View source for Spin glass

{{Short description|Disordered magnetic state}}
[[File:Spin glass by Zureks.svg|thumb|Schematic representation of the ''random'' spin structure of a ''spin glass'' (top) and the ''ordered'' one of a ''ferromagnet'' (bottom)]]
{{multiple image
| direction = horizontal
| width = 150
| footer = The magnetic disorder of spin glass compared to a ferromagnet is analogous to the positional disorder of glass (left) compared to quartz (right).
| image1 = Silica.svg
| alt1 = Amorphous SiO<sub>2</sub>
| caption1 = Glass (amorphous SiO<sub>2</sub>)
| image2 = SiO² Quartz.svg
| alt2 = Crystalline SiO<sub>2</sub>)
| caption2 = Quartz (crystalline SiO<sub>2</sub>)
| align = 
| total_width = 
}}
{{Condensed matter physics}}
In [[condensed matter physics]], a '''spin glass''' is a  magnetic state characterized by randomness, besides cooperative behavior in freezing of [[Spin (physics)|spins]] at a temperature called the "freezing temperature," ''T''<sub>f</sub>.<ref name=":0">{{Cite book |last=Mydosh |first=J. A. |title=Spin Glasses: An Experimental Introduction |publisher=Taylor & Francis |year=1993 |isbn=0748400389 |id={{isbnt|9780748400386}} |location=London, Washington DC |pages=3}}</ref>  In [[Ferromagnetism|ferromagnetic]] solids, component atoms' magnetic spins all align in the same direction.  Spin glass when contrasted with a ferromagnet is defined as "[[Entropy|disordered]]" magnetic state in which spins are aligned randomly or without a regular pattern and the couplings too are random.<ref name=":0" /> A spin glass should not be confused with a "[[Doping_(semiconductor)|spin-on glass]]".  The latter is a thin film, usually based on SiO<sub>2</sub>, which is applied via [[spin coating]].

The term "glass" comes from an analogy between the ''magnetic'' disorder in a spin glass and the ''positional'' disorder of a conventional, chemical [[glass]], e.g., a window glass.  In window glass or any [[amorphous solid]] the atomic bond structure is highly irregular; in contrast, a [[crystal]] has a uniform pattern of atomic bonds.  In [[ferromagnetic]] solids, magnetic spins all align in the same direction; this is analogous to a crystal's [[Crystal lattice structure|lattice-based structure]].

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 [[antiferromagnet]]ic 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 [[Geometrical frustration|frustrated interactions]]{{snd}} 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.

There are two main aspects of spin glass. On the physical side, spin glasses are real materials with distinctive properties, a review of which was published in 1982.<ref>{{Cite journal |last=Ford |first=Peter J. |date=March 1982 |title=Spin glasses |url=http://www.tandfonline.com/doi/abs/10.1080/00107518208237073 |journal=Contemporary Physics |language=en |volume=23 |issue=2 |pages=141–168 |doi=10.1080/00107518208237073 |bibcode=1982ConPh..23..141F |issn=0010-7514}}</ref> On the mathematical side, simple statistical mechanics models, inspired by real spin glasses, are widely studied and applied.<ref name=":1" />

Spin glasses and the complex internal structures that arise within them are termed "[[Metastability|metastable]]" because they are "stuck" in stable configurations other than the [[Ground state|lowest-energy configuration]] (which would be aligned and ferromagnetic).  The mathematical complexity of these structures is difficult but fruitful to study experimentally or in [[simulation]]s; with applications to physics, chemistry, materials science and [[artificial neural network]]s in [[computer science]].

==Magnetic behavior==
{{See also|Amorphous magnet}}
It is the time dependence which distinguishes spin glasses from other magnetic systems.

Above the spin glass [[phase transition|transition temperature]], ''T''<sub>c</sub>,<ref group="note"><math>T_\text{c}</math> is identical to the so-called "freezing temperature" <math>T_\text{f}.</math></ref> the spin glass exhibits typical magnetic behaviour (such as [[paramagnetism]]).

If a [[applied magnetic field|magnetic field]] is applied as the sample is cooled to the transition temperature, magnetization of the sample increases as described by the [[Curie's law|Curie law]]. Upon reaching ''T''<sub>c</sub>, the sample becomes a spin glass, and further cooling results in little change in magnetization. This is referred to as the ''field-cooled'' magnetization.

When the external magnetic field is removed, the magnetization of the spin glass falls rapidly to a lower value known as the ''remanent'' magnetization.

Magnetization then decays slowly as it approaches zero (or some small fraction of the original value{{snd}} this [[List of unsolved problems in physics|remains unknown]]). This [[exponential decay|decay is non-exponential]], and no simple function can fit the curve of magnetization versus time adequately.<ref name="JPhys">{{cite journal |last1=Joy |first1=P. A. |last2=Kumar |first2=P. S. Anil |last3=Date |first3=S. K. |title=The relationship between field-cooled and zero-field-cooled susceptibilities of some ordered magnetic systems |journal=J. Phys.: Condens. Matter |date=7 October 1998 |volume=10 |issue=48 |pages=11049–11054 |doi=10.1088/0953-8984/10/48/024 |bibcode=1998JPCM...1011049J |s2cid=250734239 }}</ref> 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.{{r|JPhys}}

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 field is removed, the magnetization rapidly falls to zero, with no remanent magnetization. The decay is rapid and exponential.{{Citation needed|date=September 2011}}

If the sample is cooled below ''T''<sub>c</sub> 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 ''zero-field-cooled'' magnetization. A slow upward drift then occurs toward the field-cooled magnetization.

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,<ref name="Nordblad">{{cite journal |last1=Nordblad |first1=P. |last2=Lundgren |first2=L. |last3=Sandlund |first3=L. |title=A link between the relaxation of the zero field cooled and the thermoremanent magnetizations in spin glasses |journal=Journal of Magnetism and Magnetic Materials |date=February 1986 |volume=54–57 |issue=1 |pages=185–186 |doi=10.1016/0304-8853(86)90543-3 |bibcode=1986JMMM...54..185N }}</ref> at least in the limit of very small external fields.

==Edwards–Anderson model==
This is similar to the [[Ising model]]. In this model, we have spins arranged on a <math>d</math>-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.<ref name=nishimori>{{cite book|last=Nishimori|first=Hidetoshi|title=Statistical Physics of Spin Glasses and Information Processing: An Introduction|year=2001|publisher=Oxford University Press|location=Oxford|isbn=9780198509400|pages=243}}</ref> The [[Hamiltonian mechanics|Hamiltonian]] for this spin system is given by:

: <math>H = -\sum_{\langle ij\rangle} J_{ij} S_i S_j,</math>

where <math>S_i</math> refers to the [[Pauli spin matrix]] for the spin-half particle at lattice point <math>i</math>, and the sum over <math>\langle ij\rangle</math> refers to summing over neighboring lattice points <math>i</math> and <math>j</math>. A negative value of <math>J_{ij}</math> denotes an antiferromagnetic type interaction between spins at points <math>i</math> and <math>j</math>. The sum runs over all nearest neighbor positions on a lattice, of any dimension. The variables <math>J_{ij}</math> representing the magnetic nature of the spin-spin interactions are called bond or link variables.

In order to determine the [[Partition function (statistical mechanics)|partition function]] for this system, one needs to average the [[Helmholtz free energy|free energy]] <math>f\left[J_{ij}\right] = -\frac{1}{\beta} \ln\mathcal{Z}\left[J_{ij}\right]</math> where <math>\mathcal{Z}\left[J_{ij}\right] = \operatorname{Tr}_S \left(e^{-\beta H}\right)</math>, over all possible values of <math>J_{ij}</math>. The distribution of values of <math>J_{ij}</math> is taken to be a Gaussian with a mean <math>J_0</math> and a variance <math>J^2</math>:

: <math>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\}.</math>

Solving for the free energy using the [[replica trick|replica method]], 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 <math>m = 0</math> along with a non-vanishing value of the two point correlation function between spins at the same lattice point but at two different replicas:

: <math>q = \sum_{i=1}^N S^\alpha_i S^\beta_i \neq 0,</math>

where <math>\alpha, \beta</math> are replica indices. The [[order parameter]] for the ferromagnetic to spin glass phase transition is therefore <math>q</math>, and that for paramagnetic to spin glass is again <math>q</math>. Hence the new set of order parameters describing the three magnetic phases consists of both <math>m</math> and <math>q</math>.

Under the assumption of replica symmetry, the mean-field free energy is given by the expression:{{r|nishimori}}

: <math>\begin{align}
  \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{align}</math>

==Sherrington–Kirkpatrick model==
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 [[mean-field theory]] based on a set of [[replica trick|replicas]] of the [[partition function (statistical mechanics)|partition function]] of the system.

An important, exactly solvable model of a spin glass was introduced by [[David Sherrington (physicist)|David Sherrington]] and [[Scott Kirkpatrick]] in 1975. It is an [[Ising model]] with long range frustrated ferro- as well as antiferromagnetic couplings. It corresponds to a [[mean-field theory|mean-field approximation]] of spin glasses describing the slow dynamics of the magnetization and the complex non-ergodic equilibrium state.

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 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:

: <math>
H = - \frac 1\sqrt N \sum_{i<j} J_{ij} S_i S_j
</math>

where <math>J_{ij}, S_i, S_j</math> 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 [[Giorgio Parisi]] in 1979 with the replica method. The subsequent work of interpretation of the Parisi solution—by [[Marc Mézard|M. Mezard]], [[Giorgio Parisi|G. Parisi]], [[Miguel Ángel Virasoro (physicist)|M.A. Virasoro]] 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 [[cavity method]], which allowed study of the low temperature phase without replicas. A rigorous proof of the Parisi solution has been provided in the work of [[Francesco Guerra]] and [[Michel Talagrand]].<ref>{{cite book |last1=Talagrand |first1=Michel |title=Mean Field Models for Spin Glasses |date=10 November 2010 |publisher=Springer Berlin |location=Heidelberg |isbn=978-3-642-15202-3 |url=https://doi.org/10.1007/978-3-642-15202-3 |access-date=14 January 2025}}</ref>

=== Phase diagram ===
[[File:De Almeida-Thouless line.svg|thumb|de Almeida-Thouless curve.]]
When there is a uniform external magnetic field of magnitude <math>
M
</math>, the energy function becomes<math display="block">
H = - \frac 1\sqrt N \sum_{i<j} J_{ij} S_i S_j - M \sum_i S_i
</math>Let all couplings <math>
J_{ij}
</math> are IID samples from the gaussian distribution of mean 0 and variance <math>
J^2
</math>. In 1979, J.R.L. de Almeida and [[David J. Thouless|David Thouless]]<ref name=":2" /> 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.

The stability region on the phase diagram of the SK model is determined by two dimensionless parameters <math>
x := \frac{kT}{J}, \quad y := \frac{M}{J}
</math>. Its phase diagram has two parts, divided by the ''de Almeida-Thouless curve'', The curve is the solution set to the equations<ref name=":2">{{Cite journal |last1=Almeida |first1=J R L de |last2=Thouless |first2=D J |date=May 1978 |title=Stability of the Sherrington-Kirkpatrick solution of a spin glass model |url=https://iopscience.iop.org/article/10.1088/0305-4470/11/5/028 |journal=Journal of Physics A: Mathematical and General |volume=11 |issue=5 |pages=983–990 |doi=10.1088/0305-4470/11/5/028 |bibcode=1978JPhA...11..983D |issn=0305-4470}}</ref><math display="block">
\begin{aligned}
& x^2 = \frac{1}{(2 \pi)^{1 / 2}} \int \mathrm{d} z\; \mathrm{e}^{-\frac 12 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}
</math>The phase transition occurs at <math>x = 1</math>. Just below it, we have<math display="block">
y^2 \approx \frac 43 ( 1-x)^3.
</math>At low temperature, high magnetic field limit, the line is<math display="block">
x \approx \frac{4}{3\sqrt{2\pi}} e^{-\frac 12 y^2}
</math>

==Infinite-range model==
This is also called the "p-spin model".<ref name=":1">{{Cite book |last1=Mézard |first1=Marc |title=Information, physics, and computation |last2=Montanari |first2=Andrea |date=2009 |publisher=Oxford university press |isbn=978-0-19-857083-7 |series=Oxford graduate texts |location=Oxford}}</ref> The infinite-range model is a generalization of the [[#Sherrington–Kirkpatrick model|Sherrington–Kirkpatrick model]] where we not only consider two-spin interactions but <math>p</math>-spin interactions, where <math>p \leq N</math> and <math>N</math> 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:

: <math>
H = -\sum_{i_1 < i_2 < \cdots < i_p} J_{i_1 \dots i_p} S_{i_1}\cdots S_{i_p}
</math>

where <math>J_{i_1\dots i_p}, S_{i_1},\dots, S_{i_p}</math> have similar meanings as in the EA model. The <math>p\to \infty</math> limit of this model is known as the [[random energy model]]. 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 [[central limit theorem]]. The Gaussian distribution function, with mean <math>\frac{J_0}{N} </math> and variance <math>\frac{J^2}{N}</math>, is given as:

: <math>
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\}
</math>

The order parameters for this system are given by the magnetization <math>m</math> and the two point spin correlation between spins at the same site <math>q</math>, 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<ref name="nishimori" /> in terms of <math>m</math> and <math>q</math>, under the assumption of replica symmetry as well as 1-Replica Symmetry Breaking.<ref name="nishimori" />

: <math>\begin{align}
  \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 p m^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 p m^{p-1}\right)\right)\, \mathrm{d}z
\end{align}</math>

==Non-ergodic behavior and applications==
A thermodynamic system is [[ergodic]] 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 <math>T_\text{f}</math>, 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 [[energy landscape]]; the distances between minima are given by an [[ultrametric]], with tall energy barriers between minima.<ref group="note">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, ..., etc."</ref> The [[participation ratio]] counts the number of states that are accessible from a given instance, that is, the number of states that participate in the [[ground state]]. The ergodic aspect of spin glass was instrumental in the awarding of half the [[List of Nobel laureates in Physics|2021 Nobel Prize in Physics]] to [[Giorgio Parisi]].<ref name="Geddes Nobel for climate work">{{cite web | last=Geddes | first=Linda | title=Trio of scientists win Nobel prize for physics for climate work | website=The Guardian | date=2021-10-05 | url=https://www.theguardian.com/books/2021/oct/05/nobel-prize-physics-scientists-sykuro-manabe-klaus-hasselmann-giorgio-parisi-win-climate | access-date=2023-12-23}}</ref><ref name="2021 Physics Nobel - popular exposition">{{cite web | title=The Nobel Prize in Physics 2021 - Popular Science Background | url=https://www.nobelprize.org/uploads/2021/10/popular-physicsprize2021-2.pdf | access-date=2023-12-23}}</ref><ref>{{Cite web |date=5 October 2021 |title=Scientific Background for the Nobel Prize in Physics 2021 |url=https://www.nobelprize.org/uploads/2021/10/sciback_fy_en_21.pdf |access-date=3 November 2023 |website=[[Nobel Committee for Physics]]}}</ref>

For physical systems, such as dilute manganese in copper, the freezing temperature is typically as low as 30 [[kelvin]]s (−240&nbsp;°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 [[Artificial neural network|neural networks]], including [[Hopfield network]]s, as well as many problems in [[computer science]] [[optimization (mathematics)|optimization]] and [[genetics]].

==Spin-glass without structural disorder==

Elemental crystalline neodymium is [[Paramagnetism|paramagnetic]] at room temperature and becomes an [[Antiferromagnetism|antiferromagnet]] with incommensurate order upon cooling below 19.9&nbsp;K.<ref>{{cite book|author1=Andrej Szytula|author2=Janusz Leciejewicz|title=Handbook of Crystal Structures and Magnetic Properties of Rare Earth Intermetallics|url=https://books.google.com/books?id=-tgM8oAQcdcC&pg=PA1|date=8 March 1994|publisher=CRC Press|isbn=978-0-8493-4261-5|page=1}}</ref> Below this transition temperature it exhibits a complex set of magnetic phases<ref>{{cite journal | last1=Zochowski | first1=S W | last2=McEwen | first2=K A | last3=Fawcett | first3=E | title=Magnetic phase diagrams of neodymium | journal=Journal of Physics: Condensed Matter | volume=3 | issue=41 | date=1991 | issn=0953-8984 | doi=10.1088/0953-8984/3/41/007 | pages=8079–8094| bibcode=1991JPCM....3.8079Z }}</ref><ref>{{cite journal | last1=Lebech | first1=B | last2=Wolny | first2=J | last3=Moon | first3=R M | title=Magnetic phase transitions in double hexagonal close packed neodymium metal-commensurate in two dimensions | journal=Journal of Physics: Condensed Matter | volume=6 | issue=27 | date=1994 | issn=0953-8984 | doi=10.1088/0953-8984/6/27/029 | pages=5201–5222| bibcode=1994JPCM....6.5201L }}</ref> that have long spin relaxation times and spin-glass behavior that does not rely on structural disorder.<ref>{{cite journal | last1=Kamber | first1=Umut | last2=Bergman | first2=Anders | last3=Eich | first3=Andreas | last4=Iuşan | first4=Diana | last5=Steinbrecher | first5=Manuel | last6=Hauptmann | first6=Nadine | last7=Nordström | first7=Lars | last8=Katsnelson | first8=Mikhail I. | last9=Wegner | first9=Daniel | last10=Eriksson | first10=Olle | last11=Khajetoorians | first11=Alexander A. | title=Self-induced spin glass state in elemental and crystalline neodymium | journal=Science | volume=368 | issue=6494 | date=2020 | issn=0036-8075 | doi=10.1126/science.aay6757 | page=| pmid=32467362 | arxiv=1907.02295 }}</ref>

==History==

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 [[Philip Warren Anderson|Philip W. Anderson]] in ''[[Physics Today]]''.<ref>
{{cite journal
 |title=Spin Glass I: A Scaling Law Rescued
 |journal = Physics Today|volume = 41|pages = 9–11|author=Philip W. Anderson
 |year=1988
 |issue = 1|doi=10.1063/1.2811268
 |url=http://www.physics.rutgers.edu/~pchandra/physics681/sglass1.pdf
 |bibcode=1988PhT....41a...9A
 }}</ref><ref>
{{cite journal
 |title=Spin Glass II: Is There a Phase Transition?
 |journal = Physics Today|volume = 41|issue = 3|pages = 9|author=Philip W. Anderson
 |year=1988
 |doi=10.1063/1.2811336
 |url=http://www.physics.rutgers.edu/~pchandra/physics681/sglass2.pdf |bibcode=1988PhT....41c...9A
 }}</ref><ref>
{{cite journal
 |title=Spin Glass III: Theory Raises its Head
 |journal = Physics Today|volume = 41|issue = 6|pages = 9–11|author=Philip W. Anderson
 |year=1988
 |doi=10.1063/1.2811440
 |url=http://www.physics.rutgers.edu/~pchandra/physics681/sglass3.pdf |bibcode=1988PhT....41f...9A
 }}</ref><ref>
{{cite journal
 |title=Spin Glass IV: Glimmerings of Trouble
 |journal = Physics Today|volume = 41|issue = 9|pages = 9–11|author=Philip W. Anderson
 |year=1988
 |doi=10.1063/1.881135
 |url=http://www.physics.rutgers.edu/~pchandra/physics681/sglass4.pdf |bibcode=1988PhT....41i...9A
 }}</ref><ref>
{{cite journal
 |title=Spin Glass V: Real Power Brought to Bear
 |journal = Physics Today|volume = 42|issue = 7|pages = 9–11|author=Philip W. Anderson
 |year=1989
 |doi=10.1063/1.2811073
 |url=http://www.physics.rutgers.edu/~pchandra/physics681/sglass5.pdf |bibcode=1989PhT....42g...9A
 }}</ref><ref>
{{cite journal
 |title=Spin Glass VI: Spin Glass As Cornucopia
 |journal = Physics Today|volume = 42|issue = 9|pages = 9–11|author=Philip W. Anderson
 |year=1989
 |doi=10.1063/1.2811137
 |url=http://www.physics.rutgers.edu/~pchandra/physics681/sglass6.pdf |bibcode=1989PhT....42i...9A
 }}</ref><ref>
{{cite journal
 |title=Spin Glass VII: Spin Glass as Paradigm
 |journal = Physics Today|volume = 43|issue = 3|pages = 9–11|author=Philip W. Anderson
 |year=1990
 |doi=10.1063/1.2810479
 |url=http://www.physics.rutgers.edu/~pchandra/physics681/sglass7.pdf |bibcode=1990PhT....43c...9A
 }}</ref><ref>[https://web.archive.org/web/20240206131352/https://pitp.phas.ubc.ca/confs/7pines2009/readings/Stamp-PWA-SpinGl-refF-PhysT.pdf All of them combined.]</ref>

=== Discovery ===
In 1930s, material scientists discovered the [[Kondo effect]], 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 contains a very small fraction of magnetic atoms (i.e., at high dilution).

Unusual behavior was observed in iron-in-gold alloy (Au''Fe'') and manganese-in-copper alloy (Cu''Mn'') at around 1 to 10 [[Atomic ratio|atom percent]]. Cannella and Mydosh observed in 1972<ref>{{Cite journal |last1=Cannella |first1=V. |last2=Mydosh |first2=J. A. |date=1972-12-01 |title=Magnetic Ordering in Gold-Iron Alloys |url=https://link.aps.org/doi/10.1103/PhysRevB.6.4220 |journal=Physical Review B |volume=6 |issue=11 |pages=4220–4237 |doi=10.1103/PhysRevB.6.4220|bibcode=1972PhRvB...6.4220C }}</ref> that Au''Fe'' had an unexpected cusplike peak in the [[Magnetic susceptibility|a.c. susceptibility]] at a well defined temperature, which would later be termed ''spin glass freezing temperature''.<ref>{{Cite journal |last1=Mulder |first1=C. A. M. |last2=van Duyneveldt |first2=A. J. |last3=Mydosh |first3=J. A. |date=1981-02-01 |title=Susceptibility of the $\mathrm{Cu}\mathrm{Mn}$ spin-glass: Frequency and field dependences |url=https://link.aps.org/doi/10.1103/PhysRevB.23.1384 |journal=Physical Review B |volume=23 |issue=3 |pages=1384–1396 |doi=10.1103/PhysRevB.23.1384}}</ref>

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 (<math>J > 0</math>) and antiferromagnetic (<math>J < 0</math>) 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.

=== Sherrington–Kirkpatrick model ===

Sherrington and Kirkpatrick proposed the SK model in 1975, and solved it by the replica method.<ref>{{Cite journal |last1=Sherrington |first1=David |last2=Kirkpatrick |first2=Scott |date=1975-12-29 |title=Solvable Model of a Spin-Glass |url=http://dx.doi.org/10.1103/physrevlett.35.1792 |journal=Physical Review Letters |volume=35 |issue=26 |pages=1792–1796 |doi=10.1103/physrevlett.35.1792 |bibcode=1975PhRvL..35.1792S |issn=0031-9007}}</ref> 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.

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 of [[Giorgio Parisi]]. At the full replica breaking ansatz, infinitely many order parameters are required to characterize a stable solution.<ref>{{Cite journal |last=Parisi |first=G. |date=1979-12-03 |title=Infinite Number of Order Parameters for Spin-Glasses |url=https://link.aps.org/doi/10.1103/PhysRevLett.43.1754 |journal=Physical Review Letters |language=en |volume=43 |issue=23 |pages=1754–1756 |doi=10.1103/PhysRevLett.43.1754 |bibcode=1979PhRvL..43.1754P |issn=0031-9007}}</ref>

== Applications ==
The formalism of replica mean-field theory has also been applied in the study of [[Artificial neural network|neural networks]], where it has enabled calculations of properties such as the storage capacity of simple neural network architectures without requiring a training algorithm (such as [[backpropagation]]) to be designed or implemented.<ref name="Gardner">{{cite journal |last1=Gardner |first1=E |last2=Deridda |first2=B |date=7 January 1988 |title=Optimal storage properties of neural network models |url=https://hal.archives-ouvertes.fr/hal-03285587/file/Optimal%20storage%20properties%20of%20neural%20network%20models.pdf |journal=J. Phys. A |volume=21 |pages=271 |bibcode=1988JPhA...21..271G |doi=10.1088/0305-4470/21/1/031 |number=1}}</ref>

More realistic spin glass models with short range frustrated interactions and disorder, like the Gaussian model where the couplings between neighboring spins follow a [[Gaussian distribution]], have been studied extensively as well, especially using [[Monte Carlo simulation]]s. These models display spin glass phases bordered by sharp [[phase transition]]s.

Besides its relevance in condensed matter physics, spin glass theory has acquired a strongly interdisciplinary character, with applications to [[Artificial neural network|neural network]] theory, computer science, theoretical biology, [[econophysics]] etc.

Spin glass models were adapted to the [[folding funnel]] model of [[protein folding]].

==See also==
{{Div col}}
*[[Amorphous magnet]]
*[[Antiferromagnetic interaction]]
*[[Cavity method]]
*[[Crystal structure]]
*[[Geometrical frustration]]
*[[Orientational glass]]
*[[Phase transition]]
*[[Quenched disorder]]
*[[Random energy model]]
*[[Replica trick]]
*[[Solid-state physics]]
*[[Spin ice]]
{{Div col end}}

==Notes==
{{Reflist|group="note"}}

==References==
{{Reflist}}

==Literature==

=== Expositions ===

* {{Cite book |last1=Stein |first1=Daniel L. |title=Spin glasses and complexity |last2=Newman |first2=Charles M. |date=2013 |publisher=Princeton University Press |isbn=978-0-691-14733-8 |series=Primers in complex systems |location=Princeton}} Popular exposition, with a minimal amount of mathematics.
* {{Cite journal |last1=Montanari |first1=Andrea |last2=Sen |first2=Subhabrata |date=2024-01-09 |title=A Friendly Tutorial on Mean-Field Spin Glass Techniques for Non-Physicists |url=https://www.nowpublishers.com/article/Details/MAL-105 |journal=Foundations and Trends in Machine Learning |language=English |volume=17 |issue=1 |pages=1–173 |doi=10.1561/2200000105 |issn=1935-8237|arxiv=2204.02909 }} A practical tutorial introduction.
* {{cite book|last1=Mézard|first1=Marc|last2=Montanari|first2=Andrea|title=Information, Physics, and Computation|date=2009|publisher=Oxford University Press|location=Oxford, U.K.|isbn=9780198570837|oclc=234430714|url=https://books.google.com/books?id=jhCM7i0a6UUC}} [http://www.stat.ucla.edu/~ywu/research/documents/BOOKS/MontanariInformationPhysicsComputation.pdf 1st 15 chapters of 2008 draft version, available at www.stat.ucla.edu] Textbook that focuses on the [[cavity method]] and the applications to computer science, especially [[constraint satisfaction problem]]s.
* {{Cite book |last=Nishimori |first=Hidetoshi |url=https://www.worldcat.org/title/ocm47063323 |title=Statistical physics of spin glasses and information processing: an introduction |date=2001 |publisher=Oxford University Press |isbn=978-0-19-850940-0 |series=International series of monographs on physics |location=Oxford; New York |oclc=ocm47063323}} Introduction focused on computer science applications, including neural networks.
* {{Cite book |last=Mydosh |first=J. A. |title=Spin glasses: an experimental introduction |date=1993 |publisher=Taylor & Francis |isbn=978-0-7484-0038-6 |location=London; Washington, DC}} Focuses on the experimentally measurable properties of spin glasses (such as copper-manganese alloy).
* {{Cite book |last1=Fischer |first1=K. H. |title=Spin glasses |last2=Hertz |first2=John |date=1991 |publisher=Cambridge University Press |isbn=978-0-521-34296-4 |series=Cambridge studies in magnetism |location=Cambridge; New York, NY, USA}} Covers [[Mean-field theory|mean field theory]], experimental data, and numerical simulations.
* {{citation
 | last1 = Mezard
 | first1= Marc
 | last2=Parisi|first2=Giorgio|author2-link=Giorgio Parisi
 | last3=Virasoro|first3=Miguel Angel|author3-link=Miguel Ángel Virasoro (physicist)
 | year = 1987
 | title = Spin glass theory and beyond
 | publisher = World Scientific
 | location = Singapore
 | isbn =  978-9971-5-0115-0
}}. Early exposition containing the pre-1990 breakthroughs, such as the [[replica trick]].
* {{Cite book |last1=De Dominicis |first1=Cirano |url=https://www.worldcat.org/title/ocm70764844 |title=Random fields and spin glasses: a field theory approach |last2=Giardina |first2=Irene |date=2006 |publisher=Cambridge University Press |isbn=978-0-521-84783-4 |location=Cambridge, UK; New York |oclc=ocm70764844}} Approach via [[statistical field theory]].
* {{Cite book |last=Talagrand |first=Michel |title=Mean field models for spin glasses. 1: Basic examples |date=2010 |publisher=Springer |isbn=978-3-642-26598-3 |edition=Softcover repr. of the harcover 1st ed. 2010 |series=Ergebnisse der Mathematik und ihrer Grenzgebiete |location=Berlin Heidelberg}} and {{Cite book |last=Talagrand |first=Michel |url=https://www.worldcat.org/title/733249730 |title=Mean field models for spin glasses |date=2011 |publisher=Springer |isbn=978-3-642-15201-6 |series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge = A series of modern surveys in mathematics |location=Heidelberg; New York |oclc=733249730}}. Compendium of rigorously provable results.

=== Primary sources ===
*{{citation|first1=S.F.|last1=Edwards|first2=P.W.|last2=Anderson|journal=Journal of Physics F: Metal Physics|title=Theory of spin glasses|volume=5|issue=5|pages=965–974|year=1975|doi=10.1088/0305-4608/5/5/017|bibcode=1975JPhF....5..965E}}. [http://iopscience.iop.org/article/10.1088/0305-4608/5/5/017/meta;jsessionid=4B8D9A38523A828CD28C8CE67DD973E8.c5.iopscience.cld.iop.org ShieldSquare Captcha]
*{{citation|first1=David|last1=Sherrington|first2=Scott|last2=Kirkpatrick|journal=Physical Review Letters|title=Solvable model of a spin-glass|volume=35|pages=1792–1796|doi=10.1103/PhysRevLett.35.1792|issue=26|year=1975|bibcode=1975PhRvL..35.1792S}}. [https://archive.today/20130415143828/http://papercore.org/Sherrington1975 Papercore Summary http://papercore.org/Sherrington1975]
*{{citation|first1=P.|last1=Nordblad|first2=L.|last2=Lundgren|first3=L.|last3=Sandlund|title=A link between the relaxation of the zero field cooled and the thermoremanent magnetizations in spin glasses|journal=Journal of Magnetism and Magnetic Materials|volume=54|pages=185–186|year=1986|doi=10.1016/0304-8853(86)90543-3|bibcode = 1986JMMM...54..185N }}.
*{{citation|author1-link=Kurt Binder|first1=K.|last1=Binder|first2=A. P.|last2=Young|title=Spin glasses: Experimental facts, theoretical concepts, and open questions|journal=Reviews of Modern Physics|volume=58|issue=4|pages=801–976|year=1986|doi=10.1103/RevModPhys.58.801|bibcode=1986RvMP...58..801B}}.
*{{citation|last1=Bryngelson|first1=Joseph D.|first2=Peter G.|last2=Wolynes|title=Spin glasses and the statistical mechanics of protein folding|journal=[[Proceedings of the National Academy of Sciences]]|volume=84|issue=21|pages=7524–7528|year=1987|doi=10.1073/pnas.84.21.7524|pmid=3478708|bibcode = 1987PNAS...84.7524B |pmc=299331|doi-access=free}}....
*{{citation| first=G.| last=Parisi|title=The order parameter for spin glasses: a function on the interval 0-1|journal=J. Phys. A: Math. Gen.| volume= 13| issue=3| pages=1101–1112| year=1980| doi=10.1088/0305-4470/13/3/042|bibcode = 1980JPhA...13.1101P |url=https://www.openaccessrepository.it/record/19057/files/LNF_79_038%28P%29.pdf}} [https://archive.today/20130415190815/http://papercore.org/Parisi1980 Papercore Summary http://papercore.org/Parisi1980].
*{{citation|author-link=Michel Talagrand|first=Michel|last=Talagrand|journal=Annals of Probability|volume=28|pages=1018–1062|year=2000|jstor=2652978|title=Replica symmetry breaking and exponential inequalities for the Sherrington–Kirkpatrick model|issue=3|doi=10.1214/aop/1019160325|doi-access=free}}.
*{{citation|first1=F.|last1=Guerra|first2=F. L.|last2=Toninelli|title=The thermodynamic limit in mean field spin glass models|journal=Communications in Mathematical Physics|volume=230|issue=1|pages=71–79|year=2002|doi=10.1007/s00220-002-0699-y|arxiv = cond-mat/0204280 |bibcode = 2002CMaPh.230...71G |s2cid=16833848}}

==External links==
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=2&index1=125728 Statistics of frequency of the term "Spin glass" in arxiv.org]
{{magnetic states}}

{{Authority control}}

{{DEFAULTSORT:Spin Glass}}
<!--Categories-->
[[Category:Magnetic ordering]]
[[Category:Mathematical physics]]