1164:
1616:
330:
793:
57:
1291:
1159:{\displaystyle H=\sum _{k,\sigma }\epsilon _{k}c_{k\sigma }^{\dagger }c_{k\sigma }+\sum _{j,\sigma }\epsilon _{f}f_{j\sigma }^{\dagger }f_{j\sigma }+U\sum _{j}f_{j\uparrow }^{\dagger }f_{j\uparrow }f_{j\downarrow }^{\dagger }f_{j\downarrow }+\sum _{j,k,\sigma }V_{jk}(e^{ikx_{j}}f_{j\sigma }^{\dagger }c_{k\sigma }+e^{-ikx_{j}}c_{k\sigma }^{\dagger }f_{j\sigma })}
51:. In its simplest form, the model contains a term describing the kinetic energy of the conduction electrons, a two-level term with an on-site Coulomb repulsion that models the impurity energy levels, and a hybridization term that couples conduction and impurity orbitals. For a single impurity, the Hamiltonian takes the form
325:{\displaystyle H=\sum _{k,\sigma }\epsilon _{k}c_{k\sigma }^{\dagger }c_{k\sigma }+\sum _{\sigma }\epsilon _{\sigma }d_{\sigma }^{\dagger }d_{\sigma }+Ud_{\uparrow }^{\dagger }d_{\uparrow }d_{\downarrow }^{\dagger }d_{\downarrow }+\sum _{k,\sigma }V_{k}(d_{\sigma }^{\dagger }c_{k\sigma }+c_{k\sigma }^{\dagger }d_{\sigma })}
1611:{\displaystyle H=\sum _{k,\sigma }\epsilon _{k}c_{k\sigma }^{\dagger }c_{k\sigma }+\sum _{i,\sigma }\epsilon _{d}d_{i\sigma }^{\dagger }d_{i\sigma }+\sum _{i,\sigma ,i'\sigma '}{\frac {U}{2}}n_{i\sigma }n_{i'\sigma '}+\sum _{i,k,\sigma }V_{k}(d_{i\sigma }^{\dagger }c_{k\sigma }+c_{k\sigma }^{\dagger }d_{i\sigma })}
769:
701:
653:
604:
556:
506:
1666:
422:
1195:
1688:
1640:
1255:
1235:
1215:
466:
446:
398:
378:
354:
1281:
Anderson model, which is used to describe impurities which have an orbital, as well as a spin, degree of freedom. This is relevant in
713:
775:
In the local moment regime, the magnetic moment is present at the impurity site. However, for low enough temperature, the moment is
787:
For heavy-fermion systems, a lattice of impurities is described by the periodic
Anderson model. The one-dimensional model is
357:
658:
616:
1857:
561:
519:
1852:
28:
1265:
electrons in heavy fermion systems to interact, although they are separated by a distance greater than the
1282:
1783:
Schrieffer, J.R.; Wolff, P.A. (September 1966). "Relation between the
Anderson and Kondo Hamiltonians".
476:
The model yields several regimes that depend on the relationship of the impurity energy levels to the
1266:
482:
24:
1719:
1278:
407:
1792:
1754:
1173:
8:
1796:
1758:
1646:
1808:
1673:
1625:
1240:
1220:
1200:
451:
431:
383:
363:
339:
32:
1812:
1800:
1762:
1692:
48:
1742:
1262:
425:
1846:
1704:
44:
1804:
1669:
label the orbital degree of freedom (which can take one of two values), and
1766:
1709:
776:
40:
1714:
477:
1257:
by convention for heavy-fermion systems). The hybridization term allows
401:
764:{\displaystyle \epsilon _{d}\ll E_{\rm {F}}\ll \epsilon _{d}+U}
36:
1277:
There are other variants of the
Anderson model, such as the
1782:
779:
screened to give non-magnetic many-body singlet state.
1676:
1649:
1628:
1294:
1243:
1223:
1203:
1176:
796:
716:
661:
619:
564:
522:
485:
454:
434:
410:
386:
366:
342:
60:
1237:is the impurity creation operator (used instead of
1778:
1776:
1682:
1660:
1634:
1610:
1249:
1229:
1209:
1189:
1158:
763:
696:{\displaystyle \epsilon _{d}+U\approx E_{\rm {F}}}
695:
647:
598:
550:
500:
460:
440:
416:
392:
372:
348:
324:
1285:systems. The SU(4) Anderson model Hamiltonian is
771:, which yields a magnetic moment at the impurity.
1844:
648:{\displaystyle \epsilon _{d}\approx E_{\rm {F}}}
1773:
1745:(1961). "Localized Magnetic States in Metals".
380:is the annihilation operator for the impurity,
1737:
1735:
599:{\displaystyle \epsilon _{d}+U\gg E_{\rm {F}}}
39:. It is often applied to the description of
1826:
1824:
1822:
551:{\displaystyle \epsilon _{d}\gg E_{\rm {F}}}
1732:
1819:
1741:
782:
1835:. New York: Cambridge University Press.
1845:
1830:
1833:The Kondo Problem to Heavy Fermions
428:. The on–site Coulomb repulsion is
16:Hamiltonian used in quantum physics
13:
736:
687:
639:
590:
542:
492:
14:
1869:
1272:
1197:is the position of impurity site
1605:
1537:
1153:
1036:
993:
975:
962:
944:
360:of a conduction electron, and
319:
257:
223:
208:
198:
183:
1:
1725:
606:, which has no local moment.
7:
1698:
1283:carbon nanotube quantum dot
501:{\displaystyle E_{\rm {F}}}
400:is the conduction electron
10:
1874:
471:
468:gives the hybridization.
31:that is used to describe
1858:Condensed matter physics
43:-type problems, such as
1805:10.1103/PhysRev.149.491
417:{\displaystyle \sigma }
21:Anderson impurity model
1853:Quantum lattice models
1831:Hewson, A. C. (1993).
1767:10.1103/PhysRev.124.41
1684:
1662:
1636:
1612:
1251:
1231:
1211:
1191:
1160:
765:
697:
649:
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552:
502:
462:
442:
418:
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326:
25:Philip Warren Anderson
1720:Anderson localization
1685:
1663:
1637:
1613:
1252:
1232:
1212:
1192:
1190:{\displaystyle x_{j}}
1161:
783:Heavy-fermion systems
766:
698:
650:
601:
553:
503:
463:
443:
419:
395:
375:
358:annihilation operator
351:
327:
1674:
1647:
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1201:
1174:
794:
714:
659:
617:
562:
520:
483:
452:
432:
408:
384:
364:
340:
58:
1797:1966PhRv..149..491S
1759:1961PhRv..124...41A
1591:
1557:
1404:
1344:
1139:
1079:
984:
953:
906:
846:
308:
274:
217:
192:
161:
110:
33:magnetic impurities
1695:for the impurity.
1680:
1661:{\displaystyle i'}
1658:
1632:
1608:
1574:
1540:
1526:
1455:
1387:
1376:
1327:
1316:
1247:
1227:
1207:
1187:
1156:
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1062:
1022:
967:
936:
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818:
761:
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645:
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548:
498:
458:
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370:
346:
322:
291:
260:
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203:
178:
147:
136:
93:
82:
1683:{\displaystyle n}
1635:{\displaystyle i}
1505:
1464:
1421:
1361:
1301:
1250:{\displaystyle d}
1230:{\displaystyle f}
1210:{\displaystyle j}
1001:
926:
863:
803:
461:{\displaystyle V}
441:{\displaystyle U}
393:{\displaystyle k}
373:{\displaystyle d}
349:{\displaystyle c}
231:
127:
67:
1865:
1837:
1836:
1828:
1817:
1816:
1780:
1771:
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1739:
1689:
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1501:
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1465:
1457:
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1417:
1416:
1403:
1398:
1386:
1385:
1375:
1357:
1356:
1343:
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1325:
1315:
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1248:
1236:
1234:
1233:
1228:
1216:
1214:
1213:
1208:
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1193:
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1035:
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1021:
997:
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828:
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817:
770:
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762:
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741:
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629:
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557:
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554:
549:
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532:
531:
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467:
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459:
447:
445:
444:
439:
423:
421:
420:
415:
399:
397:
396:
391:
379:
377:
376:
371:
356:operator is the
355:
353:
352:
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331:
329:
328:
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318:
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302:
287:
286:
273:
268:
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227:
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186:
171:
170:
160:
155:
146:
145:
135:
123:
122:
109:
104:
92:
91:
81:
49:Kondo insulators
1873:
1872:
1868:
1867:
1866:
1864:
1863:
1862:
1843:
1842:
1841:
1840:
1829:
1820:
1785:Physical Review
1781:
1774:
1743:Anderson, P. W.
1740:
1733:
1728:
1701:
1693:number operator
1691:represents the
1675:
1672:
1671:
1650:
1648:
1645:
1644:
1627:
1624:
1623:
1596:
1592:
1586:
1578:
1562:
1558:
1552:
1544:
1531:
1527:
1509:
1492:
1484:
1483:
1479:
1470:
1466:
1456:
1446:
1438:
1425:
1409:
1405:
1399:
1391:
1381:
1377:
1365:
1349:
1345:
1339:
1331:
1321:
1317:
1305:
1293:
1290:
1289:
1275:
1242:
1239:
1238:
1222:
1219:
1218:
1202:
1199:
1198:
1181:
1177:
1175:
1172:
1171:
1144:
1140:
1134:
1126:
1114:
1110:
1100:
1096:
1084:
1080:
1074:
1066:
1054:
1050:
1043:
1039:
1027:
1023:
1005:
989:
985:
979:
971:
958:
954:
948:
940:
930:
911:
907:
901:
893:
883:
879:
867:
851:
847:
841:
833:
823:
819:
807:
795:
792:
791:
785:
749:
745:
735:
734:
730:
721:
717:
715:
712:
711:
686:
685:
681:
666:
662:
660:
657:
656:
638:
637:
633:
624:
620:
618:
615:
614:
589:
588:
584:
569:
565:
563:
560:
559:
541:
540:
536:
527:
523:
521:
518:
517:
491:
490:
486:
484:
481:
480:
474:
453:
450:
449:
433:
430:
429:
409:
406:
405:
385:
382:
381:
365:
362:
361:
341:
338:
337:
313:
309:
303:
295:
279:
275:
269:
264:
251:
247:
235:
222:
218:
212:
207:
197:
193:
187:
182:
166:
162:
156:
151:
141:
137:
131:
115:
111:
105:
97:
87:
83:
71:
59:
56:
55:
17:
12:
11:
5:
1871:
1861:
1860:
1855:
1839:
1838:
1818:
1791:(2): 491–492.
1772:
1730:
1729:
1727:
1724:
1723:
1722:
1717:
1712:
1707:
1700:
1697:
1679:
1656:
1653:
1631:
1620:
1619:
1607:
1602:
1599:
1595:
1589:
1584:
1581:
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1573:
1568:
1565:
1561:
1555:
1550:
1547:
1543:
1539:
1534:
1530:
1524:
1521:
1518:
1515:
1512:
1508:
1504:
1498:
1495:
1490:
1487:
1482:
1476:
1473:
1469:
1463:
1460:
1452:
1449:
1444:
1441:
1437:
1434:
1431:
1428:
1424:
1420:
1415:
1412:
1408:
1402:
1397:
1394:
1390:
1384:
1380:
1374:
1371:
1368:
1364:
1360:
1355:
1352:
1348:
1342:
1337:
1334:
1330:
1324:
1320:
1314:
1311:
1308:
1304:
1300:
1297:
1274:
1273:Other variants
1271:
1246:
1226:
1206:
1184:
1180:
1168:
1167:
1155:
1150:
1147:
1143:
1137:
1132:
1129:
1125:
1117:
1113:
1109:
1106:
1103:
1099:
1095:
1090:
1087:
1083:
1077:
1072:
1069:
1065:
1057:
1053:
1049:
1046:
1042:
1038:
1033:
1030:
1026:
1020:
1017:
1014:
1011:
1008:
1004:
1000:
995:
992:
988:
982:
977:
974:
970:
964:
961:
957:
951:
946:
943:
939:
933:
929:
925:
922:
917:
914:
910:
904:
899:
896:
892:
886:
882:
876:
873:
870:
866:
862:
857:
854:
850:
844:
839:
836:
832:
826:
822:
816:
813:
810:
806:
802:
799:
784:
781:
773:
772:
760:
757:
752:
748:
744:
738:
733:
729:
724:
720:
704:
689:
684:
680:
677:
674:
669:
665:
641:
636:
632:
627:
623:
607:
592:
587:
583:
580:
577:
572:
568:
544:
539:
535:
530:
526:
494:
489:
473:
470:
457:
437:
413:
389:
369:
345:
334:
333:
321:
316:
312:
306:
301:
298:
294:
290:
285:
282:
278:
272:
267:
263:
259:
254:
250:
244:
241:
238:
234:
230:
225:
221:
215:
210:
206:
200:
196:
190:
185:
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177:
174:
169:
165:
159:
154:
150:
144:
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134:
130:
126:
121:
118:
114:
108:
103:
100:
96:
90:
86:
80:
77:
74:
70:
66:
63:
23:, named after
15:
9:
6:
4:
3:
2:
1870:
1859:
1856:
1854:
1851:
1850:
1848:
1834:
1827:
1825:
1823:
1814:
1810:
1806:
1802:
1798:
1794:
1790:
1786:
1779:
1777:
1768:
1764:
1760:
1756:
1752:
1748:
1744:
1738:
1736:
1731:
1721:
1718:
1716:
1713:
1711:
1708:
1706:
1705:Hubbard model
1703:
1702:
1696:
1694:
1690:
1677:
1668:
1654:
1651:
1629:
1600:
1597:
1593:
1587:
1582:
1579:
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1532:
1528:
1522:
1519:
1516:
1513:
1510:
1506:
1502:
1496:
1493:
1488:
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1480:
1474:
1471:
1467:
1461:
1458:
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1442:
1439:
1435:
1432:
1429:
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1413:
1410:
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1400:
1395:
1392:
1388:
1382:
1378:
1372:
1369:
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1346:
1340:
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1332:
1328:
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1312:
1309:
1306:
1302:
1298:
1295:
1288:
1287:
1286:
1284:
1280:
1270:
1268:
1264:
1260:
1244:
1224:
1204:
1182:
1178:
1148:
1145:
1141:
1135:
1130:
1127:
1123:
1115:
1111:
1107:
1104:
1101:
1097:
1093:
1088:
1085:
1081:
1075:
1070:
1067:
1063:
1055:
1051:
1047:
1044:
1040:
1031:
1028:
1024:
1018:
1015:
1012:
1009:
1006:
1002:
998:
990:
986:
980:
972:
968:
959:
955:
949:
941:
937:
931:
927:
923:
920:
915:
912:
908:
902:
897:
894:
890:
884:
880:
874:
871:
868:
864:
860:
855:
852:
848:
842:
837:
834:
830:
824:
820:
814:
811:
808:
804:
800:
797:
790:
789:
788:
780:
778:
758:
755:
750:
746:
742:
731:
727:
722:
718:
709:
705:
682:
678:
675:
672:
667:
663:
634:
630:
625:
621:
612:
608:
585:
581:
578:
575:
570:
566:
537:
533:
528:
524:
515:
514:empty orbital
511:
510:
509:
487:
479:
469:
455:
435:
427:
411:
403:
387:
367:
359:
343:
314:
310:
304:
299:
296:
292:
288:
283:
280:
276:
270:
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261:
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219:
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204:
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163:
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132:
128:
124:
119:
116:
112:
106:
101:
98:
94:
88:
84:
78:
75:
72:
68:
64:
61:
54:
53:
52:
50:
46:
45:heavy fermion
42:
38:
34:
30:
26:
22:
1832:
1788:
1784:
1753:(1): 41–53.
1750:
1746:
1710:Kondo effect
1670:
1643:
1621:
1276:
1258:
1169:
786:
774:
708:local moment
707:
611:intermediate
610:
513:
475:
335:
47:systems and
41:Kondo effect
35:embedded in
20:
18:
1715:Kondo model
710:regime for
613:regime for
516:regime for
478:Fermi level
424:labels the
29:Hamiltonian
1847:Categories
1726:References
1267:Hill limit
402:wavevector
336:where the
1747:Phys. Rev
1601:σ
1588:†
1583:σ
1567:σ
1554:†
1549:σ
1523:σ
1507:∑
1494:σ
1475:σ
1448:σ
1433:σ
1423:∑
1414:σ
1401:†
1396:σ
1379:ϵ
1373:σ
1363:∑
1354:σ
1341:†
1336:σ
1319:ϵ
1313:σ
1303:∑
1149:σ
1136:†
1131:σ
1102:−
1089:σ
1076:†
1071:σ
1019:σ
1003:∑
994:↓
981:†
976:↓
963:↑
950:†
945:↑
928:∑
916:σ
903:†
898:σ
881:ϵ
875:σ
865:∑
856:σ
843:†
838:σ
821:ϵ
815:σ
805:∑
747:ϵ
743:≪
728:≪
719:ϵ
679:≈
664:ϵ
631:≈
622:ϵ
582:≫
567:ϵ
534:≫
525:ϵ
412:σ
315:σ
305:†
300:σ
284:σ
271:†
266:σ
243:σ
233:∑
224:↓
214:†
209:↓
199:↑
189:†
184:↑
168:σ
158:†
153:σ
143:σ
139:ϵ
133:σ
129:∑
120:σ
107:†
102:σ
85:ϵ
79:σ
69:∑
1813:55838235
1699:See also
1655:′
1497:′
1489:′
1451:′
1443:′
1793:Bibcode
1755:Bibcode
1263:orbital
472:Regimes
27:, is a
1811:
1622:where
1217:, and
1170:where
448:, and
404:, and
37:metals
1809:S2CID
1279:SU(4)
777:Kondo
1642:and
706:The
609:The
512:The
426:spin
19:The
1801:doi
1789:149
1763:doi
1751:124
655:or
558:or
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1359:+
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1333:k
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1307:k
1299:=
1296:H
1261:-
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1225:f
1205:j
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1032:k
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1013:k
1010:,
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999:+
991:j
987:f
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969:f
960:j
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924:U
921:+
913:j
909:f
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891:f
885:f
872:,
869:j
861:+
853:k
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835:k
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825:k
812:,
809:k
801:=
798:H
759:U
756:+
751:d
737:F
732:E
723:d
703:.
688:F
683:E
676:U
673:+
668:d
640:F
635:E
626:d
591:F
586:E
579:U
576:+
571:d
543:F
538:E
529:d
493:F
488:E
456:V
436:U
388:k
368:d
344:c
332:,
320:)
311:d
297:k
293:c
289:+
281:k
277:c
262:d
258:(
253:k
249:V
240:,
237:k
229:+
220:d
205:d
195:d
180:d
176:U
173:+
164:d
149:d
125:+
117:k
113:c
99:k
95:c
89:k
76:,
73:k
65:=
62:H
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