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," Tf. 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. A spin glass should not be confused with a "spin-on glass". The latter is a thin film, usually based on SiO2, 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 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 antiferromagnetic 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 frustrated interactions – 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. On the mathematical side, simple statistical mechanics models, inspired by real spin glasses, are widely studied and applied. Spin glasses and the complex internal structures that arise within them are termed "metastable" because they are "stuck" in stable configurations other than the lowest-energy configuration (which would be aligned and ferromagnetic). The mathematical complexity of these structures is difficult but fruitful to study experimentally or in simulations; with applications to physics, chemistry, materials science and artificial neural networks in computer science.
Magnetic behavior
The key methods of determining whether a material is a spin glass are rooted in its magnetic behavior. Because of the underlying disordered structure of the magnetic moments, a spin glass is expected to exhibit a unique combination of features. A list of necessary criteria is proposed by Mydosh in a review paper:
The ac susceptibility χ a c = χ ′ + i χ ″ = ∂ M ∂ B a c {\displaystyle \chi _{\mathrm {ac} }=\chi '+i\chi ''={\frac {\partial M}{\partial B_{ac}}}} is the response of the spin glass to an alternating magnetic field. It consists of an (reactive) in-phase component χ ′ {\displaystyle \chi '} and an (absorptive) out-of-phase component χ ″ {\displaystyle \chi ''} . The expectation for a spin glass is that χ ″ ( T ) {\displaystyle \chi ''(T)} peaks sharply at the freezing temperature T f {\displaystyle T_{f}} . 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 χ ″ {\displaystyle \chi ''} . The temperature dependent magnetization M ( T ) {\displaystyle M(T)} in the frozen state must depend on the magnetic history: A pronounced splitting between the magnetization in heating must occur where a constant M {\displaystyle M} is expected up to T f {\displaystyle T_{f}} 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 c m {\displaystyle c_{m}} shows a broad, field-dependent maximum at T f {\displaystyle T_{f}} , 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 S ( t ) = ( 1 / H ) ∂ M ∂ ln t {\displaystyle S(t)=(1/H){\frac {\partial M}{\partial \ln t}}} . Above the spin glass transition temperature, Tc, 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 d {\displaystyle d} -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
H = − ∑ ⟨ i j ⟩ J i j S i S j , {\displaystyle H=-\sum _{\langle ij\rangle }J_{ij}S_{i}S_{j},}
where S i {\displaystyle S_{i}} refers to the Pauli spin matrix for the spin-half particle at lattice point i {\displaystyle i} and the sum over ⟨ i j ⟩ {\displaystyle \langle ij\rangle } includes all nearest-neighbor lattice points i {\displaystyle i} and j {\displaystyle j} . The variables J i j {\displaystyle J_{ij}} characterize the magnetic interactions between neighboring spins and are called bond or link variables. The interaction is antiferromagnetic for negative J i j {\displaystyle J_{ij}} and ferromagnetic for positive J i j {\displaystyle J_{ij}} . In order to determine the partition function for this system, one needs to average the free energy f [ J i j ] = − 1 β ln Z [ J i j ] {\displaystyle f\left[J_{ij}\right]=-{\frac {1}{\beta }}\ln {\mathcal {Z}}\left[J_{ij}\right]} where Z [ J i j ] = Tr S ( e − β H ) {\displaystyle {\mathcal {Z}}\left[J_{ij}\right]=\operatorname {Tr} _{S}\left(e^{-\beta H}\right)} , over all possible values of J i j {\displaystyle J_{ij}} . The distribution of values of J i j {\displaystyle J_{ij}} is taken to be a Gaussian with a mean J 0 {\displaystyle J_{0}} and a variance J 2 {\displaystyle J^{2}} :
P ( J i j ) = N 2 π J 2 exp { − N 2 J 2 ( J i j − J 0 N ) 2 } . {\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\}.}
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 exp ( − 2 β J ) = ( 1 − p ) / p {\displaystyle \exp(-2\beta J)=(1-p)/p} . 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 m = 0 {\displaystyle m=0} along with a non-vanishing value of the two point correlation function between spins at the same lattice point but at two different replicas:
q = ∑ i = 1 N S i α S i β ≠ 0 , {\displaystyle q=\sum _{i=1}^{N}S_{i}^{\alpha }S_{i}^{\beta }\neq 0,}
where α , β {\displaystyle \alpha ,\beta } are replica indices. The order parameter for the ferromagnetic to spin glass phase transition is therefore q {\displaystyle q} , and that for paramagnetic to spin glass is again q {\displaystyle q} . Hence the new set of order parameters describing the three magnetic phases consists of both m {\displaystyle m} and q {\displaystyle q} . Under the assumption of replica symmetry, the mean-field free energy is given by the expression:
β f =
− β 2 J 2 4 ( 1 − q ) 2 + β J 0 m 2 2 − ∫ exp ( − z 2 2 ) log ( 2 cosh ( β J z + β J 0 m ) ) d z . {\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}}}
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
H = − 1 N ∑ i < j J i j S i S j , {\displaystyle H=-{\frac {1}{\sqrt {N}}}\sum _{i<j}J_{ij}S_{i}S_{j},}
where J i j , S i , S j {\displaystyle J_{ij},S_{i},S_{j}} 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
When there is a uniform external magnetic field of magnitude M {\displaystyle M} , the energy function becomes H = − 1 N ∑ i < j J i j S i S j − M ∑ i S i {\displaystyle H=-{\frac {1}{\sqrt {N}}}\sum _{i<j}J_{ij}S_{i}S_{j}-M\sum _{i}S_{i}} Let all couplings J i j {\displaystyle J_{ij}} are IID samples from the gaussian distribution of mean 0 and variance J 2 {\displaystyle J^{2}} . In 1979, J.R.L. de Almeida and David Thouless 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 x := k T J , y := M J {\displaystyle x:={\frac {kT}{J}},\quad y:={\frac {M}{J}}} . Its phase diagram has two parts, divided by the de Almeida-Thouless curve, The curve is the solution set to the equations x 2 = 1 ( 2 π ) 1 / 2 ∫ d z e − 1 2 z 2 sech 4 ( q 1 / 2 z + y x ) , q = 1 ( 2 π ) 1 / 2 ∫ d z e − 1 2 z 2 tanh 2 ( q 1 / 2 z + y x ) . {\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}}} The phase transition occurs at x = 1 {\displaystyle x=1} . Just below it, we have y 2 ≈ 4 3 ( 1 − x ) 3 . {\displaystyle y^{2}\approx {\frac {4}{3}}(1-x)^{3}.} At low temperature, high magnetic field limit, the line is x ≈ 4 3 2 π e − 1 2 y 2 {\displaystyle x\approx {\frac {4}{3{\sqrt {2\pi }}}}e^{-{\frac {1}{2}}y^{2}}}
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 p {\displaystyle p} -spin interactions, where p ≤ N {\displaystyle p\leq N} and N {\displaystyle N} 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:
H = − ∑ i 1 < i 2 < ⋯ < i p J i 1 … i p S i 1 ⋯ S i p {\displaystyle H=-\sum _{i_{1}<i_{2}<\cdots <i_{p}}J_{i_{1}\dots i_{p}}S_{i_{1}}\cdots S_{i_{p}}}
where J i 1 … i p , S i 1 , … , S i p {\displaystyle J_{i_{1}\dots i_{p}},S_{i_{1}},\dots ,S_{i_{p}}} have similar meanings as in the EA model. The p → ∞ {\displaystyle p\to \infty } 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 J 0 N {\displaystyle {\frac {J_{0}}{N}}} and variance J 2 N {\displaystyle {\frac {J^{2}}{N}}} , is given as:
P ( J i 1 ⋯ i p ) = N p − 1 J 2 π p ! exp { − N p − 1 J 2 p ! ( J i 1 ⋯ i p − J 0 p ! 2 N p − 1 ) } {\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\}}
The order parameters for this system are given by the magnetization m {\displaystyle m} and the two point spin correlation between spins at the same site q {\displaystyle q} , 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 m {\displaystyle m} and q {\displaystyle q} , under the assumption of replica symmetry as well as 1-Replica Symmetry Breaking.
β f =
1 4 β 2 J 2 q p − 1 2 p β 2 J 2 q p − 1 4 β 2 J 2 + 1 2 β J 0 p m p + 1 4 2 π p β 2 J 2 q p − 1 +
∫ exp ( − 1 2 z 2 ) log ( 2 cosh ( β J z 1 2 p q p − 1 + 1 2 β J 0 p m p − 1 ) ) d z {\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}}}
Non-ergodic behavior and applications A thermodynamic system is ergodic when, given any (equilibrium) instance of the system, it eventually vis