721:
4449:
4009:
2488:
98:
3082:
20:
65:
734:
4444:{\displaystyle {\begin{aligned}\beta f={}&{\frac {1}{4}}\beta ^{2}J^{2}q^{p}-{\frac {1}{2}}p\beta ^{2}J^{2}q^{p}-{\frac {1}{4}}\beta ^{2}J^{2}+{\frac {1}{2}}\beta J_{0}pm^{p}+{\frac {1}{4{\sqrt {2\pi }}}}p\beta ^{2}J^{2}q^{p-1}+{}\\&\int \exp \left(-{\frac {1}{2}}z^{2}\right)\log \left(2\cosh \left(\beta Jz{\sqrt {{\frac {1}{2}}pq^{p-1}}}+{\frac {1}{2}}\beta J_{0}pm^{p-1}\right)\right)\,\mathrm {d} z\end{aligned}}}
2757:
49:
2255:
3598:. In this limit, the probability of the spin glass existing in a particular state depends only on the energy of that state and not on the individual spin configurations in it. A Gaussian distribution of magnetic bonds across the lattice is assumed usually to solve this model. Any other distribution is expected to give the same result, as a consequence of the
3918:
3077:{\displaystyle {\begin{aligned}&x^{2}={\frac {1}{(2\pi )^{1/2}}}\int \mathrm {d} z\;\mathrm {e} ^{-{\frac {1}{2}}z^{2}}\operatorname {sech} ^{4}\left({\frac {q^{1/2}z+y}{x}}\right),\\&q={\frac {1}{(2\pi )^{1/2}}}\int \mathrm {d} z\;\mathrm {e} ^{-{\frac {1}{2}}z^{2}}\tanh ^{2}\left({\frac {q^{1/2}z+y}{x}}\right).\end{aligned}}}
4670:) interactions, leading to their disordered magnetic structure. This term fell out of favor as the theoretical understanding of spin glasses evolved, recognizing that the magnetic frustration arises not just from a simple mixture of ferro- and antiferromagnetic interactions, but from their randomness and frustration in the system.
2018:
4682:
It was then discovered that the replica method was correct, but the problem lies in that the low-temperature broken symmetry in the SK model cannot be purely characterized by the
Edwards-Anderson order parameter. Instead, further order parameters are necessary, which leads to replica breaking ansatz
2300:
Unlike the
Edwards–Anderson (EA) model, in the system though only two-spin interactions are considered, the range of each interaction can be potentially infinite (of the order of the size of the lattice). Therefore, we see that any two spins can be linked with a ferromagnetic or an antiferromagnetic
949:
Spin glasses differ from ferromagnetic materials by the fact that after the external magnetic field is removed from a ferromagnetic substance, the magnetization remains indefinitely at the remanent value. Paramagnetic materials differ from spin glasses by the fact that, after the external magnetic
1775:
5384:
Kamber, Umut; Bergman, Anders; Eich, Andreas; Iuşan, Diana; Steinbrecher, Manuel; Hauptmann, Nadine; Nordström, Lars; Katsnelson, Mikhail I.; Wegner, Daniel; Eriksson, Olle; Khajetoorians, Alexander A. (2020). "Self-induced spin glass state in elemental and crystalline neodymium".
4678:
Sherrington and
Kirkpatrick proposed the SK model in 1975, and solved it by the replica method. They discovered that at low temperatures, its entropy becomes negative, which they thought was because the replica method is a heuristic method that does not apply at low temperatures.
850:
There are two main aspects of spin glass. On the physical side, spin glasses are real materials with distinctive properties, a review of which is. On the mathematical side, simple statistical mechanics models, inspired by real spin glasses, are widely studied and applied.
4491:, instances are trapped in a "non-ergodic" set of states: the system may fluctuate between several states, but cannot transition to other states of equivalent energy. Intuitively, one can say that the system cannot escape from deep minima of the hierarchically disordered
967:
Surprisingly, the sum of the two complicated functions of time (the zero-field-cooled and remanent magnetizations) is a constant, namely the field-cooled value, and thus both share identical functional forms with time, at least in the limit of very small external fields.
3676:
3470:
946:, and no simple function can fit the curve of magnetization versus time adequately. This slow decay is particular to spin glasses. Experimental measurements on the order of days have shown continual changes above the noise level of instrumentation.
2250:{\displaystyle {\begin{aligned}\beta f={}-{\frac {\beta ^{2}J^{2}}{4}}(1-q)^{2}+{\frac {\beta J_{0}m^{2}}{2}}-\int \exp \left(-{\frac {z^{2}}{2}}\right)\log \left(2\cosh \left(\beta Jz+\beta J_{0}m\right)\right)\,\mathrm {d} z.\end{aligned}}}
4557:
with incommensurate order upon cooling below 19.9 K. Below this transition temperature it exhibits a complex set of magnetic phases that have long spin relaxation times and spin-glass behavior that does not rely on structural disorder.
4588:, where the resistivity of nominally pure gold reaches a minimum at 10 K, and similarly for nominally pure Cu at 2 K. It was later understood that the Kondo effect occurs when a nonmagnetic metal is infused with dilute magnetic atoms.
4522:(−240 °C), and so the spin-glass magnetism appears to be practically without applications in daily life. The non-ergodic states and rugged energy landscapes are, however, quite useful in understanding the behavior of certain
2626:
4889:
The hierarchical disorder of the energy landscape may be verbally characterized by a single sentence: in this landscape there are "(random) valleys within still deeper (random) valleys within still deeper (random) valleys, ...,
1495:
847: – distortions in the geometry of atomic bonds compared to what would be seen in a regular, fully aligned solid. They may also create situations where more than one geometric arrangement of atoms is stable.
3248:
4463:
when, given any (equilibrium) instance of the system, it eventually visits every other possible (equilibrium) state (of the same energy). One characteristic of spin glass systems is that, below the freezing temperature
2265:
In addition to unusual experimental properties, spin glasses are the subject of extensive theoretical and computational investigations. A substantial part of early theoretical work on spin glasses dealt with a form of
2389:
1894:
1617:
1410:
3566:
1086:
2466:
and many others—revealed the complex nature of a glassy low temperature phase characterized by ergodicity breaking, ultrametricity and non-selfaverageness. Further developments led to the creation of the
2748:
3913:{\displaystyle P\left(J_{i_{1}\cdots i_{p}}\right)={\sqrt {\frac {N^{p-1}}{J^{2}\pi p!}}}\exp \left\{-{\frac {N^{p-1}}{J^{2}p!}}\left(J_{i_{1}\cdots i_{p}}-{\frac {J_{0}p!}{2N^{p-1}}}\right)\right\}}
3173:
4014:
2762:
2023:
843:
bonds (where neighbors have exactly the opposite orientation: north and south poles are flipped 180 degrees). These patterns of aligned and misaligned atomic magnets create what are known as
1000:-dimensional lattice with only nearest neighbor interactions. This model can be solved exactly for the critical temperatures and a glassy phase is observed to exist at low temperatures. The
1784:, below a certain temperature, a new magnetic phase called the spin glass phase (or glassy phase) of the system is found to exist which is characterized by a vanishing magnetization
960:
in the absence of an external magnetic field, and a magnetic field is applied after the transition to the spin glass phase, there is a rapid initial increase to a value called the
3668:
3634:
3327:
is the total number of spins. Unlike the
Edwards–Anderson model, but similar to the SK model, the interaction range is infinite. The Hamiltonian for this model is described by:
2448:
1169:
3333:
5021:
Joy, P. A.; Kumar, P. S. Anil; Date, S. K. (7 October 1998). "The relationship between field-cooled and zero-field-cooled susceptibilities of some ordered magnetic systems".
2518:
1923:
4879:
4849:
4489:
3592:
5064:
Nordblad, P.; Lundgren, L.; Sandlund, L. (February 1986). "A link between the relaxation of the zero field cooled and the thermoremanent magnetizations in spin glasses".
6283:
5193:
4706:
More realistic spin glass models with short range frustrated interactions and disorder, like the
Gaussian model where the couplings between neighboring spins follow a
4699:, where it has enabled calculations of properties such as the storage capacity of simple neural network architectures without requiring a training algorithm (such as
3305:
4668:
4642:
2656:
1555:
1525:
1309:
1239:
3178:
2683:
1609:
1582:
1116:
3108:
1808:
6321:
Nordblad, P.; Lundgren, L.; Sandlund, L. (1986), "A link between the relaxation of the zero field cooled and the thermoremanent magnetizations in spin glasses",
5731:
4001:
3981:
3961:
3941:
3325:
3279:
2513:
2007:
1987:
1967:
1947:
1279:
1259:
1209:
1189:
1140:
998:
4618:
It was also called "mictomagnet" (micto- is Greek for "mixed"). The term arose from the observation that these materials often contain a mix of ferromagnetic (
2450:
have same meanings as in the EA model. The equilibrium solution of the model, after some initial attempts by
Sherrington, Kirkpatrick and others, was found by
2689:
found that, as in the case of the Ising model, the mean-field solution to the SK model becomes unstable when under low-temperature, low-magnetic field state.
4500:
2301:
bond and the distribution of these is given exactly as in the case of
Edwards–Anderson model. The Hamiltonian for SK model is very similar to the EA model:
1415:
839:
The individual atomic bonds in a spin glass are a mixture of roughly equal numbers of ferromagnetic bonds (where neighbors have the same orientation) and
5908:
3963:, in two different replicas, which are the same as for the SK model. This infinite range model can be solved explicitly for the free energy in terms of
765:
5341:
Lebech, B; Wolny, J; Moon, R M (1994). "Magnetic phase transitions in double hexagonal close packed neodymium metal-commensurate in two dimensions".
862:(which would be aligned and ferromagnetic). The mathematical complexity of these structures is difficult but fruitful to study experimentally or in
6384:
80:
The magnetic disorder of spin glass compared to a ferromagnet is analogous to the positional disorder of glass (left) compared to quartz (right).
6226:. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge = A series of modern surveys in mathematics. Heidelberg ; New York: Springer.
4721:
Besides its relevance in condensed matter physics, spin glass theory has acquired a strongly interdisciplinary character, with applications to
2471:, which allowed study of the low temperature phase without replicas. A rigorous proof of the Parisi solution has been provided in the work of
1810:
along with a non-vanishing value of the two point correlation function between spins at the same lattice point but at two different replicas:
2307:
1770:{\displaystyle P(J_{ij})={\sqrt {\frac {N}{2\pi J^{2}}}}\exp \left\{-{\frac {N}{2J^{2}}}\left(J_{ij}-{\frac {J_{0}}{N}}\right)^{2}\right\}.}
3113:
1816:
6206:. Ergebnisse der Mathematik und ihrer Grenzgebiete (Softcover repr. of the harcover 1st ed. 2010 ed.). Berlin Heidelberg: Springer.
4566:
A detailed account of the history of spin glasses from the early 1960s to the late 1980s can be found in a series of popular articles by
1324:
2275:
1315:
801:
solids, component atoms' magnetic spins all align in the same direction. Spin glass when contrasted with a ferromagnet is defined as "
6585:
3478:
6048:
1010:
2695:
950:
field is removed, the magnetization rapidly falls to zero, with no remanent magnetization. The decay is rapid and exponential.
758:
6231:
6211:
6179:
6154:
6122:
6102:
6074:
5967:
5282:
5002:
4503:
counts the number of states that are accessible from a given instance, that is, the number of states that participate in the
931:
When the external magnetic field is removed, the magnetization of the spin glass falls rapidly to a lower value known as the
924:, the sample becomes a spin glass, and further cooling results in little change in magnetization. This is referred to as the
938:
Magnetization then decays slowly as it approaches zero (or some small fraction of the original value – this
913:
is applied as the sample is cooled to the transition temperature, magnetization of the sample increases as described by the
939:
6142:
2463:
4687:. At the full replica breaking ansatz, infinitely many order parameters are required to characterize a stable solution.
4508:
6033:
5111:
4929:
751:
738:
805:" magnetic state in which spins are aligned randomly or without a regular pattern and the couplings too are random.
6382:
Bryngelson, Joseph D.; Wolynes, Peter G. (1987), "Spin glasses and the statistical mechanics of protein folding",
720:
6056:
2297:
of spin glasses describing the slow dynamics of the magnetization and the complex non-ergodic equilibrium state.
6434:
4923:
503:
6733:
6578:
4755:
4518:
For physical systems, such as dilute manganese in copper, the freezing temperature is typically as low as 30
2282:
6315:
5217:
6673:
678:
158:
3465:{\displaystyle H=-\sum _{i_{1}<i_{2}<\cdots <i_{p}}J_{i_{1}\dots i_{p}}S_{i_{1}}\cdots S_{i_{p}}}
5248:
6558:
6475:(2000), "Replica symmetry breaking and exponential inequalities for the Sherrington–Kirkpatrick model",
3639:
3605:
2692:
The stability region on the phase diagram of the SK model is determined by two dimensionless parameters
2397:
1145:
6728:
6465:
4534:
683:
308:
6353:; Young, A. P. (1986), "Spin glasses: Experimental facts, theoretical concepts, and open questions",
6109:
Focuses on the experimentally measurable properties of spin glasses (such as copper-manganese alloy).
4722:
4696:
4523:
867:
833:
573:
248:
1902:
6571:
6195:
4854:
779:
568:
563:
89:
6117:. Cambridge studies in magnetism. Cambridge ; New York, NY, USA: Cambridge University Press.
6069:. International series of monographs on physics. Oxford ; New York: Oxford University Press.
5691:
5649:
5607:
5565:
5523:
5481:
5439:
5240:
4827:
4467:
3571:
1311:
representing the magnetic nature of the spin-spin interactions are called bond or link variables.
653:
6630:
4770:
4608:
2454:
in 1979 with the replica method. The subsequent work of interpretation of the Parisi solution—by
2012:
Under the assumption of replica symmetry, the mean-field free energy is given by the expression:
1281:. The sum runs over all nearest neighbor positions on a lattice, of any dimension. The variables
844:
258:
6506:
Guerra, F.; Toninelli, F. L. (2002), "The thermodynamic limit in mean field spin glass models",
5146:
663:
4944:
4711:
4567:
910:
648:
588:
558:
508:
228:
118:
5784:"Susceptibility of the $ \mathrm{Cu}\mathrm{Mn}$ spin-glass: Frequency and field dependences"
2293:
with long range frustrated ferro- as well as antiferromagnetic couplings. It corresponds to a
1969:. Hence the new set of order parameters describing the three magnetic phases consists of both
786:
is a magnetic state characterized by randomness, besides cooperative behavior in freezing of
4707:
3599:
3284:
1319:
1001:
688:
303:
288:
5272:
4647:
4621:
3257:
This is also called the "p-spin model". The infinite-range model is a generalization of the
2631:
1530:
1500:
1284:
1214:
6525:
6449:
6393:
6362:
6330:
6298:
6266:
5923:
5873:
5826:
5756:
5706:
5664:
5622:
5580:
5538:
5496:
5454:
5350:
5307:
5158:
5069:
5030:
4956:
2661:
1587:
1560:
1094:
278:
168:
8:
6668:
4800:
4790:
4775:
3595:
3087:
1787:
832:
solids, magnetic spins all align in the same direction; this is analogous to a crystal's
518:
328:
178:
6529:
6453:
6397:
6366:
6334:
6302:
6270:
5979:
5927:
5877:
5830:
5760:
5710:
5668:
5626:
5584:
5542:
5500:
5458:
5354:
5311:
5298:
Zochowski, S W; McEwen, K A; Fawcett, E (1991). "Magnetic phase diagrams of neodymium".
5162:
5073:
5034:
4960:
888:
It is the time dependence which distinguishes spin glasses from other magnetic systems.
6653:
6625:
6541:
6515:
6494:
5991:
5861:
5394:
5046:
4554:
3986:
3966:
3946:
3926:
3310:
3264:
2621:{\displaystyle H=-{\frac {1}{\sqrt {N}}}\sum _{i<j}J_{ij}S_{i}S_{j}-M\sum _{i}S_{i}}
2498:
1992:
1972:
1952:
1932:
1264:
1244:
1194:
1174:
1125:
983:
698:
658:
633:
381:
372:
6416:
5042:
964:
magnetization. A slow upward drift then occurs toward the field-cooled magnetization.
6613:
6461:
6421:
6342:
6237:
6227:
6207:
6185:
6175:
6150:
6130:
6118:
6098:
6080:
6070:
6039:
6029:
6009:
5963:
5935:
5889:
5842:
5420:
5412:
5366:
5362:
5323:
5319:
5278:
5174:
5170:
5107:
5081:
5050:
4998:
4972:
4919:
4785:
4765:
2686:
2294:
2286:
2267:
1119:
943:
628:
473:
363:
283:
6545:
6278:
5783:
5128:
1490:{\displaystyle {\mathcal {Z}}\left=\operatorname {Tr} _{S}\left(e^{-\beta H}\right)}
854:
Spin glasses and the complex internal structures that arise within them are termed "
6700:
6533:
6484:
6472:
6457:
6411:
6401:
6370:
6338:
6306:
6274:
6001:
5931:
5881:
5834:
5795:
5764:
5744:
5714:
5672:
5630:
5588:
5546:
5504:
5462:
5404:
5358:
5315:
5274:
Handbook of
Crystal Structures and Magnetic Properties of Rare Earth Intermetallics
5166:
5077:
5038:
4964:
4780:
4750:
4715:
4531:
4527:
4492:
4003:, under the assumption of replica symmetry as well as 1-Replica Symmetry Breaking.
2476:
2472:
892:
883:
871:
333:
298:
293:
253:
223:
193:
153:
113:
6289:
Sherrington, David; Kirkpatrick, Scott (1975), "Solvable model of a spin-glass",
6089:
Introduction focused on computer science applications, including neural networks.
6023:
5814:
4737:
4700:
4695:
The formalism of replica mean-field theory has also been applied in the study of
1926:
840:
821:
643:
593:
463:
218:
130:
6310:
5885:
5838:
4507:. The ergodic aspect of spin glass was instrumental in the awarding of half the
6690:
6663:
6658:
6648:
6138:
6066:
Statistical physics of spin glasses and information processing: an introduction
5104:
Statistical
Physics of Spin Glasses and Information Processing: An Introduction
4733:
4684:
4512:
2459:
2451:
914:
798:
787:
725:
693:
673:
668:
623:
543:
478:
376:
263:
108:
6537:
6374:
4968:
3243:{\displaystyle x\approx {\frac {4}{3{\sqrt {2\pi }}}}e^{-{\frac {1}{2}}y^{2}}}
2455:
6722:
6685:
6680:
6620:
6489:
6241:
6169:
6161:
6064:
6052:
6043:
6013:
5893:
5846:
5799:
5416:
5370:
5327:
5178:
4976:
4795:
4760:
4572:
4550:
2468:
2271:
1781:
903:
855:
829:
404:
385:
367:
268:
188:
6406:
6189:
6084:
5980:"A Friendly Tutorial on Mean-Field Spin Glass Techniques for Non-Physicists"
5768:
5408:
598:
6608:
6221:
5424:
4726:
4600:
4585:
4504:
2487:
859:
618:
608:
578:
538:
533:
513:
358:
338:
198:
6425:
6520:
6350:
5147:"Stability of the Sherrington-Kirkpatrick solution of a spin glass model"
4496:
2290:
977:
703:
638:
613:
583:
528:
523:
455:
97:
6498:
6005:
5210:
2384:{\displaystyle H=-{\frac {1}{\sqrt {N}}}\sum _{i<j}J_{ij}S_{i}S_{j}}
2281:
An important, exactly solvable model of a spin glass was introduced by
863:
548:
183:
19:
6435:"The order parameter for spin glasses: a function on the interval 0-1"
6160:. Early exposition containing the pre-1990 breakthroughs, such as the
5718:
5676:
5634:
5550:
5508:
5466:
2658:
are IID samples from the gaussian distribution of mean 0 and variance
1241:
denotes an antiferromagnetic type interaction between spins at points
6594:
6049:
1st 15 chapters of 2008 draft version, available at www.stat.ucla.edu
5962:. Primers in complex systems. Princeton: Princeton University Press.
5782:
Mulder, C. A. M.; van
Duyneveldt, A. J.; Mydosh, J. A. (1981-02-01).
5592:
1889:{\displaystyle q=\sum _{i=1}^{N}S_{i}^{\alpha }S_{i}^{\beta }\neq 0,}
603:
553:
426:
273:
173:
3923:
The order parameters for this system are given by the magnetization
6705:
5996:
5399:
4805:
4538:
858:" because they are "stuck" in stable configurations other than the
163:
64:
3943:
and the two point spin correlation between spins at the same site
1929:
for the ferromagnetic to spin glass phase transition is therefore
1405:{\displaystyle f\left=-{\frac {1}{\beta }}\ln {\mathcal {Z}}\left}
5194:"Trio of scientists win Nobel prize for physics for climate work"
4460:
866:; with applications to physics, chemistry, materials science and
825:
802:
483:
468:
431:
422:
417:
6257:
Edwards, S.F.; Anderson, P.W. (1975), "Theory of spin glasses",
5270:
4519:
3561:{\displaystyle J_{i_{1}\dots i_{p}},S_{i_{1}},\dots ,S_{i_{p}}}
436:
412:
143:
5218:"The Nobel Prize in Physics 2021 - Popular Science Background"
1081:{\displaystyle H=-\sum _{\langle ij\rangle }J_{ij}S_{i}S_{j},}
902:, the spin glass exhibits typical magnetic behaviour (such as
824:
the atomic bond structure is highly irregular; in contrast, a
6559:
Statistics of frequency of the term "Spin glass" in arxiv.org
6174:. Cambridge, UK ; New York: Cambridge University Press.
5185:
2495:
When there is a uniform external magnetic field of magnitude
817:
441:
138:
6563:
5241:"Scientific Background for the Nobel Prize in Physics 2021"
5129:
Mean Field Models for Spin Glasses Volume I: Basic Examples
4714:. These models display spin glass phases bordered by sharp
4611:
at a well defined temperature, which would later be termed
5781:
4997:. Oxford graduate texts. Oxford: Oxford university press.
4918:. London, Washington DC: Taylor & Francis. p. 3.
4710:, have been studied extensively as well, especially using
4591:
Unusual behavior was observed in of iron-in-gold alloy (Au
3175:
At low temperature, high magnetic field limit, the line is
2743:{\displaystyle x:={\frac {kT}{J}},\quad y:={\frac {M}{J}}}
5974:
Popular exposition, with a minimal amount of mathematics.
4544:
148:
48:
6320:
5063:
4454:
6171:
Random fields and spin glasses: a field theory approach
6316:
Papercore Summary http://papercore.org/Sherrington1975
6288:
6097:. London ; Washington, DC: Taylor & Francis.
5862:"Infinite Number of Order Parameters for Spin-Glasses"
5812:
5383:
5297:
6204:
Mean field models for spin glasses. 1: Basic examples
6055:
and the applications to computer science, especially
5909:"Optimal storage properties of neural network models"
5813:
Sherrington, David; Kirkpatrick, Scott (1975-12-29).
4857:
4851:
is identical to the so-called "freezing temperature"
4830:
4650:
4624:
4470:
4012:
3989:
3969:
3949:
3929:
3679:
3642:
3608:
3574:
3481:
3336:
3313:
3287:
3267:
3261:
where we not only consider two-spin interactions but
3181:
3168:{\displaystyle y^{2}\approx {\frac {4}{3}}(1-x)^{3}.}
3116:
3090:
2760:
2698:
2664:
2634:
2521:
2501:
2400:
2310:
2021:
1995:
1975:
1955:
1935:
1905:
1819:
1790:
1620:
1590:
1563:
1533:
1503:
1418:
1327:
1287:
1267:
1247:
1217:
1197:
1177:
1148:
1128:
1097:
1013:
986:
6167:
6136:
5271:Andrej Szytula; Janusz Leciejewicz (8 March 1994).
1949:, and that for paramagnetic to spin glass is again
808:The term "glass" comes from an analogy between the
5689:
5647:
5605:
5563:
5521:
5479:
5437:
4873:
4843:
4662:
4636:
4483:
4443:
3995:
3975:
3955:
3935:
3912:
3662:
3628:
3586:
3560:
3464:
3319:
3299:
3273:
3258:
3242:
3167:
3102:
3076:
2750:. Its phase diagram has two parts, divided by the
2742:
2677:
2650:
2620:
2507:
2442:
2383:
2249:
2001:
1981:
1961:
1941:
1917:
1888:
1802:
1769:
1603:
1576:
1549:
1519:
1489:
1404:
1303:
1273:
1253:
1233:
1203:
1183:
1171:refers to summing over neighboring lattice points
1163:
1134:
1110:
1080:
992:
6466:Papercore Summary http://papercore.org/Parisi1980
6381:
5978:Montanari, Andrea; Sen, Subhabrata (2024-01-09).
6720:
5106:. Oxford: Oxford University Press. p. 243.
4673:
4499:, with tall energy barriers between minima. The
3602:. The Gaussian distribution function, with mean
2754:, The curve is the solution set to the equations
2260:
820:, e.g., a window glass. In window glass or any
6505:
6385:Proceedings of the National Academy of Sciences
6256:
6133:, experimental data, and numerical simulations.
6021:
5340:
5144:
4992:
4725:theory, computer science, theoretical biology,
4495:; the distances between minima are given by an
790:at a temperature called "freezing temperature"
6168:De Dominicis, Cirano; Giardina, Irene (2006).
5906:
5742:
5151:Journal of Physics A: Mathematical and General
4603:. Cannella and Mydosh observed in 1972 that Au
3568:have similar meanings as in the EA model. The
6579:
5958:Stein, Daniel L.; Newman, Charles M. (2013).
5482:"Spin Glass II: Is There a Phase Transition?"
5145:Almeida, J R L de; Thouless, D J (May 1978).
4584:In 1930s, material scientists discovered the
980:. In this model, we have spins arranged on a
759:
6246:. Compendium of rigorously provable results.
6112:
5977:
5020:
1158:
1149:
1122:for the spin-half particle at lattice point
1037:
1028:
6349:
6323:Journal of Magnetism and Magnetic Materials
5957:
5097:
5095:
5093:
5091:
5066:Journal of Magnetism and Magnetic Materials
828:has a uniform pattern of atomic bonds. In
6586:
6572:
6095:Spin glasses: an experimental introduction
5984:Foundations and Trends in Machine Learning
5743:Cannella, V.; Mydosh, J. A. (1972-12-01).
5608:"Spin Glass V: Real Power Brought to Bear"
4916:Spin Glasses: An Experimental Introduction
2977:
2826:
1318:for this system, one needs to average the
766:
752:
96:
6519:
6488:
6471:
6415:
6405:
6219:
6201:
6062:
6028:. Oxford, U.K.: Oxford University Press.
5995:
5907:Gardner, E; Deridda, B (7 January 1988).
5650:"Spin Glass VI: Spin Glass As Cornucopia"
5398:
5101:
4428:
2231:
971:
6022:Mézard, Marc; Montanari, Andrea (2009).
5692:"Spin Glass VII: Spin Glass as Paradigm"
5524:"Spin Glass III: Theory Raises its Head"
5088:
4993:Mézard, Marc; Montanari, Andrea (2009).
2486:
18:
5900:
5745:"Magnetic Ordering in Gold-Iron Alloys"
5566:"Spin Glass IV: Glimmerings of Trouble"
5057:
5016:
5014:
4607:had an unexpected cusplike peak in the
3252:
6721:
6508:Communications in Mathematical Physics
6432:
6092:
5859:
5191:
4913:
4732:Spin glass models were adapted to the
4545:Spin-glass without structural disorder
1780:Solving for the free energy using the
1557:is taken to be a Gaussian with a mean
6567:
6025:Information, Physics, and Computation
5440:"Spin Glass I: A Scaling Law Rescued"
4995:Information, physics, and computation
4455:Non-ergodic behavior and applications
816:disorder of a conventional, chemical
6113:Fischer, K. H.; Hertz, John (1991).
5343:Journal of Physics: Condensed Matter
5300:Journal of Physics: Condensed Matter
5140:
5138:
5011:
4988:
4986:
4942:
4909:
4907:
3594:limit of this model is known as the
877:
6259:Journal of Physics F: Metal Physics
4595:) and manganese-in-copper alloy (Cu
4553:at room temperature and becomes an
4549:Elemental crystalline neodymium is
13:
6250:
6223:Mean field models for spin glasses
6018:A practical tutorial introduction.
4430:
3663:{\displaystyle {\frac {J^{2}}{N}}}
3629:{\displaystyle {\frac {J_{0}}{N}}}
3581:
2980:
2970:
2829:
2819:
2443:{\displaystyle J_{ij},S_{i},S_{j}}
2233:
1421:
1376:
1164:{\displaystyle \langle ij\rangle }
1004:for this spin system is given by:
14:
6745:
6552:
5135:
4983:
4904:
4703:) to be designed or implemented.
2685:. In 1979, J.R.L. de Almeida and
812:disorder in a spin glass and the
6057:constraint satisfaction problems
5815:"Solvable Model of a Spin-Glass"
2482:
1527:. The distribution of values of
733:
732:
719:
63:
47:
23:Schematic representation of the
6149:, Singapore: World Scientific,
5853:
5806:
5775:
5736:
5725:
5683:
5641:
5599:
5557:
5515:
5473:
5431:
5377:
5334:
5291:
5264:
5233:
4883:
4690:
4613:spin glass freezing temperature
3084:The phase transition occurs at
2723:
5951:
5120:
4936:
4819:
4530:, as well as many problems in
3578:
3153:
3140:
2946:
2936:
2795:
2785:
2082:
2069:
1918:{\displaystyle \alpha ,\beta }
1640:
1624:
1497:, over all possible values of
953:If the sample is cooled below
1:
6593:
6063:Nishimori, Hidetoshi (2001).
6051:Textbook that focuses on the
5946:
5102:Nishimori, Hidetoshi (2001).
4943:Ford, Peter J. (March 1982).
4897:
4874:{\displaystyle T_{\text{f}}.}
4756:Antiferromagnetic interaction
4674:Sherrington-Kirkpatrick model
3259:Sherrington–Kirkpatrick model
2515:, the energy function becomes
2261:Sherrington–Kirkpatrick model
6674:ferromagnetic superconductor
6343:10.1016/0304-8853(86)90543-3
6147:Spin glass theory and beyond
5192:Geddes, Linda (2021-10-05).
5082:10.1016/0304-8853(86)90543-3
4844:{\displaystyle T_{\text{c}}}
4579:
4484:{\displaystyle T_{\text{f}}}
3587:{\displaystyle p\to \infty }
7:
6311:10.1103/PhysRevLett.35.1792
5960:Spin glasses and complexity
5886:10.1103/PhysRevLett.43.1754
5839:10.1103/physrevlett.35.1792
5690:Philip W. Anderson (1990).
5648:Philip W. Anderson (1989).
5606:Philip W. Anderson (1989).
5564:Philip W. Anderson (1988).
5522:Philip W. Anderson (1988).
5480:Philip W. Anderson (1988).
5438:Philip W. Anderson (1988).
5249:Nobel Committee for Physics
5043:10.1088/0953-8984/10/48/024
4743:
4509:2021 Nobel Prize in Physics
860:lowest-energy configuration
10:
6750:
6462:10.1088/0305-4470/13/3/042
6220:Talagrand, Michel (2011).
6202:Talagrand, Michel (2010).
5936:10.1088/0305-4470/21/1/031
5363:10.1088/0953-8984/6/27/029
5320:10.1088/0953-8984/3/41/007
5171:10.1088/0305-4470/11/5/028
4561:
4459:A thermodynamic system is
3281:-spin interactions, where
2491:de Almeida-Thouless curve.
1314:In order to determine the
881:
868:artificial neural networks
309:Spin gapless semiconductor
6641:
6601:
6538:10.1007/s00220-002-0699-y
6375:10.1103/RevModPhys.58.801
6355:Reviews of Modern Physics
6279:10.1088/0305-4608/5/5/017
5860:Parisi, G. (1979-12-03).
5023:J. Phys.: Condens. Matter
4969:10.1080/00107518208237073
4644:) and antiferromagnetic (
2752:de Almeida-Thouless curve
1925:are replica indices. The
249:Electronic band structure
16:Disordered magnetic state
6196:statistical field theory
5800:10.1103/PhysRevB.23.1384
5277:. CRC Press. p. 1.
4812:
3110:. Just below it, we have
2295:mean-field approximation
944:decay is non-exponential
780:condensed matter physics
159:Bose–Einstein condensate
90:Condensed matter physics
6631:Van Vleck paramagnetism
6407:10.1073/pnas.84.21.7524
6291:Physical Review Letters
5866:Physical Review Letters
5819:Physical Review Letters
5769:10.1103/PhysRevB.6.4220
5409:10.1126/science.aay6757
4771:Geometrical frustration
4712:Monte Carlo simulations
3300:{\displaystyle p\leq N}
976:This is similar to the
845:frustrated interactions
834:lattice-based structure
70:Quartz (crystalline SiO
6490:10.1214/aop/1019160325
6442:J. Phys. A: Math. Gen.
6143:Virasoro, Miguel Angel
6093:Mydosh, J. A. (1993).
5068:. 54–57 (1): 185–186.
4914:Mydosh, J. A. (1993).
4875:
4845:
4664:
4663:{\displaystyle J<0}
4638:
4637:{\displaystyle J>0}
4485:
4445:
3997:
3977:
3957:
3937:
3914:
3664:
3630:
3588:
3562:
3466:
3321:
3301:
3275:
3244:
3169:
3104:
3078:
2744:
2679:
2652:
2651:{\displaystyle J_{ij}}
2622:
2509:
2492:
2444:
2385:
2251:
2003:
1983:
1963:
1943:
1919:
1890:
1846:
1804:
1771:
1605:
1578:
1551:
1550:{\displaystyle J_{ij}}
1521:
1520:{\displaystyle J_{ij}}
1491:
1406:
1305:
1304:{\displaystyle J_{ij}}
1275:
1255:
1235:
1234:{\displaystyle J_{ij}}
1211:. A negative value of
1205:
1185:
1165:
1136:
1112:
1082:
994:
972:Edwards–Anderson model
893:transition temperature
40:
6477:Annals of Probability
5732:All of them combined.
4876:
4846:
4708:Gaussian distribution
4665:
4639:
4486:
4446:
3998:
3978:
3958:
3938:
3915:
3665:
3631:
3600:central limit theorem
3589:
3563:
3467:
3322:
3302:
3276:
3245:
3170:
3105:
3079:
2745:
2680:
2678:{\displaystyle J^{2}}
2653:
2623:
2510:
2490:
2445:
2386:
2252:
2004:
1984:
1964:
1944:
1920:
1891:
1826:
1805:
1772:
1606:
1604:{\displaystyle J^{2}}
1579:
1577:{\displaystyle J_{0}}
1552:
1522:
1492:
1407:
1306:
1276:
1256:
1236:
1206:
1186:
1166:
1137:
1113:
1111:{\displaystyle S_{i}}
1083:
995:
891:Above the spin glass
304:Topological insulator
22:
6734:Mathematical physics
6284:ShieldSquare Captcha
4949:Contemporary Physics
4855:
4828:
4648:
4622:
4599:) at around 1 to 10
4468:
4010:
3987:
3967:
3947:
3927:
3677:
3640:
3606:
3572:
3479:
3334:
3311:
3285:
3265:
3253:Infinite-range model
3179:
3114:
3088:
2758:
2696:
2662:
2632:
2519:
2499:
2398:
2308:
2019:
1993:
1973:
1953:
1933:
1903:
1817:
1788:
1618:
1588:
1561:
1531:
1501:
1416:
1325:
1285:
1265:
1245:
1215:
1195:
1175:
1146:
1126:
1095:
1011:
984:
322:Electronic phenomena
169:Fermionic condensate
54:Glass (amorphous SiO
27:spin structure of a
6701:amorphous magnetism
6669:superferromagnetism
6530:2002CMaPh.230...71G
6454:1980JPhA...13.1101P
6433:Parisi, G. (1980),
6398:1987PNAS...84.7524B
6367:1986RvMP...58..801B
6335:1986JMMM...54..185N
6303:1975PhRvL..35.1792S
6271:1975JPhF....5..965E
5928:1988JPhA...21..271G
5878:1979PhRvL..43.1754P
5831:1975PhRvL..35.1792S
5761:1972PhRvB...6.4220C
5711:1990PhT....43c...9A
5669:1989PhT....42i...9A
5627:1989PhT....42g...9A
5585:1988PhT....41i...9A
5543:1988PhT....41f...9A
5501:1988PhT....41c...9A
5459:1988PhT....41a...9A
5355:1994JPCM....6.5201L
5312:1991JPCM....3.8079Z
5163:1978JPhA...11..983D
5074:1986JMMM...54..185N
5035:1998JPCM...1011049J
5029:(48): 11049–11054.
4961:1982ConPh..23..141F
4801:Solid-state physics
4791:Random energy model
4776:Orientational glass
4609:a.c. susceptibility
4501:participation ratio
3596:random energy model
3103:{\displaystyle x=1}
1876:
1861:
1803:{\displaystyle m=0}
1142:, and the sum over
329:Quantum Hall effect
6654:antiferromagnetism
6626:superparamagnetism
6006:10.1561/2200000105
5126:Michel Talagrand,
4871:
4841:
4660:
4634:
4568:Philip W. Anderson
4481:
4441:
4439:
3993:
3973:
3953:
3933:
3910:
3660:
3626:
3584:
3558:
3462:
3394:
3317:
3297:
3271:
3240:
3165:
3100:
3074:
3072:
2740:
2675:
2648:
2628:Let all couplings
2618:
2607:
2558:
2505:
2493:
2440:
2381:
2347:
2289:in 1975. It is an
2276:partition function
2270:based on a set of
2247:
2245:
1999:
1979:
1959:
1939:
1915:
1886:
1862:
1847:
1800:
1767:
1601:
1574:
1547:
1517:
1487:
1402:
1316:partition function
1301:
1271:
1251:
1231:
1201:
1181:
1161:
1132:
1108:
1078:
1041:
990:
726:Physics portal
41:
6729:Magnetic ordering
6716:
6715:
6614:superdiamagnetism
6602:Magnetic response
6473:Talagrand, Michel
6392:(21): 7524–7528,
6297:(26): 1792–1796,
6233:978-3-642-15201-6
6213:978-3-642-26598-3
6181:978-0-521-84783-4
6156:978-9971-5-0115-0
6131:mean field theory
6124:978-0-521-34296-4
6104:978-0-7484-0038-6
6076:978-0-19-850940-0
5969:978-0-691-14733-8
5872:(23): 1754–1756.
5825:(26): 1792–1796.
5788:Physical Review B
5755:(11): 4220–4237.
5749:Physical Review B
5719:10.1063/1.2810479
5677:10.1063/1.2811137
5635:10.1063/1.2811073
5551:10.1063/1.2811440
5509:10.1063/1.2811336
5467:10.1063/1.2811268
5349:(27): 5201–5222.
5306:(41): 8079–8094.
5284:978-0-8493-4261-5
5004:978-0-19-857083-7
4865:
4838:
4786:Quenched disorder
4766:Crystal structure
4716:phase transitions
4528:Hopfield networks
4478:
4384:
4371:
4350:
4289:
4211:
4208:
4160:
4127:
4081:
4038:
3996:{\displaystyle q}
3976:{\displaystyle m}
3956:{\displaystyle q}
3936:{\displaystyle m}
3898:
3817:
3765:
3764:
3658:
3624:
3346:
3320:{\displaystyle N}
3274:{\displaystyle p}
3226:
3208:
3205:
3138:
3061:
2996:
2964:
2910:
2845:
2813:
2738:
2718:
2598:
2543:
2541:
2540:
2508:{\displaystyle M}
2332:
2330:
2329:
2287:Scott Kirkpatrick
2283:David Sherrington
2268:mean-field theory
2161:
2124:
2067:
2002:{\displaystyle q}
1982:{\displaystyle m}
1962:{\displaystyle q}
1942:{\displaystyle q}
1746:
1707:
1671:
1670:
1366:
1274:{\displaystyle j}
1254:{\displaystyle i}
1204:{\displaystyle j}
1184:{\displaystyle i}
1135:{\displaystyle i}
1120:Pauli spin matrix
1023:
993:{\displaystyle d}
962:zero-field-cooled
878:Magnetic behavior
841:antiferromagnetic
776:
775:
474:Granular material
242:Electronic phases
6741:
6588:
6581:
6574:
6565:
6564:
6548:
6523:
6521:cond-mat/0204280
6501:
6492:
6483:(3): 1018–1062,
6464:
6448:(3): 1101–1112,
6439:
6428:
6419:
6409:
6377:
6345:
6313:
6281:
6245:
6217:
6193:
6159:
6128:
6108:
6088:
6047:
6017:
5999:
5973:
5940:
5939:
5913:
5904:
5898:
5897:
5857:
5851:
5850:
5810:
5804:
5803:
5794:(3): 1384–1396.
5779:
5773:
5772:
5740:
5734:
5729:
5723:
5722:
5696:
5687:
5681:
5680:
5654:
5645:
5639:
5638:
5612:
5603:
5597:
5596:
5593:10.1063/1.881135
5570:
5561:
5555:
5554:
5528:
5519:
5513:
5512:
5486:
5477:
5471:
5470:
5444:
5435:
5429:
5428:
5402:
5381:
5375:
5374:
5338:
5332:
5331:
5295:
5289:
5288:
5268:
5262:
5261:
5259:
5257:
5252:. 5 October 2021
5245:
5237:
5231:
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5208:
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5205:
5204:
5189:
5183:
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5133:
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5118:
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5099:
5086:
5085:
5061:
5055:
5054:
5018:
5009:
5008:
4990:
4981:
4980:
4940:
4934:
4933:
4911:
4891:
4887:
4881:
4880:
4878:
4877:
4872:
4867:
4866:
4863:
4850:
4848:
4847:
4842:
4840:
4839:
4836:
4823:
4781:Phase transition
4751:Amorphous magnet
4669:
4667:
4666:
4661:
4643:
4641:
4640:
4635:
4532:computer science
4493:energy landscape
4490:
4488:
4487:
4482:
4480:
4479:
4476:
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4448:
4447:
4442:
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4433:
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4305:
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4209:
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4095:
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4074:
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4049:
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4031:
4027:
4002:
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3999:
3994:
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3974:
3962:
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3954:
3942:
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3808:
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3763:
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3742:
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3716:
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3713:
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3700:
3669:
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3666:
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3644:
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3083:
3081:
3080:
3075:
3073:
3066:
3062:
3057:
3047:
3046:
3042:
3028:
3019:
3018:
3009:
3008:
3007:
3006:
2997:
2989:
2983:
2973:
2965:
2963:
2962:
2961:
2957:
2931:
2922:
2915:
2911:
2906:
2896:
2895:
2891:
2877:
2868:
2867:
2858:
2857:
2856:
2855:
2846:
2838:
2832:
2822:
2814:
2812:
2811:
2810:
2806:
2780:
2775:
2774:
2764:
2749:
2747:
2746:
2741:
2739:
2731:
2719:
2714:
2706:
2684:
2682:
2681:
2676:
2674:
2673:
2657:
2655:
2654:
2649:
2647:
2646:
2627:
2625:
2624:
2619:
2617:
2616:
2606:
2591:
2590:
2581:
2580:
2571:
2570:
2557:
2542:
2536:
2532:
2514:
2512:
2511:
2506:
2477:Michel Talagrand
2473:Francesco Guerra
2449:
2447:
2446:
2441:
2439:
2438:
2426:
2425:
2413:
2412:
2390:
2388:
2387:
2382:
2380:
2379:
2370:
2369:
2360:
2359:
2346:
2331:
2325:
2321:
2256:
2254:
2253:
2248:
2246:
2236:
2230:
2226:
2225:
2221:
2217:
2216:
2167:
2163:
2162:
2157:
2156:
2147:
2125:
2120:
2119:
2118:
2109:
2108:
2095:
2090:
2089:
2068:
2063:
2062:
2061:
2052:
2051:
2041:
2036:
2008:
2006:
2005:
2000:
1988:
1986:
1985:
1980:
1968:
1966:
1965:
1960:
1948:
1946:
1945:
1940:
1924:
1922:
1921:
1916:
1895:
1893:
1892:
1887:
1875:
1870:
1860:
1855:
1845:
1840:
1809:
1807:
1806:
1801:
1776:
1774:
1773:
1768:
1763:
1759:
1758:
1757:
1752:
1748:
1747:
1742:
1741:
1732:
1727:
1726:
1708:
1706:
1705:
1704:
1688:
1672:
1669:
1668:
1667:
1648:
1647:
1639:
1638:
1610:
1608:
1607:
1602:
1600:
1599:
1583:
1581:
1580:
1575:
1573:
1572:
1556:
1554:
1553:
1548:
1546:
1545:
1526:
1524:
1523:
1518:
1516:
1515:
1496:
1494:
1493:
1488:
1486:
1482:
1481:
1459:
1458:
1446:
1442:
1441:
1425:
1424:
1411:
1409:
1408:
1403:
1401:
1397:
1396:
1380:
1379:
1367:
1359:
1351:
1347:
1346:
1310:
1308:
1307:
1302:
1300:
1299:
1280:
1278:
1277:
1272:
1260:
1258:
1257:
1252:
1240:
1238:
1237:
1232:
1230:
1229:
1210:
1208:
1207:
1202:
1190:
1188:
1187:
1182:
1170:
1168:
1167:
1162:
1141:
1139:
1138:
1133:
1117:
1115:
1114:
1109:
1107:
1106:
1087:
1085:
1084:
1079:
1074:
1073:
1064:
1063:
1054:
1053:
1040:
999:
997:
996:
991:
917:. Upon reaching
884:Amorphous magnet
872:computer science
768:
761:
754:
741:
736:
735:
728:
724:
723:
334:Spin Hall effect
224:Phase transition
194:Luttinger liquid
131:States of matter
114:Phase transition
100:
86:
85:
67:
51:
6749:
6748:
6744:
6743:
6742:
6740:
6739:
6738:
6719:
6718:
6717:
6712:
6642:Magnetic states
6637:
6597:
6592:
6555:
6437:
6253:
6251:Primary sources
6234:
6214:
6182:
6157:
6139:Parisi, Giorgio
6125:
6105:
6077:
6036:
5970:
5954:
5949:
5944:
5943:
5911:
5905:
5901:
5858:
5854:
5811:
5807:
5780:
5776:
5741:
5737:
5730:
5726:
5694:
5688:
5684:
5652:
5646:
5642:
5610:
5604:
5600:
5568:
5562:
5558:
5526:
5520:
5516:
5484:
5478:
5474:
5442:
5436:
5432:
5382:
5378:
5339:
5335:
5296:
5292:
5285:
5269:
5265:
5255:
5253:
5243:
5239:
5238:
5234:
5225:
5223:
5220:
5216:
5215:
5211:
5202:
5200:
5190:
5186:
5143:
5136:
5125:
5121:
5114:
5100:
5089:
5062:
5058:
5019:
5012:
5005:
4991:
4984:
4941:
4937:
4926:
4912:
4905:
4900:
4895:
4894:
4888:
4884:
4862:
4858:
4856:
4853:
4852:
4835:
4831:
4829:
4826:
4825:
4824:
4820:
4815:
4810:
4746:
4738:protein folding
4701:backpropagation
4697:neural networks
4693:
4676:
4649:
4646:
4645:
4623:
4620:
4619:
4582:
4564:
4555:antiferromagnet
4547:
4524:neural networks
4475:
4471:
4469:
4466:
4465:
4457:
4438:
4437:
4429:
4406:
4402:
4393:
4389:
4376:
4359:
4355:
4342:
4340:
4330:
4326:
4316:
4312:
4295:
4291:
4281:
4277:
4273:
4258:
4257:
4255:
4240:
4236:
4230:
4226:
4220:
4216:
4200:
4196:
4191:
4182:
4178:
4169:
4165:
4152:
4143:
4139:
4133:
4129:
4119:
4110:
4106:
4100:
4096:
4090:
4086:
4073:
4064:
4060:
4054:
4050:
4044:
4040:
4030:
4028:
4026:
4013:
4011:
4008:
4007:
3988:
3985:
3984:
3968:
3965:
3964:
3948:
3945:
3944:
3928:
3925:
3924:
3885:
3881:
3877:
3864:
3860:
3859:
3857:
3846:
3842:
3833:
3829:
3828:
3824:
3823:
3819:
3804:
3800:
3799:
3787:
3783:
3781:
3777:
3773:
3748:
3744:
3743:
3731:
3727:
3724:
3709:
3705:
3696:
3692:
3691:
3687:
3683:
3678:
3675:
3674:
3670:, is given as:
3649:
3645:
3643:
3641:
3638:
3637:
3615:
3611:
3609:
3607:
3604:
3603:
3573:
3570:
3569:
3550:
3546:
3545:
3541:
3524:
3520:
3519:
3515:
3504:
3500:
3491:
3487:
3486:
3482:
3480:
3477:
3476:
3454:
3450:
3449:
3445:
3434:
3430:
3429:
3425:
3417:
3413:
3404:
3400:
3399:
3395:
3387:
3383:
3368:
3364:
3355:
3351:
3350:
3335:
3332:
3331:
3312:
3309:
3308:
3286:
3283:
3282:
3266:
3263:
3262:
3255:
3232:
3228:
3218:
3214:
3210:
3197:
3193:
3188:
3180:
3177:
3176:
3156:
3152:
3130:
3121:
3117:
3115:
3112:
3111:
3089:
3086:
3085:
3071:
3070:
3038:
3034:
3030:
3029:
3027:
3023:
3014:
3010:
3002:
2998:
2988:
2984:
2979:
2978:
2969:
2953:
2949:
2945:
2935:
2930:
2920:
2919:
2887:
2883:
2879:
2878:
2876:
2872:
2863:
2859:
2851:
2847:
2837:
2833:
2828:
2827:
2818:
2802:
2798:
2794:
2784:
2779:
2770:
2766:
2761:
2759:
2756:
2755:
2730:
2707:
2705:
2697:
2694:
2693:
2669:
2665:
2663:
2660:
2659:
2639:
2635:
2633:
2630:
2629:
2612:
2608:
2602:
2586:
2582:
2576:
2572:
2563:
2559:
2547:
2531:
2520:
2517:
2516:
2500:
2497:
2496:
2485:
2434:
2430:
2421:
2417:
2405:
2401:
2399:
2396:
2395:
2375:
2371:
2365:
2361:
2352:
2348:
2336:
2320:
2309:
2306:
2305:
2278:of the system.
2263:
2244:
2243:
2232:
2212:
2208:
2192:
2188:
2178:
2174:
2152:
2148:
2146:
2142:
2138:
2114:
2110:
2104:
2100:
2096:
2094:
2085:
2081:
2057:
2053:
2047:
2043:
2042:
2040:
2035:
2022:
2020:
2017:
2016:
1994:
1991:
1990:
1974:
1971:
1970:
1954:
1951:
1950:
1934:
1931:
1930:
1927:order parameter
1904:
1901:
1900:
1871:
1866:
1856:
1851:
1841:
1830:
1818:
1815:
1814:
1789:
1786:
1785:
1753:
1737:
1733:
1731:
1719:
1715:
1714:
1710:
1709:
1700:
1696:
1692:
1687:
1683:
1679:
1663:
1659:
1652:
1646:
1631:
1627:
1619:
1616:
1615:
1595:
1591:
1589:
1586:
1585:
1584:and a variance
1568:
1564:
1562:
1559:
1558:
1538:
1534:
1532:
1529:
1528:
1508:
1504:
1502:
1499:
1498:
1471:
1467:
1463:
1454:
1450:
1434:
1430:
1426:
1420:
1419:
1417:
1414:
1413:
1389:
1385:
1381:
1375:
1374:
1358:
1339:
1335:
1331:
1326:
1323:
1322:
1292:
1288:
1286:
1283:
1282:
1266:
1263:
1262:
1246:
1243:
1242:
1222:
1218:
1216:
1213:
1212:
1196:
1193:
1192:
1176:
1173:
1172:
1147:
1144:
1143:
1127:
1124:
1123:
1102:
1098:
1096:
1093:
1092:
1069:
1065:
1059:
1055:
1046:
1042:
1027:
1012:
1009:
1008:
985:
982:
981:
974:
959:
940:remains unknown
935:magnetization.
928:magnetization.
923:
901:
886:
880:
822:amorphous solid
796:
772:
731:
718:
717:
710:
709:
708:
498:
490:
489:
488:
464:Amorphous solid
458:
448:
447:
446:
425:
407:
397:
396:
395:
384:
382:Antiferromagnet
375:
373:Superparamagnet
366:
353:
352:Magnetic phases
345:
344:
343:
323:
315:
314:
313:
243:
235:
234:
233:
219:Order parameter
213:
212:Phase phenomena
205:
204:
203:
133:
123:
84:
83:
82:
81:
77:
76:
75:
73:
68:
60:
59:
57:
52:
17:
12:
11:
5:
6747:
6737:
6736:
6731:
6714:
6713:
6711:
6710:
6709:
6708:
6703:
6693:
6691:mictomagnetism
6688:
6683:
6678:
6677:
6676:
6671:
6664:ferromagnetism
6661:
6659:ferrimagnetism
6656:
6651:
6649:altermagnetism
6645:
6643:
6639:
6638:
6636:
6635:
6634:
6633:
6628:
6618:
6617:
6616:
6605:
6603:
6599:
6598:
6591:
6590:
6583:
6576:
6568:
6562:
6561:
6554:
6553:External links
6551:
6550:
6549:
6503:
6469:
6430:
6379:
6361:(4): 801–976,
6347:
6318:
6286:
6265:(5): 965–974,
6252:
6249:
6248:
6247:
6232:
6212:
6199:
6180:
6165:
6155:
6137:Mezard, Marc;
6134:
6123:
6110:
6103:
6090:
6075:
6060:
6034:
6019:
5975:
5968:
5953:
5950:
5948:
5945:
5942:
5941:
5899:
5852:
5805:
5774:
5735:
5724:
5682:
5640:
5598:
5556:
5514:
5472:
5430:
5376:
5333:
5290:
5283:
5263:
5232:
5209:
5184:
5157:(5): 983–990.
5134:
5119:
5112:
5087:
5056:
5010:
5003:
4982:
4955:(2): 141–168.
4945:"Spin glasses"
4935:
4924:
4902:
4901:
4899:
4896:
4893:
4892:
4882:
4870:
4861:
4834:
4817:
4816:
4814:
4811:
4809:
4808:
4803:
4798:
4793:
4788:
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4747:
4745:
4742:
4734:folding funnel
4723:neural network
4692:
4689:
4685:Giorgio Parisi
4675:
4672:
4659:
4656:
4653:
4633:
4630:
4627:
4601:atomic percent
4581:
4578:
4563:
4560:
4546:
4543:
4513:Giorgio Parisi
4474:
4456:
4453:
4452:
4451:
4436:
4432:
4426:
4421:
4415:
4412:
4409:
4405:
4401:
4396:
4392:
4388:
4383:
4380:
4375:
4368:
4365:
4362:
4358:
4354:
4349:
4346:
4339:
4336:
4333:
4329:
4325:
4322:
4319:
4315:
4311:
4308:
4304:
4298:
4294:
4288:
4285:
4280:
4276:
4272:
4269:
4266:
4263:
4261:
4259:
4254:
4249:
4246:
4243:
4239:
4233:
4229:
4223:
4219:
4215:
4207:
4204:
4199:
4195:
4190:
4185:
4181:
4177:
4172:
4168:
4164:
4159:
4156:
4151:
4146:
4142:
4136:
4132:
4126:
4123:
4118:
4113:
4109:
4103:
4099:
4093:
4089:
4085:
4080:
4077:
4072:
4067:
4063:
4057:
4053:
4047:
4043:
4037:
4034:
4029:
4025:
4022:
4019:
4016:
4015:
3992:
3972:
3952:
3932:
3921:
3920:
3908:
3903:
3894:
3891:
3888:
3884:
3880:
3875:
3872:
3867:
3863:
3856:
3849:
3845:
3841:
3836:
3832:
3827:
3822:
3815:
3812:
3807:
3803:
3796:
3793:
3790:
3786:
3780:
3776:
3772:
3769:
3762:
3759:
3756:
3751:
3747:
3740:
3737:
3734:
3730:
3723:
3719:
3712:
3708:
3704:
3699:
3695:
3690:
3686:
3682:
3657:
3652:
3648:
3623:
3618:
3614:
3583:
3580:
3577:
3553:
3549:
3544:
3540:
3537:
3534:
3527:
3523:
3518:
3514:
3507:
3503:
3499:
3494:
3490:
3485:
3473:
3472:
3457:
3453:
3448:
3444:
3437:
3433:
3428:
3420:
3416:
3412:
3407:
3403:
3398:
3390:
3386:
3382:
3379:
3376:
3371:
3367:
3363:
3358:
3354:
3349:
3345:
3342:
3339:
3316:
3296:
3293:
3290:
3270:
3254:
3251:
3235:
3231:
3225:
3222:
3217:
3213:
3204:
3201:
3196:
3192:
3187:
3184:
3164:
3159:
3155:
3151:
3148:
3145:
3142:
3137:
3134:
3129:
3124:
3120:
3099:
3096:
3093:
3069:
3065:
3060:
3056:
3053:
3050:
3045:
3041:
3037:
3033:
3026:
3022:
3017:
3013:
3005:
3001:
2995:
2992:
2987:
2982:
2976:
2972:
2968:
2960:
2956:
2952:
2948:
2944:
2941:
2938:
2934:
2929:
2926:
2923:
2921:
2918:
2914:
2909:
2905:
2902:
2899:
2894:
2890:
2886:
2882:
2875:
2871:
2866:
2862:
2854:
2850:
2844:
2841:
2836:
2831:
2825:
2821:
2817:
2809:
2805:
2801:
2797:
2793:
2790:
2787:
2783:
2778:
2773:
2769:
2765:
2763:
2737:
2734:
2729:
2726:
2722:
2717:
2713:
2710:
2704:
2701:
2687:David Thouless
2672:
2668:
2645:
2642:
2638:
2615:
2611:
2605:
2601:
2597:
2594:
2589:
2585:
2579:
2575:
2569:
2566:
2562:
2556:
2553:
2550:
2546:
2539:
2535:
2530:
2527:
2524:
2504:
2484:
2481:
2452:Giorgio Parisi
2437:
2433:
2429:
2424:
2420:
2416:
2411:
2408:
2404:
2392:
2391:
2378:
2374:
2368:
2364:
2358:
2355:
2351:
2345:
2342:
2339:
2335:
2328:
2324:
2319:
2316:
2313:
2262:
2259:
2258:
2257:
2242:
2239:
2235:
2229:
2224:
2220:
2215:
2211:
2207:
2204:
2201:
2198:
2195:
2191:
2187:
2184:
2181:
2177:
2173:
2170:
2166:
2160:
2155:
2151:
2145:
2141:
2137:
2134:
2131:
2128:
2123:
2117:
2113:
2107:
2103:
2099:
2093:
2088:
2084:
2080:
2077:
2074:
2071:
2066:
2060:
2056:
2050:
2046:
2039:
2034:
2031:
2028:
2025:
2024:
1998:
1978:
1958:
1938:
1914:
1911:
1908:
1897:
1896:
1885:
1882:
1879:
1874:
1869:
1865:
1859:
1854:
1850:
1844:
1839:
1836:
1833:
1829:
1825:
1822:
1799:
1796:
1793:
1782:replica method
1778:
1777:
1766:
1762:
1756:
1751:
1745:
1740:
1736:
1730:
1725:
1722:
1718:
1713:
1703:
1699:
1695:
1691:
1686:
1682:
1678:
1675:
1666:
1662:
1658:
1655:
1651:
1645:
1642:
1637:
1634:
1630:
1626:
1623:
1598:
1594:
1571:
1567:
1544:
1541:
1537:
1514:
1511:
1507:
1485:
1480:
1477:
1474:
1470:
1466:
1462:
1457:
1453:
1449:
1445:
1440:
1437:
1433:
1429:
1423:
1400:
1395:
1392:
1388:
1384:
1378:
1373:
1370:
1365:
1362:
1357:
1354:
1350:
1345:
1342:
1338:
1334:
1330:
1298:
1295:
1291:
1270:
1250:
1228:
1225:
1221:
1200:
1180:
1160:
1157:
1154:
1151:
1131:
1118:refers to the
1105:
1101:
1089:
1088:
1077:
1072:
1068:
1062:
1058:
1052:
1049:
1045:
1039:
1036:
1033:
1030:
1026:
1022:
1019:
1016:
989:
973:
970:
957:
921:
911:magnetic field
899:
879:
876:
794:
774:
773:
771:
770:
763:
756:
748:
745:
744:
743:
742:
729:
712:
711:
707:
706:
701:
696:
691:
686:
681:
676:
671:
666:
661:
656:
651:
646:
641:
636:
631:
626:
621:
616:
611:
606:
601:
596:
591:
586:
581:
576:
571:
566:
561:
556:
551:
546:
541:
536:
531:
526:
521:
516:
511:
506:
500:
499:
496:
495:
492:
491:
487:
486:
481:
479:Liquid crystal
476:
471:
466:
460:
459:
454:
453:
450:
449:
445:
444:
439:
434:
429:
420:
415:
409:
408:
405:Quasiparticles
403:
402:
399:
398:
394:
393:
388:
379:
370:
364:Superdiamagnet
361:
355:
354:
351:
350:
347:
346:
342:
341:
336:
331:
325:
324:
321:
320:
317:
316:
312:
311:
306:
301:
296:
291:
289:Thermoelectric
286:
284:Superconductor
281:
276:
271:
266:
264:Mott insulator
261:
256:
251:
245:
244:
241:
240:
237:
236:
232:
231:
226:
221:
215:
214:
211:
210:
207:
206:
202:
201:
196:
191:
186:
181:
176:
171:
166:
161:
156:
151:
146:
141:
135:
134:
129:
128:
125:
124:
122:
121:
116:
111:
105:
102:
101:
93:
92:
79:
78:
71:
69:
62:
61:
55:
53:
46:
45:
44:
43:
42:
31:(top) and the
15:
9:
6:
4:
3:
2:
6746:
6735:
6732:
6730:
6727:
6726:
6724:
6707:
6704:
6702:
6699:
6698:
6697:
6694:
6692:
6689:
6687:
6686:metamagnetism
6684:
6682:
6681:helimagnetism
6679:
6675:
6672:
6670:
6667:
6666:
6665:
6662:
6660:
6657:
6655:
6652:
6650:
6647:
6646:
6644:
6640:
6632:
6629:
6627:
6624:
6623:
6622:
6621:paramagnetism
6619:
6615:
6612:
6611:
6610:
6607:
6606:
6604:
6600:
6596:
6589:
6584:
6582:
6577:
6575:
6570:
6569:
6566:
6560:
6557:
6556:
6547:
6543:
6539:
6535:
6531:
6527:
6522:
6517:
6513:
6509:
6504:
6500:
6496:
6491:
6486:
6482:
6478:
6474:
6470:
6467:
6463:
6459:
6455:
6451:
6447:
6443:
6436:
6431:
6427:
6423:
6418:
6413:
6408:
6403:
6399:
6395:
6391:
6387:
6386:
6380:
6376:
6372:
6368:
6364:
6360:
6356:
6352:
6348:
6344:
6340:
6336:
6332:
6328:
6324:
6319:
6317:
6312:
6308:
6304:
6300:
6296:
6292:
6287:
6285:
6280:
6276:
6272:
6268:
6264:
6260:
6255:
6254:
6243:
6239:
6235:
6229:
6225:
6224:
6215:
6209:
6205:
6200:
6197:
6194:Approach via
6191:
6187:
6183:
6177:
6173:
6172:
6166:
6163:
6162:replica trick
6158:
6152:
6148:
6144:
6140:
6135:
6132:
6126:
6120:
6116:
6111:
6106:
6100:
6096:
6091:
6086:
6082:
6078:
6072:
6068:
6067:
6061:
6058:
6054:
6053:cavity method
6050:
6045:
6041:
6037:
6035:9780198570837
6031:
6027:
6026:
6020:
6015:
6011:
6007:
6003:
5998:
5993:
5989:
5985:
5981:
5976:
5971:
5965:
5961:
5956:
5955:
5937:
5933:
5929:
5925:
5921:
5917:
5910:
5903:
5895:
5891:
5887:
5883:
5879:
5875:
5871:
5867:
5863:
5856:
5848:
5844:
5840:
5836:
5832:
5828:
5824:
5820:
5816:
5809:
5801:
5797:
5793:
5789:
5785:
5778:
5770:
5766:
5762:
5758:
5754:
5750:
5746:
5739:
5733:
5728:
5720:
5716:
5712:
5708:
5704:
5700:
5699:Physics Today
5693:
5686:
5678:
5674:
5670:
5666:
5662:
5658:
5657:Physics Today
5651:
5644:
5636:
5632:
5628:
5624:
5620:
5616:
5615:Physics Today
5609:
5602:
5594:
5590:
5586:
5582:
5578:
5574:
5573:Physics Today
5567:
5560:
5552:
5548:
5544:
5540:
5536:
5532:
5531:Physics Today
5525:
5518:
5510:
5506:
5502:
5498:
5494:
5490:
5489:Physics Today
5483:
5476:
5468:
5464:
5460:
5456:
5452:
5448:
5447:Physics Today
5441:
5434:
5426:
5422:
5418:
5414:
5410:
5406:
5401:
5396:
5392:
5388:
5380:
5372:
5368:
5364:
5360:
5356:
5352:
5348:
5344:
5337:
5329:
5325:
5321:
5317:
5313:
5309:
5305:
5301:
5294:
5286:
5280:
5276:
5275:
5267:
5251:
5250:
5242:
5236:
5219:
5213:
5199:
5195:
5188:
5180:
5176:
5172:
5168:
5164:
5160:
5156:
5152:
5148:
5141:
5139:
5131:
5130:
5123:
5115:
5113:9780198509400
5109:
5105:
5098:
5096:
5094:
5092:
5083:
5079:
5075:
5071:
5067:
5060:
5052:
5048:
5044:
5040:
5036:
5032:
5028:
5024:
5017:
5015:
5006:
5000:
4996:
4989:
4987:
4978:
4974:
4970:
4966:
4962:
4958:
4954:
4950:
4946:
4939:
4931:
4930:9780748400386
4927:
4921:
4917:
4910:
4908:
4903:
4886:
4868:
4859:
4832:
4822:
4818:
4807:
4804:
4802:
4799:
4797:
4796:Replica trick
4794:
4792:
4789:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4761:Cavity method
4759:
4757:
4754:
4752:
4749:
4748:
4741:
4739:
4735:
4730:
4728:
4724:
4719:
4717:
4713:
4709:
4704:
4702:
4698:
4688:
4686:
4680:
4671:
4657:
4654:
4651:
4631:
4628:
4625:
4616:
4614:
4610:
4606:
4602:
4598:
4594:
4589:
4587:
4577:
4575:
4574:
4573:Physics Today
4569:
4559:
4556:
4552:
4542:
4540:
4536:
4533:
4529:
4525:
4521:
4516:
4514:
4510:
4506:
4502:
4498:
4494:
4472:
4462:
4434:
4424:
4419:
4413:
4410:
4407:
4403:
4399:
4394:
4390:
4386:
4381:
4378:
4373:
4366:
4363:
4360:
4356:
4352:
4347:
4344:
4337:
4334:
4331:
4327:
4323:
4320:
4317:
4313:
4309:
4306:
4302:
4296:
4292:
4286:
4283:
4278:
4274:
4270:
4267:
4264:
4262:
4252:
4247:
4244:
4241:
4237:
4231:
4227:
4221:
4217:
4213:
4205:
4202:
4197:
4193:
4188:
4183:
4179:
4175:
4170:
4166:
4162:
4157:
4154:
4149:
4144:
4140:
4134:
4130:
4124:
4121:
4116:
4111:
4107:
4101:
4097:
4091:
4087:
4083:
4078:
4075:
4070:
4065:
4061:
4055:
4051:
4045:
4041:
4035:
4032:
4023:
4020:
4017:
4006:
4005:
4004:
3990:
3970:
3950:
3930:
3906:
3901:
3892:
3889:
3886:
3882:
3878:
3873:
3870:
3865:
3861:
3854:
3847:
3843:
3839:
3834:
3830:
3825:
3820:
3813:
3810:
3805:
3801:
3794:
3791:
3788:
3784:
3778:
3774:
3770:
3767:
3760:
3757:
3754:
3749:
3745:
3738:
3735:
3732:
3728:
3721:
3717:
3710:
3706:
3702:
3697:
3693:
3688:
3684:
3680:
3673:
3672:
3671:
3655:
3650:
3646:
3636:and variance
3621:
3616:
3612:
3601:
3597:
3575:
3551:
3547:
3542:
3538:
3535:
3532:
3525:
3521:
3516:
3512:
3505:
3501:
3497:
3492:
3488:
3483:
3455:
3451:
3446:
3442:
3435:
3431:
3426:
3418:
3414:
3410:
3405:
3401:
3396:
3388:
3384:
3380:
3377:
3374:
3369:
3365:
3361:
3356:
3352:
3347:
3343:
3340:
3337:
3330:
3329:
3328:
3314:
3294:
3291:
3288:
3268:
3260:
3250:
3233:
3229:
3223:
3220:
3215:
3211:
3202:
3199:
3194:
3190:
3185:
3182:
3162:
3157:
3149:
3146:
3143:
3135:
3132:
3127:
3122:
3118:
3097:
3094:
3091:
3067:
3063:
3058:
3054:
3051:
3048:
3043:
3039:
3035:
3031:
3024:
3020:
3015:
3011:
3003:
2999:
2993:
2990:
2985:
2974:
2966:
2958:
2954:
2950:
2942:
2939:
2932:
2927:
2924:
2916:
2912:
2907:
2903:
2900:
2897:
2892:
2888:
2884:
2880:
2873:
2869:
2864:
2860:
2852:
2848:
2842:
2839:
2834:
2823:
2815:
2807:
2803:
2799:
2791:
2788:
2781:
2776:
2771:
2767:
2753:
2735:
2732:
2727:
2724:
2720:
2715:
2711:
2708:
2702:
2699:
2690:
2688:
2670:
2666:
2643:
2640:
2636:
2613:
2609:
2603:
2599:
2595:
2592:
2587:
2583:
2577:
2573:
2567:
2564:
2560:
2554:
2551:
2548:
2544:
2537:
2533:
2528:
2525:
2522:
2502:
2489:
2483:Phase diagram
2480:
2478:
2474:
2470:
2469:cavity method
2465:
2464:M.A. Virasoro
2461:
2457:
2453:
2435:
2431:
2427:
2422:
2418:
2414:
2409:
2406:
2402:
2376:
2372:
2366:
2362:
2356:
2353:
2349:
2343:
2340:
2337:
2333:
2326:
2322:
2317:
2314:
2311:
2304:
2303:
2302:
2298:
2296:
2292:
2288:
2284:
2279:
2277:
2273:
2269:
2240:
2237:
2227:
2222:
2218:
2213:
2209:
2205:
2202:
2199:
2196:
2193:
2189:
2185:
2182:
2179:
2175:
2171:
2168:
2164:
2158:
2153:
2149:
2143:
2139:
2135:
2132:
2129:
2126:
2121:
2115:
2111:
2105:
2101:
2097:
2091:
2086:
2078:
2075:
2072:
2064:
2058:
2054:
2048:
2044:
2037:
2032:
2029:
2026:
2015:
2014:
2013:
2010:
1996:
1976:
1956:
1936:
1928:
1912:
1909:
1906:
1883:
1880:
1877:
1872:
1867:
1863:
1857:
1852:
1848:
1842:
1837:
1834:
1831:
1827:
1823:
1820:
1813:
1812:
1811:
1797:
1794:
1791:
1783:
1764:
1760:
1754:
1749:
1743:
1738:
1734:
1728:
1723:
1720:
1716:
1711:
1701:
1697:
1693:
1689:
1684:
1680:
1676:
1673:
1664:
1660:
1656:
1653:
1649:
1643:
1635:
1632:
1628:
1621:
1614:
1613:
1612:
1596:
1592:
1569:
1565:
1542:
1539:
1535:
1512:
1509:
1505:
1483:
1478:
1475:
1472:
1468:
1464:
1460:
1455:
1451:
1447:
1443:
1438:
1435:
1431:
1427:
1398:
1393:
1390:
1386:
1382:
1371:
1368:
1363:
1360:
1355:
1352:
1348:
1343:
1340:
1336:
1332:
1328:
1321:
1317:
1312:
1296:
1293:
1289:
1268:
1248:
1226:
1223:
1219:
1198:
1178:
1155:
1152:
1129:
1121:
1103:
1099:
1075:
1070:
1066:
1060:
1056:
1050:
1047:
1043:
1034:
1031:
1024:
1020:
1017:
1014:
1007:
1006:
1005:
1003:
987:
979:
969:
965:
963:
956:
951:
947:
945:
941:
936:
934:
929:
927:
920:
916:
912:
907:
905:
904:paramagnetism
898:
894:
889:
885:
875:
873:
869:
865:
861:
857:
852:
848:
846:
842:
837:
835:
831:
830:ferromagnetic
827:
823:
819:
815:
811:
806:
804:
800:
799:ferromagnetic
793:
789:
785:
781:
769:
764:
762:
757:
755:
750:
749:
747:
746:
740:
730:
727:
722:
716:
715:
714:
713:
705:
702:
700:
697:
695:
692:
690:
687:
685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
645:
642:
640:
637:
635:
632:
630:
627:
625:
622:
620:
617:
615:
612:
610:
607:
605:
602:
600:
597:
595:
592:
590:
587:
585:
582:
580:
577:
575:
572:
570:
567:
565:
562:
560:
557:
555:
552:
550:
547:
545:
542:
540:
537:
535:
532:
530:
527:
525:
522:
520:
517:
515:
512:
510:
507:
505:
504:Van der Waals
502:
501:
494:
493:
485:
482:
480:
477:
475:
472:
470:
467:
465:
462:
461:
457:
452:
451:
443:
440:
438:
435:
433:
430:
428:
424:
421:
419:
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340:
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332:
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327:
326:
319:
318:
310:
307:
305:
302:
300:
299:Ferroelectric
297:
295:
294:Piezoelectric
292:
290:
287:
285:
282:
280:
277:
275:
272:
270:
269:Semiconductor
267:
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262:
260:
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230:
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190:
189:Superfluidity
187:
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6609:diamagnetism
6514:(1): 71–79,
6511:
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6383:
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6115:Spin glasses
6114:
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4691:Applications
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4590:
4586:Kondo effect
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4535:optimization
4526:, including
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4505:ground state
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634:von Klitzing
390:
339:Kondo effect
199:Time crystal
179:Fermi liquid
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28:
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6329:: 185–186,
5952:Expositions
5705:(3): 9–11.
5663:(9): 9–11.
5621:(7): 9–11.
5579:(9): 9–11.
5537:(6): 9–11.
5453:(1): 9–11.
4497:ultrametric
2291:Ising model
1320:free energy
1002:Hamiltonian
978:Ising model
864:simulations
456:Soft matter
377:Ferromagnet
37:ferromagnet
6723:Categories
6696:spin glass
6351:Binder, K.
5997:2204.02909
5947:Literature
5922:(1): 271.
5916:J. Phys. A
5400:1907.02295
5256:3 November
5226:2023-12-23
5203:2023-12-23
4925:0748400389
4898:References
882:See also:
856:metastable
814:positional
803:disordered
784:spin glass
599:Louis Néel
589:Schrieffer
497:Scientists
391:Spin glass
386:Metamagnet
368:Paramagnet
184:Supersolid
29:spin glass
6595:Magnetism
6242:733249730
6044:234430714
6014:1935-8237
5894:0031-9007
5847:0031-9007
5417:0036-8075
5371:0953-8984
5328:0953-8984
5179:0305-4470
5051:250734239
4977:0010-7514
4736:model of
4580:Discovery
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4364:−
4332:β
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2456:M. Mezard
2334:∑
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1356:−
1159:⟩
1150:⟨
1038:⟩
1029:⟨
1025:∑
1021:−
915:Curie law
699:Wetterich
679:Abrikosov
594:Josephson
564:Van Vleck
554:Luttinger
427:Polariton
359:Diamagnet
279:Conductor
274:Semimetal
259:Insulator
174:Fermi gas
35:one of a
6706:spin ice
6546:16833848
6190:70764844
6145:(1987),
6085:47063323
5495:(3): 9.
5425:32467362
5393:(6494).
4806:Spin ice
4744:See also
4539:genetics
2272:replicas
942:). This
933:remanent
810:magnetic
739:Category
684:Ginzburg
659:Laughlin
619:Kadanoff
574:Shockley
559:Anderson
514:von Laue
164:Bose gas
39:(bottom)
6526:Bibcode
6499:2652978
6450:Bibcode
6426:3478708
6394:Bibcode
6363:Bibcode
6331:Bibcode
6299:Bibcode
6267:Bibcode
6129:Covers
5924:Bibcode
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5827:Bibcode
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5707:Bibcode
5665:Bibcode
5623:Bibcode
5581:Bibcode
5539:Bibcode
5497:Bibcode
5455:Bibcode
5387:Science
5351:Bibcode
5308:Bibcode
5159:Bibcode
5070:Bibcode
5031:Bibcode
4957:Bibcode
4562:History
4520:kelvins
4461:ergodic
2274:of the
826:crystal
689:Leggett
664:Störmer
649:Bednorz
609:Giaever
579:Bardeen
569:Hubbard
544:Peierls
534:Onsager
484:Polymer
469:Colloid
432:Polaron
423:Plasmon
418:Exciton
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694:Parisi
654:Müller
644:Rohrer
639:Binnig
629:Wilson
624:Fisher
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549:Landau
437:Magnon
413:Phonon
254:Plasma
154:Plasma
144:Liquid
109:Phases
25:random
6542:S2CID
6516:arXiv
6495:JSTOR
6438:(PDF)
5992:arXiv
5912:(PDF)
5695:(PDF)
5653:(PDF)
5611:(PDF)
5569:(PDF)
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5443:(PDF)
5395:arXiv
5244:(PDF)
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4890:etc."
4813:Notes
4729:etc.
909:If a
818:glass
788:spins
604:Esaki
529:Bloch
524:Debye
519:Bragg
509:Onnes
442:Roton
139:Solid
6429:....
6422:PMID
6238:OCLC
6228:ISBN
6218:and
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6176:ISBN
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6119:ISBN
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6081:OCLC
6071:ISBN
6040:OCLC
6030:ISBN
6010:ISSN
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753:v
74:)
72:2
58:)
56:2
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