434:, and the MBL phenomenon was shown to survive. In 2005 and 2006, this was extended to high orders in perturbation theory in high dimensional systems. MBL was argued to survive at least at low energy density. A series of numerical works provided further evidence for the phenomenon in one dimensional systems, at all energy densities (“infinite temperature”). Finally, in 2014 Imbrie presented a proof of MBL for certain one dimensional spin chains with strong disorder, with the localization being stable to arbitrary local perturbations – i.e. the systems were shown to be in a many body localized phase.
27:
381:. A system in a many body localized phase exhibits MBL, and continues to exhibit MBL even when subject to arbitrary local perturbations. Eigenstates of systems exhibiting MBL do not obey the ETH, and generically follow an "area law" for entanglement entropy (i.e. the entanglement entropy scales with the surface area of subregion
99:). The process of thermalization erases local memory of the initial conditions. In textbooks, thermalization is ensured by coupling the system to an external environment or "reservoir," with which the system can exchange energy. What happens if the system is isolated from the environment, and evolves according to its own
892:
418:
MBL was first proposed by P.W. Anderson in 1958 as a possibility that could arise in strongly disordered quantum systems. The basic idea was that if particles all live in a random energy landscape, then any rearrangement of particles would change the energy of the system. Since energy is a conserved
106:
Quantum mechanical time evolution is unitary and formally preserves all information about the initial condition in the quantum state at all times. However, a quantum system generically contains a macroscopic number of degrees of freedom, but can only be probed through few-body measurements which are
445:
Many body localized systems exhibit a phenomenon known as emergent integrability. In a non-interacting
Anderson insulator, the occupation number of each localized single particle orbital is separately a local integral of motion. It was conjectured (and proven by Imbrie) that a similar extensive set
217:
approaches a thermal density matrix at a temperature set by the energy density in the state, then the system has "thermalized," and no local information about the initial condition can be extracted from local measurements. This process of "quantum thermalization" may be understood in terms of
3082:
Choi, Soonwon; Choi, Joonhee; Landig, Renate; Kucsko, Georg; Zhou, Hengyun; Isoya, Junichi; Jelezko, Fedor; Onoda, Shinobu; Sumiya, Hitoshi; Khemani, Vedika; von
Keyserlingk, Curt; Yao, Norman Y.; Demler, Eugene; Lukin, Mikhail D. (2017).
686:
405:
In thermalizing systems, entanglement entropy grows as a power law in time starting from low entanglement initial conditions. In MBL systems, entanglement entropy grows logarithmically in time starting from low entanglement initial
3217:
Schreiber, M.; Hodgman, S. S.; Bordia, P.; Luschen, H. P.; Fischer, M. H.; Vosk, R.; Altman, E.; Schneider, U.; Bloch, I. (2015). "Observation of many-body localization of interacting fermions in a quasirandom optical lattice".
426:), the existence of the phenomenon for many particle systems remained a conjecture for decades. In 1980 Fleishman and Anderson demonstrated the phenomenon survived the addition of interactions to lowest order in
3012:
Zhang, J.; Hess, P. W.; Kyprianidis, A.; Becker, P.; Lee, A.; Smith, J.; Pagano, G.; Potirniche, I.-D.; Potter, A. C.; Vishwanath, A.; Yao, N. Y.; Monroe, C. (2017). "Observation of a discrete time crystal".
288:
389:
In thermalizing systems, a memory of initial conditions is not accessible in local observables at long times. In MBL systems, memory of initial conditions remains accessible in local observables at long
3279:
Choi, J.-y.; Hild, S.; Zeiher, J.; Schauss, P.; Rubio-Abadal, A.; Yefsah, T.; Khemani, V.; Huse, D. A.; Bloch, I.; Gross, C. (2016). "Exploring the many-body localization transition in two dimensions".
409:
In thermalizing systems, the dynamics of out-of-time-ordered correlators forms a linear light cone which reflects the ballistic propagation of information. In MBL systems, the light cone is logarithmic.
179:
3416:
Smith, J.; Lee, A.; Richerme, P.; Neyenhuis, B.; Hess, P. W.; Hauke, P.; Heyl, M.; Huse, D. A.; Monroe, C. (2016). "Many-body localization in a quantum simulator with programmable random disorder".
909:
operators, and the coefficients fall off exponentially with distance. This
Hamiltonian manifestly contains an extensive number of localized integrals of motion or "l-bits" (the operators
623:
373:, then the system retains forever a memory of its initial condition in local observables. This latter possibility is referred to as "many body localization," and involves
371:
335:
215:
2199:
Basko, D.M.; Aleiner, I.L.; Altshuler, B.L. (2006). "Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states".
887:{\displaystyle H=\sum _{i}h_{i}^{\prime }\tau _{i}^{z}+\sum _{ij}J_{ij}\tau _{i}^{z}\tau _{j}^{z}+\sum _{ijk}K_{ijk}\tau _{i}^{z}\tau _{j}^{z}\tau _{k}^{z}+\cdots ,}
431:
903:
are Pauli operators that are related to the physical Pauli operators by a local unitary transformation, the ... indicates additional terms which only involve
942:
A number of experiments have been reported observing the MBL phenomenon. Most of these experiments involve synthetic quantum systems, such as assemblies of
396:
In thermalizing systems, energy eigenstates have volume law entanglement entropy. In MBL systems, energy eigenstates have area law entanglement entropy.
2077:
Altshuler, Boris L.; Gefen, Yuval; Kamenev, Alex; Levitov, Leonid S. (1997). "Quasiparticle
Lifetime in a Finite System: A Nonperturbative Approach".
918:, which all commute with the Hamiltonian). If the original Hamiltonian is perturbed, the l-bits get redefined, but the integrable structure survives.
1954:
Chen, Xiao; Zhou, Tianci; Huse, David A.; Fradkin, Eduardo (July 2017). "Out-of-time-order correlations in many-body localized and thermal phases".
1516:
Nandkishore, Rahul; Gopalakrishnan, Sarang; Huse, David A. (2014). "Spectral features of a many-body-localized system weakly coupled to a bath".
1633:Žnidarič, Marko; Prosen, Tomaž; Prelovšek, Peter (February 25, 2008). "Many-body localization in the Heisenberg XXZ magnet in a random field".
437:
It is now believed that MBL can arise also in periodically driven "Floquet" systems where energy is conserved only modulo the drive frequency.
2138:
Gornyi, I. V.; Mirlin, A. D.; Polyakov, D. G. (2005). "Interacting
Electrons in Disordered Wires: Anderson Localization and Low-TTransport".
446:
of local integrals of motion should also exist in the MBL phase. Consider for specificity a one dimensional spin-1/2 chain with
Hamiltonian
1455:
Rigol, Marcos; Dunjko, Vanja; Olshanii, Maxim (April 2008). "Thermalization and its mechanism for generic isolated quantum systems".
229:
2644:
Serbyn, Maksym; Papić, Z.; Abanin, Dmitry A. (2013). "Local
Conservation Laws and the Structure of the Many-Body Localized States".
926:
MBL enables the formation of exotic forms of quantum order that could not arise in thermal equilibrium, through the phenomenon of
1159:
Deutsch, J. M.; Li, Haibin; Sharma, Auditya (April 30, 2013). "Microscopic origin of thermodynamic entropy in isolated systems".
419:
quantity in quantum mechanics, such a process can only be virtual and cannot lead to any transport of particle number or energy.
129:
1864:
He, Rong-Qiang; Lu, Zhong-Yi (February 10, 2017). "Characterizing many-body localization by out-of-time-ordered correlation".
422:
While localization for single particle systems was demonstrated already in
Anderson's original paper (coming to be known as
290:
of a thermalizing system in a pure state plays the role of thermal entropy. Thermalizing systems therefore generically have
927:
337:
fails to approach a thermal density matrix even in the long time limit, and remains instead close to its initial condition
295:
2472:
D’Alessio, Luca; Polkovnikov, Anatoli (2013). "Many-body energy localization transition in periodically driven systems".
989:
Nandkishore, Rahul; Huse, David A. (2015). "Many-Body
Localization and Thermalization in Quantum Statistical Mechanics".
107:
local in real space. The meaningful question then becomes whether accessible local measurements display thermalization.
291:
2525:
Lazarides, Achilleas; Das, Arnab; Moessner, Roderich (2015). "Fate of Many-Body
Localization Under Periodic Driving".
2705:
Huse, David A.; Nandkishore, Rahul; Oganesyan, Vadim (2014). "Phenomenology of fully many-body-localized systems".
1255:
Dymarsky, Anatoly; Lashkari, Nima; Liu, Hong (January 25, 2018). "Subsystem eigenstate thermalization hypothesis".
1569:
Kim, Hyungwon; Huse, David A. (2013). "Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System".
399:
Thermalizing systems generically have non-zero thermal conductivity. MBL systems have zero thermal conductivity.
930:. A form of localization-protected quantum order, arising only in periodically driven systems, is the Floquet
3488:
92:
1909:
Swingle, Brian; Chowdhury, Debanjan (February 21, 2017). "Slow scrambling in disordered quantum systems".
393:
In thermalizing systems, energy eigenstates obey ETH. In MBL systems, energy eigenstates do not obey ETH.
2587:
674:) that all eigenstates are localized, then there exists a local unitary transformation to new variables
3483:
452:
2366:
Serbyn, Maksym; Papić, Z.; Abanin, D. A. (2014). "Quantum quenches in the many-body localized phase".
2260:
Oganesyan, Vadim; Huse, David A. (2007). "Localization of interacting fermions at high temperature".
1050:
340:
304:
184:
950:. Experimental explorations of the phenomenon in solid state systems are still in their infancy.
294:
or "volume law" entanglement entropy at any non-zero temperature. They also generically obey the
100:
44:
1780:
385:). A brief list of properties differentiating thermalizing and MBL systems is provided below.
1738:
1020:
423:
37:
402:
Thermalizing systems have continuous local spectra. MBL systems have discrete local spectra.
3435:
3372:
3299:
3106:
3032:
2976:
2903:
2840:
2781:
2218:
2096:
2008:
1474:
1413:
1352:
1008:
83:, and retaining a memory of its initial condition in local observables for infinite times.
2758:
Huse, David A.; Nandkishore, Rahul; Oganesyan, Vadim; Pal, Arijeet; Sondhi, S. L. (2013).
8:
3085:"Observation of discrete time-crystalline order in a disordered dipolar many-body system"
427:
3439:
3376:
3303:
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3036:
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2907:
2844:
2785:
2222:
2100:
2012:
1478:
1417:
1356:
1086:
1063:
Deutsch, J M (July 26, 2010). "Thermodynamic entropy of a many-body energy eigenstate".
1012:
430:. In a 1998 study, the analysis was extended to all orders in perturbation theory, in a
3459:
3425:
3362:
3331:
3289:
3261:
3227:
3171:
3135:
3096:
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3064:
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2714:
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2428:
2401:
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2208:
2181:
2147:
2120:
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1981:
1963:
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1834:
1803:
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1691:
1660:
1642:
1612:
1578:
1551:
1525:
1498:
1464:
1437:
1403:
1376:
1343:
Deutsch, J. M. (February 1, 1991). "Quantum statistical mechanics in a closed system".
1313:
1225:
1194:
1168:
1121:
1090:
1072:
1032:
998:
122:(everything else), then all information that can be extracted by measurements made on
3451:
3398:
3390:
3323:
3315:
3253:
3245:
3199:
3191:
3140:
3122:
3056:
3048:
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2919:
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2732:
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2610:
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2511:
2499:
2446:
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2165:
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1985:
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1490:
1429:
1368:
1282:
1186:
1141:
1094:
1036:
1024:
110:
This question can be formalized by considering the quantum mechanical density matrix
3463:
3335:
2691:
2630:
2572:
2458:
2405:
2352:
2246:
2185:
2124:
1940:
1807:
1765:
1616:
1441:
1380:
1328:
1301:
3443:
3385:
3380:
3350:
3307:
3265:
3237:
3186:
3181:
3159:
3130:
3114:
3068:
3040:
2984:
2939:
2915:
2911:
2853:
2848:
2818:
2789:
2744:
2724:
2667:
2663:
2606:
2602:
2548:
2544:
2491:
2438:
2383:
2375:
2332:
2279:
2226:
2157:
2104:
2051:
2042:
Fleishman, L.; Anderson, P. W. (1980). "Interactions and the Anderson transition".
2016:
1973:
1926:
1918:
1883:
1844:
1795:
1753:
1706:
1701:
1679:
1652:
1592:
1588:
1535:
1502:
1482:
1421:
1360:
1323:
1272:
1264:
1235:
1178:
1131:
1082:
1016:
2161:
1198:
2817:
Khemani, Vedika; Lazarides, Achilleas; Moessner, Roderich; Sondhi, S. L. (2016).
1739:"Entanglement dynamics in critical random quantum Ising chain with perturbations"
52:
16:
Phenomenon of isolated many-body quantum systems not reaching thermal equilibrium
2313:
Pal, Arijeet; Huse, David A. (2010). "Many-body localization phase transition".
2108:
2989:
2954:
2794:
2759:
2728:
2379:
2336:
2283:
1922:
1887:
1849:
1822:
1656:
1539:
1268:
1182:
1136:
1109:
964:
943:
96:
80:
48:
2884:
Else, Dominic V.; Bauer, Bela; Nayak, Chetan (2016). "Floquet Time Crystals".
2495:
2442:
2230:
1757:
1240:
1213:
3477:
3455:
3394:
3319:
3249:
3195:
3126:
3052:
2998:
2923:
2862:
2803:
2736:
2675:
2614:
2556:
2503:
2450:
2419:
Imbrie, John Z. (2016). "On Many-Body Localization for Quantum Spin Chains".
2397:
2344:
2291:
2238:
2169:
2116:
2063:
2055:
2028:
2020:
1715:
1600:
1547:
1364:
1028:
3311:
3241:
1425:
3402:
3327:
3257:
3203:
3160:"Disorder-Induced Localization in a Strongly Correlated Atomic Hubbard Gas"
3144:
3060:
2955:"Absolute stability and spatiotemporal long-range order in Floquet systems"
2931:
2870:
2683:
2622:
2564:
2177:
1999:
Anderson, P. W. (1958). "Absence of Diffusion in Certain Random Lattices".
1977:
1799:
1723:
1608:
1494:
1286:
1190:
1145:
969:
959:
931:
1433:
1372:
2274:
2213:
2152:
2091:
1931:
1408:
1277:
947:
3118:
3044:
2586:
Ponte, Pedro; Papić, Z.; Huveneers, François; Abanin, Dmitry A. (2015).
2388:
1486:
3447:
1680:"Unbounded Growth of Entanglement in Models of Many-Body Localization"
1394:
Srednicki, Mark (August 1, 1994). "Chaos and quantum thermalization".
76:
3349:
Wei, Ken Xuan; Ramanathan, Chandrasekhar; Cappellaro, Paola (2018).
1108:
Santos, Lea F.; Polkovnikov, Anatoli; Rigol, Marcos (July 5, 2012).
3430:
3367:
3294:
3232:
3101:
3027:
2971:
2898:
2835:
1968:
1878:
1839:
1318:
1230:
3176:
2776:
2719:
2658:
2539:
2486:
2433:
2327:
1696:
1647:
1583:
1530:
1469:
1173:
1126:
1077:
1003:
1821:
Fan, Ruihua; Zhang, Pengfei; Shen, Huitao; Zhai, Hui (May 2017).
1302:"Universal eigenstate entanglement of chaotic local Hamiltonians"
1781:"Out-of-time-ordered correlators in many-body localized systems"
2953:
von Keyserlingk, C. W.; Khemani, Vedika; Sondhi, S. L. (2016).
283:{\displaystyle S=-\operatorname {Tr} (\rho _{A}\log \rho _{A})}
2816:
79:
quantum systems. It is characterized by the system failing to
3216:
1515:
1509:
655:
are random variables drawn from a distribution of some width
2952:
2076:
1678:
Bardarson, Jens H.; Pollmann, Frank; Moore, Joel E. (2012).
3158:
Kondov, S. S.; McGehee, W. R.; Xu, W.; DeMarco, B. (2015).
2757:
3011:
2946:
1823:"Out-of-time-order correlation for many-body localization"
3415:
2585:
1779:
Huang, Yichen; Zhang, Yong-Liang; Chen, Xie (July 2017).
174:{\displaystyle \rho _{A}=\operatorname {Tr} _{B}\rho (t)}
114:
of the system. If the system is divided into a subregion
2704:
2465:
3348:
2588:"Many-Body Localization in Periodically Driven Systems"
1632:
1214:"Does a Single Eigenstate Encode the Full Hamiltonian?"
3157:
1677:
1107:
3278:
2198:
2035:
1671:
1628:
1626:
689:
455:
343:
307:
232:
187:
132:
3210:
3081:
2810:
2524:
2471:
2137:
1212:
Garrison, James R.; Grover, Tarun (April 30, 2018).
2751:
2698:
1953:
1454:
1254:
2070:
1623:
982:
886:
617:
365:
329:
282:
209:
173:
86:
3342:
3151:
2643:
2365:
2253:
2041:
3475:
2518:
2192:
2131:
1908:
1820:
226:. In this perspective, the entanglement entropy
95:assumes that systems go to thermal equilibrium (
75:is a dynamical phenomenon occurring in isolated
3351:"Exploring Localization in Nuclear Spin Chains"
2637:
2359:
1158:
1110:"Weak and strong typicality in quantum systems"
988:
126:alone is encoded in the reduced density matrix
2883:
2579:
1778:
1211:
3005:
2877:
2259:
118:(the region being probed) and its complement
3409:
3272:
3075:
1992:
937:
2819:"Phase Structure of Driven Quantum Systems"
2306:
1562:
2412:
3429:
3384:
3366:
3293:
3231:
3185:
3175:
3134:
3100:
3026:
2988:
2970:
2897:
2852:
2834:
2793:
2775:
2718:
2657:
2538:
2485:
2432:
2387:
2326:
2273:
2212:
2151:
2090:
1967:
1930:
1877:
1848:
1838:
1705:
1695:
1646:
1582:
1529:
1468:
1407:
1393:
1327:
1317:
1276:
1239:
1229:
1172:
1135:
1125:
1076:
1002:
991:Annual Review of Condensed Matter Physics
440:
1998:
1021:10.1146/annurev-conmatphys-031214-014726
2312:
1568:
1342:
1062:
3476:
2760:"Localization-protected quantum order"
2418:
661:. When the disorder is strong enough (
1736:
1299:
1863:
928:localization-protected quantum order
296:eigenstate thermalization hypothesis
103:? Does the system still thermalize?
20:
1053:. Menlo Park, CA: Benjamin/Cummings
13:
716:
553:
377:failing to act as a reservoir for
14:
3500:
618:{\displaystyle H=\sum _{i}\left,}
921:
25:
1947:
1902:
1857:
1814:
1772:
1730:
1448:
1387:
1329:10.1016/j.nuclphysb.2018.09.013
87:Thermalization and localization
3386:10.1103/PhysRevLett.120.070501
3187:10.1103/PhysRevLett.114.083002
2916:10.1103/PhysRevLett.117.090402
2854:10.1103/PhysRevLett.116.250401
2668:10.1103/PhysRevLett.111.127201
2607:10.1103/PhysRevLett.114.140401
2549:10.1103/PhysRevLett.115.030402
2421:Journal of Statistical Physics
1707:10.1103/PhysRevLett.109.017202
1593:10.1103/PhysRevLett.111.127205
1336:
1300:Huang, Yichen (January 2019).
1293:
1248:
1205:
1152:
1101:
1056:
1043:
360:
354:
324:
318:
277:
248:
204:
198:
181:. If, in the long time limit,
168:
162:
1:
2162:10.1103/PhysRevLett.95.206603
1087:10.1088/1367-2630/12/7/075021
975:
93:quantum statistical mechanics
51:in tone and meet Knowledge's
366:{\displaystyle \rho _{A}(0)}
330:{\displaystyle \rho _{A}(T)}
210:{\displaystyle \rho _{A}(t)}
73:Many-body localization (MBL)
7:
2109:10.1103/PhysRevLett.78.2803
953:
10:
3505:
2990:10.1103/PhysRevB.94.085112
2795:10.1103/PhysRevB.88.014206
2729:10.1103/PhysRevB.90.174202
2380:10.1103/PhysRevB.90.174302
2337:10.1103/PhysRevB.82.174411
2284:10.1103/PhysRevB.75.155111
1923:10.1103/PhysRevB.95.060201
1888:10.1103/PhysRevB.95.054201
1850:10.1016/j.scib.2017.04.011
1737:Huang, Yichen (May 2017).
1657:10.1103/PhysRevB.77.064426
1540:10.1103/PhysRevB.90.064203
1269:10.1103/PhysRevE.97.012140
1183:10.1103/PhysRevE.87.042135
1137:10.1103/PhysRevE.86.010102
413:
222:acting as a reservoir for
2496:10.1016/j.aop.2013.02.011
2443:10.1007/s10955-016-1508-x
2231:10.1016/j.aop.2005.11.014
1758:10.1016/j.aop.2017.02.018
1241:10.1103/PhysRevX.8.021026
938:Experimental realizations
646:are Pauli operators, and
81:reach thermal equilibrium
2056:10.1103/PhysRevB.21.2366
2021:10.1103/PhysRev.109.1492
1365:10.1103/PhysRevA.43.2046
1051:Modern Quantum Mechanics
3355:Physical Review Letters
3312:10.1126/science.aaf8834
3242:10.1126/science.aaa7432
3164:Physical Review Letters
2886:Physical Review Letters
2823:Physical Review Letters
2646:Physical Review Letters
2595:Physical Review Letters
2527:Physical Review Letters
2140:Physical Review Letters
2079:Physical Review Letters
1684:Physical Review Letters
1571:Physical Review Letters
1426:10.1103/PhysRevE.50.888
432:zero-dimensional system
1978:10.1002/andp.201600332
1800:10.1002/andp.201600318
1065:New Journal of Physics
888:
619:
441:Emergent integrability
367:
331:
284:
211:
175:
889:
620:
424:Anderson localization
368:
332:
285:
212:
176:
3489:Quantum chaos theory
687:
453:
341:
305:
230:
185:
130:
101:Schrödinger equation
45:improve this article
3440:2016NatPh..12..907S
3377:2018PhRvL.120g0501W
3304:2016Sci...352.1547C
3288:(6293): 1547–1552.
3119:10.1038/nature21426
3111:2017Natur.543..221C
3045:10.1038/nature21413
3037:2017Natur.543..217Z
2981:2016PhRvB..94h5112V
2908:2016PhRvL.117i0402E
2845:2016PhRvL.116y0401K
2786:2013PhRvB..88a4206H
2223:2006AnPhy.321.1126B
2101:1997PhRvL..78.2803A
2013:1958PhRv..109.1492A
1487:10.1038/nature06838
1479:2008Natur.452..854R
1418:1994PhRvE..50..888S
1357:1991PhRvA..43.2046D
1013:2015ARCMP...6...15N
874:
859:
844:
794:
779:
735:
720:
428:perturbation theory
1956:Annalen der Physik
1788:Annalen der Physik
1049:Sakurai JJ. 1985.
884:
860:
845:
830:
813:
780:
765:
751:
721:
706:
705:
615:
471:
363:
327:
280:
207:
171:
3484:Quantum mechanics
3448:10.1038/nphys3783
3226:(6250): 842–845.
3095:(7644): 221–225.
3021:(7644): 217–220.
2959:Physical Review B
2764:Physical Review B
2707:Physical Review B
2474:Annals of Physics
2368:Physical Review B
2315:Physical Review B
2262:Physical Review B
2201:Annals of Physics
2085:(14): 2803–2806.
2044:Physical Review B
1911:Physical Review B
1866:Physical Review B
1746:Annals of Physics
1635:Physical Review B
1518:Physical Review B
1463:(7189): 854–858.
1396:Physical Review E
1345:Physical Review A
1306:Nuclear Physics B
1257:Physical Review E
1218:Physical Review X
1161:Physical Review E
1114:Physical Review E
798:
739:
696:
462:
70:
69:
53:quality standards
3496:
3468:
3467:
3433:
3413:
3407:
3406:
3388:
3370:
3346:
3340:
3339:
3297:
3276:
3270:
3269:
3235:
3214:
3208:
3207:
3189:
3179:
3155:
3149:
3148:
3138:
3104:
3079:
3073:
3072:
3030:
3009:
3003:
3002:
2992:
2974:
2950:
2944:
2943:
2901:
2881:
2875:
2874:
2856:
2838:
2814:
2808:
2807:
2797:
2779:
2755:
2749:
2748:
2722:
2702:
2696:
2695:
2661:
2641:
2635:
2634:
2592:
2583:
2577:
2576:
2542:
2522:
2516:
2515:
2489:
2469:
2463:
2462:
2436:
2416:
2410:
2409:
2391:
2363:
2357:
2356:
2330:
2310:
2304:
2303:
2277:
2275:cond-mat/0610854
2257:
2251:
2250:
2216:
2214:cond-mat/0506617
2207:(5): 1126–1205.
2196:
2190:
2189:
2155:
2153:cond-mat/0506411
2135:
2129:
2128:
2094:
2092:cond-mat/9609132
2074:
2068:
2067:
2050:(6): 2366–2377.
2039:
2033:
2032:
2007:(5): 1492–1505.
1996:
1990:
1989:
1971:
1951:
1945:
1944:
1934:
1906:
1900:
1899:
1881:
1861:
1855:
1854:
1852:
1842:
1827:Science Bulletin
1818:
1812:
1811:
1785:
1776:
1770:
1769:
1743:
1734:
1728:
1727:
1709:
1699:
1675:
1669:
1668:
1650:
1630:
1621:
1620:
1586:
1566:
1560:
1559:
1533:
1513:
1507:
1506:
1472:
1452:
1446:
1445:
1411:
1409:cond-mat/9403051
1391:
1385:
1384:
1351:(4): 2046–2049.
1340:
1334:
1333:
1331:
1321:
1297:
1291:
1290:
1280:
1252:
1246:
1245:
1243:
1233:
1209:
1203:
1202:
1176:
1156:
1150:
1149:
1139:
1129:
1105:
1099:
1098:
1080:
1060:
1054:
1047:
1041:
1040:
1006:
986:
917:
908:
902:
893:
891:
890:
885:
873:
868:
858:
853:
843:
838:
829:
828:
812:
793:
788:
778:
773:
764:
763:
750:
734:
729:
719:
714:
704:
679:
673:
660:
654:
645:
639:
633:
624:
622:
621:
616:
611:
607:
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605:
596:
595:
583:
582:
567:
566:
557:
556:
544:
540:
539:
538:
523:
522:
510:
509:
494:
493:
470:
384:
380:
376:
372:
370:
369:
364:
353:
352:
336:
334:
333:
328:
317:
316:
301:In contrast, if
289:
287:
286:
281:
276:
275:
260:
259:
225:
221:
216:
214:
213:
208:
197:
196:
180:
178:
177:
172:
155:
154:
142:
141:
125:
121:
117:
113:
65:
62:
56:
29:
28:
21:
3504:
3503:
3499:
3498:
3497:
3495:
3494:
3493:
3474:
3473:
3472:
3471:
3424:(10): 907–911.
3414:
3410:
3347:
3343:
3277:
3273:
3215:
3211:
3156:
3152:
3080:
3076:
3010:
3006:
2951:
2947:
2882:
2878:
2815:
2811:
2756:
2752:
2703:
2699:
2642:
2638:
2590:
2584:
2580:
2523:
2519:
2470:
2466:
2427:(5): 998–1048.
2417:
2413:
2364:
2360:
2311:
2307:
2258:
2254:
2197:
2193:
2136:
2132:
2075:
2071:
2040:
2036:
2001:Physical Review
1997:
1993:
1952:
1948:
1907:
1903:
1862:
1858:
1833:(10): 707–711.
1819:
1815:
1783:
1777:
1773:
1741:
1735:
1731:
1676:
1672:
1631:
1624:
1567:
1563:
1514:
1510:
1453:
1449:
1392:
1388:
1341:
1337:
1298:
1294:
1253:
1249:
1210:
1206:
1157:
1153:
1106:
1102:
1061:
1057:
1048:
1044:
987:
983:
978:
956:
944:ultracold atoms
940:
924:
916:
915:
910:
907:
904:
901:
898:
869:
864:
854:
849:
839:
834:
818:
814:
802:
789:
784:
774:
769:
756:
752:
743:
730:
725:
715:
710:
700:
688:
685:
684:
678:
675:
672:
671:
665:
662:
659:
656:
653:
652:
647:
644:
641:
638:
635:
632:
629:
601:
597:
591:
587:
572:
568:
562:
558:
552:
548:
528:
524:
518:
514:
499:
495:
489:
485:
484:
480:
476:
472:
466:
454:
451:
450:
443:
416:
382:
378:
374:
348:
344:
342:
339:
338:
312:
308:
306:
303:
302:
271:
267:
255:
251:
231:
228:
227:
223:
219:
192:
188:
186:
183:
182:
150:
146:
137:
133:
131:
128:
127:
123:
119:
115:
111:
89:
66:
60:
57:
42:
30:
26:
17:
12:
11:
5:
3502:
3492:
3491:
3486:
3470:
3469:
3418:Nature Physics
3408:
3341:
3271:
3209:
3150:
3074:
3004:
2945:
2876:
2829:(25): 250401.
2809:
2750:
2713:(17): 174202.
2697:
2652:(12): 127201.
2636:
2601:(14): 140401.
2578:
2517:
2464:
2411:
2374:(17): 174302.
2358:
2321:(17): 174411.
2305:
2268:(15): 155111.
2252:
2191:
2146:(20): 206603.
2130:
2069:
2034:
1991:
1962:(7): 1600332.
1946:
1901:
1856:
1813:
1794:(7): 1600318.
1771:
1729:
1670:
1622:
1577:(12): 127205.
1561:
1508:
1447:
1402:(2): 888–901.
1386:
1335:
1292:
1247:
1204:
1151:
1100:
1055:
1042:
980:
979:
977:
974:
973:
972:
967:
965:Thermalization
962:
955:
952:
939:
936:
923:
920:
913:
911:
905:
899:
895:
894:
883:
880:
877:
872:
867:
863:
857:
852:
848:
842:
837:
833:
827:
824:
821:
817:
811:
808:
805:
801:
797:
792:
787:
783:
777:
772:
768:
762:
759:
755:
749:
746:
742:
738:
733:
728:
724:
718:
713:
709:
703:
699:
695:
692:
676:
669:
667:
663:
657:
650:
648:
642:
636:
630:
626:
625:
614:
610:
604:
600:
594:
590:
586:
581:
578:
575:
571:
565:
561:
555:
551:
547:
543:
537:
534:
531:
527:
521:
517:
513:
508:
505:
502:
498:
492:
488:
483:
479:
475:
469:
465:
461:
458:
442:
439:
415:
412:
411:
410:
407:
403:
400:
397:
394:
391:
362:
359:
356:
351:
347:
326:
323:
320:
315:
311:
279:
274:
270:
266:
263:
258:
254:
250:
247:
244:
241:
238:
235:
206:
203:
200:
195:
191:
170:
167:
164:
161:
158:
153:
149:
145:
140:
136:
97:thermalization
88:
85:
68:
67:
33:
31:
24:
15:
9:
6:
4:
3:
2:
3501:
3490:
3487:
3485:
3482:
3481:
3479:
3465:
3461:
3457:
3453:
3449:
3445:
3441:
3437:
3432:
3427:
3423:
3419:
3412:
3404:
3400:
3396:
3392:
3387:
3382:
3378:
3374:
3369:
3364:
3361:(7): 070501.
3360:
3356:
3352:
3345:
3337:
3333:
3329:
3325:
3321:
3317:
3313:
3309:
3305:
3301:
3296:
3291:
3287:
3283:
3275:
3267:
3263:
3259:
3255:
3251:
3247:
3243:
3239:
3234:
3229:
3225:
3221:
3213:
3205:
3201:
3197:
3193:
3188:
3183:
3178:
3173:
3170:(8): 083002.
3169:
3165:
3161:
3154:
3146:
3142:
3137:
3132:
3128:
3124:
3120:
3116:
3112:
3108:
3103:
3098:
3094:
3090:
3086:
3078:
3070:
3066:
3062:
3058:
3054:
3050:
3046:
3042:
3038:
3034:
3029:
3024:
3020:
3016:
3008:
3000:
2996:
2991:
2986:
2982:
2978:
2973:
2968:
2965:(8): 085112.
2964:
2960:
2956:
2949:
2941:
2937:
2933:
2929:
2925:
2921:
2917:
2913:
2909:
2905:
2900:
2895:
2892:(9): 090402.
2891:
2887:
2880:
2872:
2868:
2864:
2860:
2855:
2850:
2846:
2842:
2837:
2832:
2828:
2824:
2820:
2813:
2805:
2801:
2796:
2791:
2787:
2783:
2778:
2773:
2770:(1): 014206.
2769:
2765:
2761:
2754:
2746:
2742:
2738:
2734:
2730:
2726:
2721:
2716:
2712:
2708:
2701:
2693:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2660:
2655:
2651:
2647:
2640:
2632:
2628:
2624:
2620:
2616:
2612:
2608:
2604:
2600:
2596:
2589:
2582:
2574:
2570:
2566:
2562:
2558:
2554:
2550:
2546:
2541:
2536:
2533:(3): 030402.
2532:
2528:
2521:
2513:
2509:
2505:
2501:
2497:
2493:
2488:
2483:
2479:
2475:
2468:
2460:
2456:
2452:
2448:
2444:
2440:
2435:
2430:
2426:
2422:
2415:
2407:
2403:
2399:
2395:
2390:
2385:
2381:
2377:
2373:
2369:
2362:
2354:
2350:
2346:
2342:
2338:
2334:
2329:
2324:
2320:
2316:
2309:
2301:
2297:
2293:
2289:
2285:
2281:
2276:
2271:
2267:
2263:
2256:
2248:
2244:
2240:
2236:
2232:
2228:
2224:
2220:
2215:
2210:
2206:
2202:
2195:
2187:
2183:
2179:
2175:
2171:
2167:
2163:
2159:
2154:
2149:
2145:
2141:
2134:
2126:
2122:
2118:
2114:
2110:
2106:
2102:
2098:
2093:
2088:
2084:
2080:
2073:
2065:
2061:
2057:
2053:
2049:
2045:
2038:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2002:
1995:
1987:
1983:
1979:
1975:
1970:
1965:
1961:
1957:
1950:
1942:
1938:
1933:
1932:1721.1/107244
1928:
1924:
1920:
1917:(6): 060201.
1916:
1912:
1905:
1897:
1893:
1889:
1885:
1880:
1875:
1872:(5): 054201.
1871:
1867:
1860:
1851:
1846:
1841:
1836:
1832:
1828:
1824:
1817:
1809:
1805:
1801:
1797:
1793:
1789:
1782:
1775:
1767:
1763:
1759:
1755:
1751:
1747:
1740:
1733:
1725:
1721:
1717:
1713:
1708:
1703:
1698:
1693:
1690:(1): 017202.
1689:
1685:
1681:
1674:
1666:
1662:
1658:
1654:
1649:
1644:
1641:(6): 064426.
1640:
1636:
1629:
1627:
1618:
1614:
1610:
1606:
1602:
1598:
1594:
1590:
1585:
1580:
1576:
1572:
1565:
1557:
1553:
1549:
1545:
1541:
1537:
1532:
1527:
1524:(6): 064203.
1523:
1519:
1512:
1504:
1500:
1496:
1492:
1488:
1484:
1480:
1476:
1471:
1466:
1462:
1458:
1451:
1443:
1439:
1435:
1431:
1427:
1423:
1419:
1415:
1410:
1405:
1401:
1397:
1390:
1382:
1378:
1374:
1370:
1366:
1362:
1358:
1354:
1350:
1346:
1339:
1330:
1325:
1320:
1315:
1311:
1307:
1303:
1296:
1288:
1284:
1279:
1278:1721.1/114450
1274:
1270:
1266:
1263:(1): 012140.
1262:
1258:
1251:
1242:
1237:
1232:
1227:
1224:(2): 021026.
1223:
1219:
1215:
1208:
1200:
1196:
1192:
1188:
1184:
1180:
1175:
1170:
1167:(4): 042135.
1166:
1162:
1155:
1147:
1143:
1138:
1133:
1128:
1123:
1120:(1): 010102.
1119:
1115:
1111:
1104:
1096:
1092:
1088:
1084:
1079:
1074:
1071:(7): 075021.
1070:
1066:
1059:
1052:
1046:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1005:
1000:
996:
992:
985:
981:
971:
968:
966:
963:
961:
958:
957:
951:
949:
945:
935:
933:
929:
922:Exotic orders
919:
881:
878:
875:
870:
865:
861:
855:
850:
846:
840:
835:
831:
825:
822:
819:
815:
809:
806:
803:
799:
795:
790:
785:
781:
775:
770:
766:
760:
757:
753:
747:
744:
740:
736:
731:
726:
722:
711:
707:
701:
697:
693:
690:
683:
682:
681:
612:
608:
602:
598:
592:
588:
584:
579:
576:
573:
569:
563:
559:
549:
545:
541:
535:
532:
529:
525:
519:
515:
511:
506:
503:
500:
496:
490:
486:
481:
477:
473:
467:
463:
459:
456:
449:
448:
447:
438:
435:
433:
429:
425:
420:
408:
404:
401:
398:
395:
392:
388:
387:
386:
357:
349:
345:
321:
313:
309:
299:
297:
293:
272:
268:
264:
261:
256:
252:
245:
242:
239:
236:
233:
201:
193:
189:
165:
159:
156:
151:
147:
143:
138:
134:
108:
104:
102:
98:
94:
84:
82:
78:
74:
64:
54:
50:
46:
40:
39:
36:reads like a
34:This article
32:
23:
22:
19:
3421:
3417:
3411:
3358:
3354:
3344:
3285:
3281:
3274:
3223:
3219:
3212:
3167:
3163:
3153:
3092:
3088:
3077:
3018:
3014:
3007:
2962:
2958:
2948:
2889:
2885:
2879:
2826:
2822:
2812:
2767:
2763:
2753:
2710:
2706:
2700:
2649:
2645:
2639:
2598:
2594:
2581:
2530:
2526:
2520:
2477:
2473:
2467:
2424:
2420:
2414:
2389:1721.1/91499
2371:
2367:
2361:
2318:
2314:
2308:
2265:
2261:
2255:
2204:
2200:
2194:
2143:
2139:
2133:
2082:
2078:
2072:
2047:
2043:
2037:
2004:
2000:
1994:
1959:
1955:
1949:
1914:
1910:
1904:
1869:
1865:
1859:
1830:
1826:
1816:
1791:
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