
1146::
1138::
1128::
1103:.
1083::
1075::
1065::
1035:.
1015::
1007::
997::
965:.
937::
929::
919::
892:.
880::
872::
845:.
825::
817::
807::
777:.
757::
749::
702:2
699:σ
693:1
690:σ
668:h
664:/
658:2
654:e
650:)
647:2
643:/
639:1
636:+
633:N
630:(
627:=
622:0
617:h
603:2
599:)
595:)
592:h
589:2
586:(
582:/
576:2
572:e
568:(
565:=
560:2
556:)
550:0
545:h
532:h
524:(
521:+
516:2
506:1
495:{
458:1
455:=
451:|
447:q
440:p
429:q
425:p
421:|
397:0
395:Γ
377:n
362:)
356:2
353:1
348:+
345:n
341:(
335:h
330:2
326:e
320:=
312:2
308:)
302:0
297:x
294:x
286:(
283:+
278:2
274:)
268:0
263:y
260:x
252:(
246:0
241:y
238:x
228:=
223:0
218:y
215:x
192:σ
183:σ
167:2
163:)
157:0
152:x
149:x
141:(
138:=
133:2
129:)
123:0
118:y
115:x
102:y
99:x
91:(
88:+
83:2
78:x
75:x
41:σ
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.