25:
540:
From the above results, many new quantum states of matter are predicted, including bosonic topological insulators (the SPT states protected by U(1) and time-reversal symmetry) and bosonic topological superconductors (the SPT states protected by time-reversal symmetry), as well as many other new SPT
144:
or mixed gauge-gravity anomaly for the symmetry group. As a result, the boundary of a SPT state is either gapless or degenerate, regardless how we cut the sample to form the boundary. A gapped non-degenerate boundary is impossible for a non-trivial SPT state. If the boundary is a gapped degenerate
151:
in non-trivial 2+1D SPT states carry non-trival statistics and fractional quantum numbers of the symmetry group. Monodromy defects are created by twisting the boundary condition along a cut by a symmetry transformation. The ends of such cut are the monodromy defects. For example, 2+1D bosonic
3362:
Mishra, Shantanu; Catarina, Gonçalo; Wu, Fupeng; Ortiz, Ricardo; Jacob, David; Eimre, Kristjan; Ma, Ji; Pignedoli, Carlo A.; Feng, Xinliang; Ruffieux, Pascal; Fernández-Rossier, Joaquín; Fasel, Roman (13 October 2021). "Observation of fractional edge excitations in nanographene spin chains".
2218:
all 1D gapped states are short-range entangled). Thus, if the
Hamiltonians have no symmetry, all their 1D gapped quantum states belong to one phase—the phase of trivial product states. On the other hand, if the Hamiltonians do have a symmetry, their 1D gapped quantum states are either
3153:
One should also note the semantical subtleness of the name SPT: "symmetry protected" does not mean that the stability of the state is conserved "because of the symmetry", but it is just meant that the symmetry is
1877:
672:
3628:
Wang, Juven C.; Gu, Zheng-Cheng; Wen, Xiao-Gang (22 January 2015). "Field-Theory
Representation of Gauge-Gravity Symmetry-Protected Topological Invariants, Group Cohomology, and Beyond".
2314:
2195:
theory. So the group (super-)cohomology theory allows us to construct many SPT orders even for interacting systems, which include interacting topological insulator/superconductor.
2182:
496:
1680:
931:
255:(due to its short-range entanglement). Note that the monodromy defects discussed above are not finite-energy excitations in the spectrum of the Hamiltonian, but defects created by
2571:
1071:
2030:
205:
3826:(23 September 2014). "Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory".
2780:
Wen, Xiao-Gang (9 August 2013). "Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders".
1983:
1783:
1031:
1364:
1199:
2963:
Lu, Yuan-Ming; Vishwanath, Ashvin (14 September 2012). "Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach".
2457:
2368:
1109:
881:
704:
535:
2719:
Pollmann, Frank; Berg, Erez; Turner, Ari M.; Oshikawa, Masaki (22 February 2012). "Symmetry protection of topological phases in one-dimensional quantum spin systems".
319:
112:
4139:
Schuch, Norbert; Pérez-García, David; Cirac, Ignacio (31 October 2011). "Classifying quantum phases using matrix product states and projected entangled pair states".
2094:
408:
1939:
1908:
1814:
1711:
307:
2598:
2508:
2426:
2341:
1739:
1624:
1554:
1526:
1302:
1253:
1137:
987:
959:
840:
2192:
1398:
1488:
2528:
1596:
1575:
1461:
1440:
1419:
1323:
1274:
1220:
1158:
812:
791:
770:
749:
191:
2+1D bosonic U(1) SPT states have a Hall conductance that is quantized as an even integer. 2+1D bosonic SO(3) SPT states have a quantized spin Hall conductance.
2226:
Such an understanding allows one to classify all 1D gapped quantum phases: All 1D gapped phases are classified by the following three mathematical objects:
2902:
Wen, Xiao-Gang (31 January 2014). "Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions".
99:
however, they all can be smoothly deformed into the same trivial product state without a phase transition, if the symmetry is broken during the deformation
93:
distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry
3024:
Liu, Zheng-Xin; Mei, Jia-Wei; Ye, Peng; Wen, Xiao-Gang (24 December 2014). "U(1)×U(1)symmetry-protected topological order in
Gutzwiller wave functions".
148:
306:, one obtains the following general picture of gapped phases at zero temperature. All gapped zero-temperature phases can be divided into two classes:
104:
The above definition works for both bosonic systems and fermionic systems, which leads to the notions of bosonic SPT order and fermionic SPT order.
43:
3170:"Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State"
275:
spin rotation symmetry. Note that
Haldane phases of even-integer-spin chain do not have SPT order. A more well known example of SPT order is the
4017:
Turner, Ari M.; Pollmann, Frank; Berg, Erez (8 February 2011). "Topological phases of one-dimensional fermions: An entanglement point of view".
3887:
Verstraete, F.; Cirac, J. I.; Latorre, J. I.; Rico, E.; Wolf, M. M. (14 April 2005). "Renormalization-Group
Transformations on Quantum States".
212:
can be robust against any local perturbations, while the gapless boundary excitations in SPT order are robust only against local perturbations
537:
is the
Abelian group formed by (d+1)D topologically ordered phases that have no non-trivial topological excitations (referred as iTO phases).
3956:
Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang (13 January 2011). "Classification of gapped symmetric phases in one-dimensional spin systems".
3761:(4 May 2015). "Construction of bosonic symmetry-protected-trivial states and their topological invariants via G×SO(∞) nonlinear σ models".
3697:
Kapustin, Anton; Thorngren, Ryan; Turzillo, Alex; Wang, Zitao (2015). "Fermionic symmetry protected topological phases and cobordisms".
3217:
Haldane, F.D.M. (1983). "Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model".
88:
are used (leading to equivalence classes corresponding to certain fixed points). The SPT order has the following defining properties:
2619:
3559:"Physics of Three-Dimensional Bosonic Topological Insulators: Surface-Deconfined Criticality and Quantized Magnetoelectric Effect"
2841:
Levin, Michael; Gu, Zheng-Cheng (10 September 2012). "Braiding statistics approach to symmetry-protected topological phases".
3431:(22 December 2011). "Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations".
1824:
2658:(26 October 2009). "Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order".
4211:
61:
3085:
Liu, Zheng-Xin; Wen, Xiao-Gang (7 February 2013). "Symmetry-Protected
Quantum Spin Hall Phases in Two Dimensions".
549:
200:
SPT states are short-range entangled while topologically ordered states are long-range entangled. Both intrinsic
39:
4216:
2229:
2220:
338:
331:
3615:
Anton
Kapustin, "Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology"
2099:
413:
294:
states are not SPT states. They are states with (intrinsic) topological order and long-range entanglements.
1635:
889:
145:
state, the degeneracy may be caused by spontaneous symmetry breaking and/or (intrinsic) topological order.
3496:(4 April 2013). "Symmetry protected topological orders and the group cohomology of their symmetry group".
4078:
Fidkowski, Lukasz; Kitaev, Alexei (8 February 2011). "Topological phases of fermions in one dimension".
141:
345:. For bosonic SPT phases with pure gauge anomalous boundary, it was shown that they are classified by
2536:
1037:
410:. For other (d+1)D SPT states with mixed gauge-gravity anomalous boundary, they can be described by
1989:
2624:
1945:
1745:
993:
4206:
1329:
1164:
2435:
2346:
1082:
847:
677:
501:
334:
phases, SPT phases, and their mix (symmetry breaking order and SPT order can appear together).
284:
3260:
Affleck, Ian; Haldane, F. D. M. (1 September 1987). "Critical theory of quantum spin chains".
2614:
2042:
356:
276:
248:
236:
85:
2198:
1914:
1883:
1789:
1686:
4158:
4097:
4036:
3975:
3906:
3845:
3780:
3716:
3647:
3580:
3515:
3450:
3382:
3320:
3269:
3226:
3181:
3104:
3043:
2982:
2921:
2860:
2799:
2738:
2677:
2576:
2573:), the 1D gapped phases are classified by the projective representations of symmetry group
2462:
2373:
2319:
2204:
1717:
1602:
1532:
1499:
1280:
1231:
1115:
965:
937:
818:
303:
108:
2188:
theory can give us various SPT phases in any dimensions with any on-site symmetry groups.
1374:
8:
2207:
and SPT order, one can obtain a complete classification of all 1D gapped quantum phases.
1467:
291:
4162:
4101:
4040:
3979:
3910:
3849:
3784:
3720:
3651:
3584:
3519:
3454:
3386:
3324:
3273:
3230:
3185:
3108:
3047:
2986:
2925:
2864:
2803:
2742:
2681:
4231:
4182:
4148:
4121:
4087:
4060:
4026:
3999:
3965:
3938:
3896:
3869:
3835:
3804:
3770:
3740:
3706:
3679:
3637:
3570:
3539:
3505:
3474:
3440:
3406:
3372:
3344:
3136:
3094:
3067:
3033:
3006:
2972:
2945:
2911:
2884:
2850:
2823:
2789:
2762:
2728:
2701:
2667:
2513:
1581:
1560:
1446:
1425:
1404:
1308:
1259:
1205:
1143:
797:
776:
755:
734:
4226:
4221:
4174:
4113:
4064:
4052:
3991:
3930:
3922:
3873:
3861:
3796:
3732:
3671:
3663:
3598:
3543:
3531:
3466:
3410:
3398:
3348:
3336:
3332:
3293:
3285:
3242:
3238:
3199:
3128:
3120:
3059:
2998:
2937:
2888:
2876:
2815:
2754:
2693:
2629:
2211:
327:
315:
244:
232:
228:
220:
are topologically protected, while the gapless boundary excitations in SPT order are
217:
209:
201:
128:
120:
4186:
3942:
3808:
3744:
3478:
3140:
3071:
3010:
2949:
2827:
2766:
2705:
4166:
4125:
4105:
4044:
4003:
3983:
3914:
3853:
3788:
3724:
3683:
3659:
3655:
3588:
3523:
3458:
3390:
3328:
3277:
3234:
3189:
3116:
3112:
3051:
2990:
2929:
2868:
2807:
2746:
2685:
2429:
2185:
346:
81:
quantum-mechanical states of matter that have a symmetry and a finite energy gap.
3918:
3728:
3194:
3169:
330:). All short-range entangled phases can be further divided into three classes:
4170:
4109:
4048:
3987:
3857:
3792:
3527:
3462:
3394:
3055:
2994:
2933:
2872:
2811:
2750:
2689:
195:
3690:
3593:
3558:
4200:
4178:
4117:
4056:
3995:
3926:
3865:
3823:
3800:
3758:
3736:
3667:
3602:
3535:
3493:
3470:
3428:
3340:
3289:
3281:
3246:
3203:
3124:
3063:
3002:
2941:
2880:
2819:
2758:
2697:
2655:
78:
3934:
3675:
3402:
3132:
342:
252:
240:
3297:
208:. The difference is subtle: the gapless boundary excitations in intrinsic
3901:
2199:
A complete classification of 1D gapped quantum phases (with interactions)
140:
The boundary effective theory of a non-trivial SPT state always has pure
124:
2609:
131:
we may also refer the SPT order as
Symmetry Protected "Trivial" order.
3311:
Affleck, I (15 May 1989). "Quantum spin chains and the
Haldane gap".
2184:. Just like group theory can give us 230 crystal structures in 3+1D,
3377:
4153:
4092:
4031:
3970:
3840:
3775:
3711:
3642:
3616:
3575:
3510:
3445:
3099:
3038:
2977:
2916:
2855:
2794:
2733:
2672:
3880:
2712:
3696:
3253:
2191:
On the other hand, the fermionic SPT orders are described by
272:
4132:
196:
Relation between SPT order and (intrinsic) topological order
280:
268:
3886:
2718:
271:
of odd-integer spin chain. It is a SPT phase protected by
297:
127:). Since short-range entangled states have only trivial
3210:
4138:
4010:
353:
are labeled by the elements in group cohomology class
279:
of non-interacting fermions, a SPT phase protected by
16:
Type of topological order in condensed matter physics
3550:
3361:
3304:
3161:
2579:
2539:
2516:
2465:
2438:
2376:
2349:
2322:
2232:
2102:
2045:
1992:
1948:
1917:
1886:
1827:
1792:
1748:
1720:
1689:
1638:
1628:
1+1D: odd-integer-spin chain; 2+1D: spin Hall effect
1605:
1584:
1563:
1535:
1502:
1470:
1449:
1428:
1407:
1377:
1332:
1311:
1283:
1262:
1234:
1208:
1167:
1146:
1118:
1085:
1040:
996:
968:
940:
892:
850:
821:
800:
779:
758:
737:
680:
552:
504:
416:
359:
3557:
Vishwanath, Ashvin; Senthil, T. (28 February 2013).
4071:
216:. So the gapless boundary excitations in intrinsic
204:, and also SPT order, can sometimes have protected
34:
may be too technical for most readers to understand
4016:
3485:
3268:(10). American Physical Society (APS): 5291–5300.
3180:(15). American Physical Society (APS): 1153–1156.
2592:
2565:
2522:
2502:
2451:
2420:
2362:
2335:
2308:
2176:
2088:
2024:
1977:
1933:
1902:
1871:
1808:
1777:
1733:
1705:
1674:
1618:
1590:
1569:
1548:
1520:
1482:
1455:
1434:
1413:
1392:
1358:
1317:
1296:
1268:
1247:
1214:
1193:
1152:
1131:
1103:
1065:
1025:
981:
953:
925:
875:
834:
806:
785:
764:
743:
698:
666:
545:A list of bosonic SPT states from group cohomology
529:
490:
402:
3556:
3158:by the interactions corresponding to the process.
1872:{\displaystyle Z_{2}\times Z_{2}\times Z_{2}^{T}}
4198:
3017:
2210:First, it is shown that there is no (intrinsic)
4077:
2956:
84:To derive the results in a most-invariant way,
4086:(7). American Physical Society (APS): 075103.
3949:
3621:
3259:
349:theory: those (d+1)D SPT states with symmetry
119:(by contrast: for long-range entanglement see
2510:classifies the projective representations of
2370:the symmetry group of the ground states, and
667:{\displaystyle H^{d+1}\oplus _{k=1}^{d}H^{k}}
168:identical elementary monodromy defects in a Z
3492:Chen, Xie; Gu, Zheng-Cheng; Liu, Zheng-Xin;
3491:
2962:
251:for finite-energy excitations, nor emergent
134:
3023:
243:. In contrast, a SPT order has no emergent
3426:
2834:
2343:is the symmetry group of the Hamiltonian,
75:Symmetry-protected topological (SPT) order
4152:
4091:
4030:
3969:
3955:
3900:
3839:
3774:
3710:
3641:
3627:
3592:
3574:
3509:
3444:
3376:
3193:
3098:
3078:
3037:
2976:
2915:
2854:
2793:
2732:
2671:
62:Learn how and when to remove this message
46:, without removing the technical details.
2649:
2647:
2645:
2620:Periodic table of topological invariants
3310:
3216:
3167:
2840:
2309:{\displaystyle (G_{H},G_{\Psi },H^{2})}
4199:
3422:
3420:
3084:
2177:{\displaystyle \oplus _{k=1}^{d}H^{k}}
541:states protected by other symmetries.
491:{\displaystyle \oplus _{k=1}^{d}H^{k}}
298:Group cohomology theory for SPT phases
267:The first example of SPT order is the
3821:
2773:
2653:
2642:
2530:.) If there is no symmetry breaking (
1675:{\displaystyle SO(3)\times Z_{2}^{T}}
926:{\displaystyle U(1)\rtimes Z_{2}^{T}}
123:, which is not related to the famous
44:make it understandable to non-experts
3313:Journal of Physics: Condensed Matter
18:
3757:
3417:
3168:Haldane, F. D. M. (11 April 1983).
2901:
2779:
2223:phases, SPT phases, and their mix.
1224:bosonic topological superconductor
13:
2545:
2444:
2395:
2355:
2280:
2254:
14:
4243:
3319:(19). IOP Publishing: 3047–3072.
2096:. The phases after "+" come from
111:, we can say that SPT states are
2039:The phases before "+" come from
706:= time-reversal-symmetry group)
156:SPT states are classified by a Z
23:
3815:
3751:
3609:
3355:
2566:{\displaystyle G_{\Psi }=G_{H}}
1066:{\displaystyle Z\oplus Z_{2}+Z}
227:We also know that an intrinsic
3699:Journal of High Energy Physics
3660:10.1103/physrevlett.114.031601
3147:
3117:10.1103/physrevlett.110.067205
2895:
2497:
2494:
2488:
2476:
2415:
2412:
2406:
2387:
2303:
2300:
2297:
2291:
2272:
2233:
2171:
2134:
2083:
2080:
2074:
2062:
2025:{\displaystyle 12Z_{2}+2Z_{2}}
1651:
1645:
1515:
1509:
1387:
1381:
1075:bosonic topological insulator
902:
896:
661:
624:
590:
587:
581:
569:
485:
448:
397:
394:
388:
376:
290:On the other hand, fractional
214:that do not break the symmetry
1:
3919:10.1103/physrevlett.94.140601
2635:
844:iTO phases with no symmetry:
86:renormalization group methods
3239:10.1016/0375-9601(83)90631-x
1978:{\displaystyle 9Z_{2}+Z_{2}}
1778:{\displaystyle 3Z_{2}+Z_{2}}
1026:{\displaystyle 2Z_{2}+Z_{2}}
206:gapless boundary excitations
7:
3427:Chen, Xie; Liu, Zheng-Xin;
3225:(9). Elsevier BV: 464–468.
3195:10.1103/physrevlett.50.1153
2603:
1359:{\displaystyle Z_{n}+Z_{n}}
1194:{\displaystyle Z_{2}+Z_{2}}
262:
184:which is not a multiple of
10:
4248:
4171:10.1103/physrevb.84.165139
4110:10.1103/physrevb.83.075103
4049:10.1103/physrevb.83.075102
3988:10.1103/physrevb.83.035107
3858:10.1103/physrevb.90.115141
3793:10.1103/physrevb.91.205101
3528:10.1103/physrevb.87.155114
3463:10.1103/physrevb.84.235141
3395:10.1038/s41586-021-03842-3
3333:10.1088/0953-8984/1/19/001
3056:10.1103/physrevb.90.235146
2995:10.1103/physrevb.86.125119
2934:10.1103/physrevb.89.035147
2873:10.1103/physrevb.86.115109
2812:10.1103/physrevd.88.045013
2751:10.1103/physrevb.85.075125
2690:10.1103/physrevb.80.155131
1492:2+1D: quantum Hall effect
3594:10.1103/physrevx.3.011016
2452:{\displaystyle G_{\Psi }}
2363:{\displaystyle G_{\Psi }}
1104:{\displaystyle Z_{2}^{T}}
876:{\displaystyle iTO^{d+1}}
699:{\displaystyle Z_{2}^{T}}
530:{\displaystyle iTO^{d+1}}
326:phases with no intrinsic
135:Characteristic properties
4212:Condensed matter physics
3282:10.1103/physrevb.36.5291
2625:Quantum spin Hall effect
341:orders are described by
3889:Physical Review Letters
3729:10.1007/jhep12(2015)052
3630:Physical Review Letters
3174:Physical Review Letters
3087:Physical Review Letters
2089:{\displaystyle H^{d+1}}
403:{\displaystyle H^{d+1}}
2594:
2567:
2524:
2504:
2453:
2422:
2364:
2337:
2310:
2193:group super-cohomology
2178:
2090:
2026:
1979:
1935:
1934:{\displaystyle 6Z_{2}}
1904:
1903:{\displaystyle 4Z_{2}}
1873:
1810:
1809:{\displaystyle 2Z_{2}}
1779:
1735:
1707:
1706:{\displaystyle 2Z_{2}}
1676:
1620:
1592:
1571:
1550:
1522:
1484:
1457:
1436:
1415:
1394:
1360:
1319:
1298:
1270:
1249:
1216:
1195:
1154:
1133:
1105:
1067:
1027:
983:
955:
927:
877:
836:
808:
787:
766:
745:
700:
668:
531:
492:
404:
337:It is well known that
314:phases with intrinsic
285:time reversal symmetry
77:is a kind of order in
2615:Topological insulator
2595:
2593:{\displaystyle G_{H}}
2568:
2525:
2505:
2503:{\displaystyle H^{2}}
2454:
2423:
2421:{\displaystyle H^{2}}
2365:
2338:
2336:{\displaystyle G_{H}}
2311:
2203:Using the notions of
2179:
2091:
2027:
1980:
1936:
1905:
1874:
1811:
1780:
1736:
1734:{\displaystyle Z_{2}}
1708:
1677:
1621:
1619:{\displaystyle Z_{2}}
1593:
1572:
1551:
1549:{\displaystyle Z_{2}}
1523:
1521:{\displaystyle SO(3)}
1485:
1458:
1437:
1416:
1395:
1361:
1320:
1299:
1297:{\displaystyle Z_{n}}
1271:
1250:
1248:{\displaystyle Z_{n}}
1217:
1196:
1155:
1134:
1132:{\displaystyle Z_{2}}
1106:
1068:
1028:
984:
982:{\displaystyle Z_{2}}
956:
954:{\displaystyle Z_{2}}
928:
878:
837:
835:{\displaystyle Z_{2}}
809:
788:
767:
746:
701:
669:
532:
493:
405:
320:short-range entangled
277:topological insulator
249:fractional statistics
237:fractional statistics
172:SPT state labeled by
113:short-range entangled
4217:Mathematical physics
2577:
2537:
2514:
2463:
2436:
2374:
2347:
2320:
2230:
2205:quantum entanglement
2100:
2043:
1990:
1946:
1915:
1884:
1825:
1790:
1746:
1718:
1687:
1636:
1603:
1582:
1561:
1533:
1500:
1468:
1447:
1426:
1405:
1393:{\displaystyle U(1)}
1375:
1330:
1309:
1281:
1260:
1232:
1206:
1165:
1144:
1116:
1083:
1038:
994:
966:
938:
890:
848:
819:
798:
777:
756:
735:
678:
550:
502:
414:
357:
308:long-range entangled
304:quantum entanglement
302:Using the notion of
176:will carry a total Z
164:. One can show that
109:quantum entanglement
107:Using the notion of
4163:2011PhRvB..84p5139S
4102:2011PhRvB..83g5103F
4041:2011PhRvB..83g5102T
3980:2011PhRvB..83c5107C
3911:2005PhRvL..94n0601V
3850:2014PhRvB..90k5141G
3785:2015PhRvB..91t5101W
3721:2015JHEP...12..052K
3652:2015PhRvL.114c1601W
3585:2013PhRvX...3a1016V
3520:2013PhRvB..87o5114C
3455:2011PhRvB..84w5141C
3387:2021Natur.598..287M
3325:1989JPCM....1.3047A
3274:1987PhRvB..36.5291A
3231:1983PhLA...93..464H
3186:1983PhRvL..50.1153H
3109:2013PhRvL.110f7205L
3048:2014PhRvB..90w5146L
2987:2012PhRvB..86l5119L
2926:2014PhRvB..89c5147W
2865:2012PhRvB..86k5109L
2804:2013PhRvD..88d5013W
2743:2012PhRvB..85g5125P
2682:2009PhRvB..80o5131G
2123:
1868:
1671:
1483:{\displaystyle Z+Z}
1100:
922:
695:
613:
437:
2590:
2563:
2520:
2500:
2449:
2418:
2360:
2333:
2306:
2174:
2103:
2086:
2022:
1975:
1931:
1900:
1869:
1854:
1806:
1775:
1731:
1703:
1672:
1657:
1616:
1588:
1567:
1546:
1518:
1480:
1453:
1432:
1411:
1390:
1356:
1315:
1294:
1266:
1245:
1212:
1191:
1150:
1129:
1101:
1086:
1063:
1023:
979:
951:
923:
908:
873:
832:
804:
783:
762:
741:
696:
681:
664:
593:
527:
488:
417:
400:
222:symmetry protected
129:topological orders
4141:Physical Review B
4080:Physical Review B
4019:Physical Review B
3958:Physical Review B
3828:Physical Review B
3822:Gu, Zheng-Cheng;
3763:Physical Review B
3563:Physical Review X
3498:Physical Review B
3433:Physical Review B
3371:(7880): 287–292.
3262:Physical Review B
3219:Physics Letters A
3026:Physical Review B
2965:Physical Review B
2904:Physical Review B
2843:Physical Review B
2782:Physical Review D
2721:Physical Review B
2660:Physical Review B
2654:Gu, Zheng-Cheng;
2630:Topological order
2523:{\displaystyle G}
2221:symmetry-breaking
2212:topological order
2037:
2036:
1591:{\displaystyle 0}
1570:{\displaystyle Z}
1456:{\displaystyle 0}
1435:{\displaystyle Z}
1414:{\displaystyle 0}
1318:{\displaystyle 0}
1269:{\displaystyle 0}
1215:{\displaystyle 0}
1153:{\displaystyle 0}
807:{\displaystyle 0}
786:{\displaystyle Z}
765:{\displaystyle 0}
744:{\displaystyle 0}
339:symmetry-breaking
332:symmetry-breaking
328:topological order
316:topological order
259:the Hamiltonian.
245:fractional charge
233:fractional charge
229:topological order
218:topological order
210:topological order
202:topological order
149:Monodromy defects
121:topological order
72:
71:
64:
4239:
4191:
4190:
4156:
4136:
4130:
4129:
4095:
4075:
4069:
4068:
4034:
4014:
4008:
4007:
3973:
3953:
3947:
3946:
3904:
3902:quant-ph/0410227
3884:
3878:
3877:
3843:
3819:
3813:
3812:
3778:
3755:
3749:
3748:
3714:
3694:
3688:
3687:
3645:
3625:
3619:
3613:
3607:
3606:
3596:
3578:
3554:
3548:
3547:
3513:
3489:
3483:
3482:
3448:
3424:
3415:
3414:
3380:
3359:
3353:
3352:
3308:
3302:
3301:
3257:
3251:
3250:
3214:
3208:
3207:
3197:
3165:
3159:
3151:
3145:
3144:
3102:
3082:
3076:
3075:
3041:
3021:
3015:
3014:
2980:
2960:
2954:
2953:
2919:
2899:
2893:
2892:
2858:
2838:
2832:
2831:
2797:
2777:
2771:
2770:
2736:
2716:
2710:
2709:
2675:
2651:
2599:
2597:
2596:
2591:
2589:
2588:
2572:
2570:
2569:
2564:
2562:
2561:
2549:
2548:
2529:
2527:
2526:
2521:
2509:
2507:
2506:
2501:
2475:
2474:
2458:
2456:
2455:
2450:
2448:
2447:
2430:group cohomology
2427:
2425:
2424:
2419:
2399:
2398:
2386:
2385:
2369:
2367:
2366:
2361:
2359:
2358:
2342:
2340:
2339:
2334:
2332:
2331:
2315:
2313:
2312:
2307:
2284:
2283:
2271:
2270:
2258:
2257:
2245:
2244:
2186:group cohomology
2183:
2181:
2180:
2175:
2170:
2169:
2133:
2132:
2122:
2117:
2095:
2093:
2092:
2087:
2061:
2060:
2031:
2029:
2028:
2023:
2021:
2020:
2005:
2004:
1984:
1982:
1981:
1976:
1974:
1973:
1961:
1960:
1940:
1938:
1937:
1932:
1930:
1929:
1909:
1907:
1906:
1901:
1899:
1898:
1878:
1876:
1875:
1870:
1867:
1862:
1850:
1849:
1837:
1836:
1815:
1813:
1812:
1807:
1805:
1804:
1784:
1782:
1781:
1776:
1774:
1773:
1761:
1760:
1740:
1738:
1737:
1732:
1730:
1729:
1712:
1710:
1709:
1704:
1702:
1701:
1681:
1679:
1678:
1673:
1670:
1665:
1625:
1623:
1622:
1617:
1615:
1614:
1597:
1595:
1594:
1589:
1576:
1574:
1573:
1568:
1555:
1553:
1552:
1547:
1545:
1544:
1527:
1525:
1524:
1519:
1489:
1487:
1486:
1481:
1462:
1460:
1459:
1454:
1441:
1439:
1438:
1433:
1420:
1418:
1417:
1412:
1399:
1397:
1396:
1391:
1365:
1363:
1362:
1357:
1355:
1354:
1342:
1341:
1324:
1322:
1321:
1316:
1303:
1301:
1300:
1295:
1293:
1292:
1275:
1273:
1272:
1267:
1254:
1252:
1251:
1246:
1244:
1243:
1221:
1219:
1218:
1213:
1200:
1198:
1197:
1192:
1190:
1189:
1177:
1176:
1159:
1157:
1156:
1151:
1138:
1136:
1135:
1130:
1128:
1127:
1110:
1108:
1107:
1102:
1099:
1094:
1072:
1070:
1069:
1064:
1056:
1055:
1032:
1030:
1029:
1024:
1022:
1021:
1009:
1008:
988:
986:
985:
980:
978:
977:
960:
958:
957:
952:
950:
949:
932:
930:
929:
924:
921:
916:
882:
880:
879:
874:
872:
871:
841:
839:
838:
833:
831:
830:
813:
811:
810:
805:
792:
790:
789:
784:
771:
769:
768:
763:
750:
748:
747:
742:
709:
708:
705:
703:
702:
697:
694:
689:
673:
671:
670:
665:
660:
659:
623:
622:
612:
607:
568:
567:
536:
534:
533:
528:
526:
525:
497:
495:
494:
489:
484:
483:
447:
446:
436:
431:
409:
407:
406:
401:
375:
374:
347:group cohomology
79:zero-temperature
67:
60:
56:
53:
47:
27:
26:
19:
4247:
4246:
4242:
4241:
4240:
4238:
4237:
4236:
4197:
4196:
4195:
4194:
4137:
4133:
4076:
4072:
4015:
4011:
3954:
3950:
3885:
3881:
3820:
3816:
3756:
3752:
3695:
3691:
3626:
3622:
3617:arXiv:1403.1467
3614:
3610:
3555:
3551:
3490:
3486:
3425:
3418:
3360:
3356:
3309:
3305:
3258:
3254:
3215:
3211:
3166:
3162:
3152:
3148:
3083:
3079:
3022:
3018:
2961:
2957:
2900:
2896:
2839:
2835:
2778:
2774:
2717:
2713:
2652:
2643:
2638:
2606:
2584:
2580:
2578:
2575:
2574:
2557:
2553:
2544:
2540:
2538:
2535:
2534:
2515:
2512:
2511:
2470:
2466:
2464:
2461:
2460:
2443:
2439:
2437:
2434:
2433:
2394:
2390:
2381:
2377:
2375:
2372:
2371:
2354:
2350:
2348:
2345:
2344:
2327:
2323:
2321:
2318:
2317:
2279:
2275:
2266:
2262:
2253:
2249:
2240:
2236:
2231:
2228:
2227:
2201:
2153:
2149:
2128:
2124:
2118:
2107:
2101:
2098:
2097:
2050:
2046:
2044:
2041:
2040:
2016:
2012:
2000:
1996:
1991:
1988:
1987:
1969:
1965:
1956:
1952:
1947:
1944:
1943:
1925:
1921:
1916:
1913:
1912:
1894:
1890:
1885:
1882:
1881:
1863:
1858:
1845:
1841:
1832:
1828:
1826:
1823:
1822:
1800:
1796:
1791:
1788:
1787:
1769:
1765:
1756:
1752:
1747:
1744:
1743:
1725:
1721:
1719:
1716:
1715:
1697:
1693:
1688:
1685:
1684:
1666:
1661:
1637:
1634:
1633:
1610:
1606:
1604:
1601:
1600:
1583:
1580:
1579:
1562:
1559:
1558:
1540:
1536:
1534:
1531:
1530:
1501:
1498:
1497:
1469:
1466:
1465:
1448:
1445:
1444:
1427:
1424:
1423:
1406:
1403:
1402:
1376:
1373:
1372:
1350:
1346:
1337:
1333:
1331:
1328:
1327:
1310:
1307:
1306:
1288:
1284:
1282:
1279:
1278:
1261:
1258:
1257:
1239:
1235:
1233:
1230:
1229:
1207:
1204:
1203:
1185:
1181:
1172:
1168:
1166:
1163:
1162:
1145:
1142:
1141:
1123:
1119:
1117:
1114:
1113:
1095:
1090:
1084:
1081:
1080:
1051:
1047:
1039:
1036:
1035:
1017:
1013:
1004:
1000:
995:
992:
991:
973:
969:
967:
964:
963:
945:
941:
939:
936:
935:
917:
912:
891:
888:
887:
861:
857:
849:
846:
845:
826:
822:
820:
817:
816:
799:
796:
795:
778:
775:
774:
757:
754:
753:
736:
733:
732:
690:
685:
679:
676:
675:
643:
639:
618:
614:
608:
597:
557:
553:
551:
548:
547:
515:
511:
503:
500:
499:
467:
463:
442:
438:
432:
421:
415:
412:
411:
364:
360:
358:
355:
354:
300:
265:
239:, and emergent
198:
180:quantum number
179:
171:
159:
155:
137:
117:with a symmetry
96:
68:
57:
51:
48:
40:help improve it
37:
28:
24:
17:
12:
11:
5:
4245:
4235:
4234:
4229:
4224:
4219:
4214:
4209:
4207:Quantum phases
4193:
4192:
4147:(16): 165139.
4131:
4070:
4009:
3948:
3895:(14): 140601.
3879:
3834:(11): 115141.
3824:Wen, Xiao-Gang
3814:
3769:(20): 205101.
3759:Wen, Xiao-Gang
3750:
3689:
3620:
3608:
3549:
3504:(15): 155114.
3494:Wen, Xiao-Gang
3484:
3439:(23): 235141.
3429:Wen, Xiao-Gang
3416:
3354:
3303:
3252:
3209:
3160:
3146:
3077:
3032:(23): 235146.
3016:
2971:(12): 125119.
2955:
2894:
2849:(11): 114109.
2833:
2772:
2711:
2666:(15): 155131.
2656:Wen, Xiao-Gang
2640:
2639:
2637:
2634:
2633:
2632:
2627:
2622:
2617:
2612:
2605:
2602:
2587:
2583:
2560:
2556:
2552:
2547:
2543:
2519:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2473:
2469:
2446:
2442:
2417:
2414:
2411:
2408:
2405:
2402:
2397:
2393:
2389:
2384:
2380:
2357:
2353:
2330:
2326:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2282:
2278:
2274:
2269:
2265:
2261:
2256:
2252:
2248:
2243:
2239:
2235:
2200:
2197:
2173:
2168:
2165:
2162:
2159:
2156:
2152:
2148:
2145:
2142:
2139:
2136:
2131:
2127:
2121:
2116:
2113:
2110:
2106:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2059:
2056:
2053:
2049:
2035:
2034:
2032:
2019:
2015:
2011:
2008:
2003:
1999:
1995:
1985:
1972:
1968:
1964:
1959:
1955:
1951:
1941:
1928:
1924:
1920:
1910:
1897:
1893:
1889:
1879:
1866:
1861:
1857:
1853:
1848:
1844:
1840:
1835:
1831:
1819:
1818:
1816:
1803:
1799:
1795:
1785:
1772:
1768:
1764:
1759:
1755:
1751:
1741:
1728:
1724:
1713:
1700:
1696:
1692:
1682:
1669:
1664:
1660:
1656:
1653:
1650:
1647:
1644:
1641:
1630:
1629:
1626:
1613:
1609:
1598:
1587:
1577:
1566:
1556:
1543:
1539:
1528:
1517:
1514:
1511:
1508:
1505:
1494:
1493:
1490:
1479:
1476:
1473:
1463:
1452:
1442:
1431:
1421:
1410:
1400:
1389:
1386:
1383:
1380:
1369:
1368:
1366:
1353:
1349:
1345:
1340:
1336:
1325:
1314:
1304:
1291:
1287:
1276:
1265:
1255:
1242:
1238:
1226:
1225:
1222:
1211:
1201:
1188:
1184:
1180:
1175:
1171:
1160:
1149:
1139:
1126:
1122:
1111:
1098:
1093:
1089:
1077:
1076:
1073:
1062:
1059:
1054:
1050:
1046:
1043:
1033:
1020:
1016:
1012:
1007:
1003:
999:
989:
976:
972:
961:
948:
944:
933:
920:
915:
911:
907:
904:
901:
898:
895:
884:
883:
870:
867:
864:
860:
856:
853:
842:
829:
825:
814:
803:
793:
782:
772:
761:
751:
740:
729:
728:
725:
722:
719:
716:
713:
712:symmetry group
693:
688:
684:
663:
658:
655:
652:
649:
646:
642:
638:
635:
632:
629:
626:
621:
617:
611:
606:
603:
600:
596:
592:
589:
586:
583:
580:
577:
574:
571:
566:
563:
560:
556:
524:
521:
518:
514:
510:
507:
487:
482:
479:
476:
473:
470:
466:
462:
459:
456:
453:
450:
445:
441:
435:
430:
427:
424:
420:
399:
396:
393:
390:
387:
384:
381:
378:
373:
370:
367:
363:
299:
296:
264:
261:
197:
194:
193:
192:
189:
177:
169:
157:
153:
146:
136:
133:
70:
69:
31:
29:
22:
15:
9:
6:
4:
3:
2:
4244:
4233:
4230:
4228:
4225:
4223:
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4204:
4202:
4188:
4184:
4180:
4176:
4172:
4168:
4164:
4160:
4155:
4150:
4146:
4142:
4135:
4127:
4123:
4119:
4115:
4111:
4107:
4103:
4099:
4094:
4089:
4085:
4081:
4074:
4066:
4062:
4058:
4054:
4050:
4046:
4042:
4038:
4033:
4028:
4025:(7): 075102.
4024:
4020:
4013:
4005:
4001:
3997:
3993:
3989:
3985:
3981:
3977:
3972:
3967:
3964:(3): 035107.
3963:
3959:
3952:
3944:
3940:
3936:
3932:
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231:has emergent
230:
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32:This article
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343:group theory
336:
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292:quantum Hall
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253:gauge theory
241:gauge theory
226:
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116:
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103:
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2428:the second
235:, emergent
125:EPR paradox
4201:Categories
3378:2105.09102
2636:References
2610:AKLT Model
4232:Emergence
4179:1098-0121
4154:1010.3732
4118:1098-0121
4093:1008.4138
4065:118491997
4057:1098-0121
4032:1008.4346
3996:1098-0121
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3927:0031-9007
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3776:1410.8477
3737:1029-8479
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3668:0031-9007
3643:1405.7689
3603:2160-3308
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2546:Ψ
2445:Ψ
2432:class of
2396:Ψ
2356:Ψ
2281:Ψ
2255:Ψ
2164:−
2105:⊕
1852:×
1839:×
1655:×
1045:⊕
906:⋊
654:−
595:⊕
478:−
419:⊕
257:modifying
52:June 2020
4227:Topology
4222:Symmetry
4187:74872240
3943:21362387
3935:15904055
3809:13950401
3745:42613274
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2706:15114579
2604:See also
2316:, where
727:comment
498:, where
322:phases (
310:phases (
263:Examples
160:integer
4159:Bibcode
4126:1201670
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115:states
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283:and
281:U(1)
97:(b)
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1187:2
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998:2
975:2
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903:)
900:1
897:(
894:U
869:1
866:+
863:d
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674:(
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657:k
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625:[
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602:=
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582:(
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573:G
570:[
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449:[
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434:d
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398:]
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392:1
389:(
386:U
383:,
380:G
377:[
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369:+
366:d
362:H
351:G
247:/
188:.
186:n
178:n
174:m
170:n
166:n
162:m
158:n
154:n
152:Z
65:)
59:(
54:)
50:(
36:.
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