thumb|Schematic representation of the random spin structure of a spin glass (top) and the ordered one of a ferromagnet (bottom)
In condensed matter physics, a spin glass is a magnetic state characterized by randomness, besides cooperative behavior in freezing of spins at a temperature called the "freezing temperature," T<sub>f</sub>. In ferromagnetic solids, component atoms' magnetic spins all align in the same direction. Spin glass when contrasted with a ferromagnet is defined as "disordered" magnetic state in which spins are aligned randomly or without a regular pattern and the couplings too are random. On the mathematical side, simple statistical mechanics models, inspired by real spin glasses, are widely studied and applied.
- The ac susceptibility <math>\chi_\mathrm{ac} = \chi' + i\chi = \frac{\partial M}{\partial B_{ac</math> is the response of the spin glass to an alternating magnetic field. It consists of an (reactive) in-phase component <math>\chi'</math> and an (absorptive) out-of-phase component <math>\chi</math>. The expectation for a spin glass is that <math>\chi(T)</math> peaks sharply at the freezing temperature <math>T_f</math>. It is, furthermore expected that the freezing temperature is only weakly dependent on the excitation frequency. An explanation for this is that during the freezing process, the fluctuations lead to a slowing down of the spin dynamics and thus a higher absorption <math>\chi</math>.
- The temperature dependent magnetization <math>M(T)</math> in the frozen state must depend on the magnetic history: A pronounced splitting between the magnetization in heating must occur where a constant <math>M</math> is expected up to <math>T_f</math> when the sample was cooled across the transition in a magnetic field (so called field cooled FC). This is opposed to a field-free cooling and application of the magnetic field only in the frozen state (zero-field cooling ZFC). Underlying this is that the glass is frozen into a (partially) magnetized state in a FC protocol.
- The magnetic specific heat <math>c_m</math> shows a broad, field-dependent maximum at <math>T_f</math>, whereas it is sharp at the transition to an ordered state (like a ferro- or antiferromagnet).
- Because spin glasses are based on disorder, there is a decay of the magnetized state, it ages. This means that a zero-field cooled magnetized state will decay back into an unmagnetized one. There will be a maximum in the quantity <math>S(t) = (1/H)\frac{\partial M}{\partial \ln t}</math>.
Above the spin glass transition temperature, T<sub>c</sub>, the spin glass exhibits typical magnetic behaviour (such as paramagnetism).
If a magnetic field is applied as the sample is cooled to the transition temperature, magnetization of the sample increases as described by the Curie law.
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.
Edwards–Anderson model
The Edwards-Anderson model is similar to the Ising model, in which spins are arranged on a <math>d</math>-dimensional lattice with only nearest neighbor interactions. Critical temperatures can be solved for exactly and a glassy phase is observed to exist at low temperatures. The 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> includes all nearest-neighbor lattice points <math>i</math> and <math>j</math>. The variables <math>J_{ij}</math> characterize the magnetic interactions between neighboring spins and are called bond or link variables. The interaction is antiferromagnetic for negative <math>J_{ij}</math> and ferromagnetic for positive <math>J_{ij}</math>.
In order to determine the partition function for this system, one needs to average the 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>
For the +J and -J random-bond Ising model, in which nearest-neighbor couplings take the values +J and -J with probabilities p and 1 - p, the Nishimori line is the locus in the temperature-disorder phase diagram satisfying <math>\exp(-2\beta J)=(1-p)/p</math>. On this line, gauge transformations give exact results such as the internal energy, making it one of the few exactly controlled cases of finite-dimensional spin-glass models. In information-processing applications, Nishimori used related spin-glass methods to show that, for certain error-correcting code ensembles, the decoding error can be minimized at a finite decoding temperature matched to the channel noise. Iba later discussed the Nishimori line from the viewpoint of Bayesian statistics.
Solving for the free energy using the 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:
: <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 replicas of the partition function of the system.
An important, exactly solvable model of a spin glass was introduced by 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 approximation of spin glasses describing the slow dynamics of the magnetization and the complex non-ergodic equilibrium state.
Unlike the Edwards–Anderson (EA) model, the range of each spin-spin interaction can be arbitrarily large (not restricted to neighboring sites). Any two spins can be linked with a ferromagnetic or an antiferromagnetic bond; the distribution of these bonds is the same as in the EA model. The SK Hamiltonian is
: <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 initial attempts by Sherrington, Kirkpatrick and others, was found by Giorgio Parisi in 1979 using the replica method. The subsequent work interpreting the Parisi solution—by M. Mezard, G. Parisi, M.A. Virasoro and many others—revealed the complex nature of glassy, low-temperature phases, 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.
Phase diagram
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 Thouless<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". The infinite-range model is a generalization of the 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 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 2021 Nobel Prize in Physics to Giorgio Parisi.
For physical systems, such as dilute manganese in copper, the freezing temperature is typically as low as 30 kelvins (−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 neural networks, including Hopfield networks, as well as many problems in computer science optimization and genetics.
Spin-glass without structural disorder
Elemental crystalline neodymium is paramagnetic at room temperature and becomes an antiferromagnet 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.
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 W. Anderson in Physics Today.
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 (AuFe) and manganese-in-copper alloy (CuMn) at around 1 to 10 atom percent. Cannella and Mydosh observed in 1972 that AuFe had an unexpected cusplike peak in the a.c. susceptibility at a well defined temperature, which would later be termed spin glass freezing temperature.
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. 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.
Applications
The formalism of replica mean-field theory has also been applied in the study of 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.
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 simulations. These models display spin glass phases bordered by sharp phase transitions.
Besides its relevance in condensed matter physics, spin glass theory has acquired a strongly interdisciplinary character, with applications to neural network theory, computer science, theoretical biology, econophysics etc.
Spin glass models were adapted to the folding funnel model of protein folding.
See also
- 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
Notes
References
Literature
Expositions
- Popular exposition, with a minimal amount of mathematics.
- A practical tutorial introduction.
- 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 problems.
- Introduction focused on computer science applications, including neural networks.
- Focuses on the experimentally measurable properties of spin glasses (such as copper-manganese alloy).
- Covers mean field theory, experimental data, and numerical simulations.
- . Early exposition containing the pre-1990 breakthroughs, such as the replica trick.
- Approach via statistical field theory.
- and . Compendium of rigorously provable results.
Primary sources
- . ShieldSquare Captcha
- . Papercore Summary http://papercore.org/Sherrington1975
- .
- .
- ....
- Papercore Summary http://papercore.org/Parisi1980.
- .
External links
- Statistics of frequency of the term "Spin glass" in arxiv.org
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