Disordered version of the Ising model, and corresponding magnetic materials showing disordered phases.

Spin glass models

A short course on mean field spin glasses – solvable model of a spin-glass – Long-Distance Behaviour of Correlation Functions in Disordered Systems – Scale Invariance and Self-averaging in disordered systems

Edwards-Anderson Hamiltonian

Frustration

Any real lattice in two or more dimensions will have a complicated network of interpenetrating frustrated loops. How does one then find the ground state? Which couplings should be chosen to be unsatisfied? Or are there possiblymanyground states not connected by any simple symmetry transformation? _{this is an open question.}

An important point is that not any disorder appears to be relevant for the thermodynamic properties of the system. It is the frustrations, which are that relevant part of the disorder, which essentially changes the behavior of the system in comparison with the corresponding one without disorder. In other words, if the disorder does not produce frustrations, then it is not relevant for the basic properties of the ground state of the system.

Spin glass phase transition

Spin glass energy landscape

Mean field theory of spin glasses: the Sherington-Kirkpatrick Hamiltonian

Related models and systems

• Ising model
• Artificial neural network

Mechanisms underlying spin glass behaviour

Direct moment-moment coupling is too weak to account for the observed behaviour.

In a metal such as copper, the outermost atomic electrons leave the individual copper atoms and more or less freely roam through the metal (thus becoming conduction electrons). So, in an alloy like copper manganese, it might be suspected that these conduction electrons are playing some role. And that suspicion is correct.

Electron spins have two properties that are crucial to their mediation role:

• The first is that electrons carry their own intrinsic magnetic moments. This means that the magnetic moments of the conduction electrons can interact with those of the localized moments on the manganese atoms, for example, through mutual spin flips as an electron passes by the manganese moment.
• The second is that, as quantum mechanical objects, electrons travel through metals as waves, meaning that, like all waves, electrons can exhibit diffraction and interference. As conduction electrons zip past and interact with the localized moment, one gets concentric spheres, centered on the localized manganese moment, of conduction electron spins polarized parallel and antiparallel to the localized moment. These bands are known as Ruderman-Kittel-Kasuya-Yosida (RKKY) oscillations

Spin glass materials

• Dilute magnetic alloy
• Insulator spin glasses. Like:
  • europium strontium sulfide (Eu_x Sr_1−x S), magnetic impurity Europium is substituted randomly for nonmagnetic Strontium, with the fraction x of europium
  • lithium holmium yttrium fluoride (LiHo_0.167 Y_0.833 F₄ ), in which holmium is the magnetic ion.

Static features of spin glasses

Four properties constitute the most prominent static features of materials we have come to call Spin glasses.

• a cusp in the magnetic susceptibility,
• a rounded maximum but no discontinuities in the specific heat,
• spin freezing below temperature $T_f$ , and
• an absence of spatial long-range order

Dynamics of spin glasses

I.e. non-equilibrium properties.

• "Remanence" behaviour
• Memory effects.

https://en.wikipedia.org/wiki/Spin_glass

http://www.birs.ca/events/2014/5-day-workshops/14w5082/videos

Courses - F. Guerra “Equilibrium and off equilibrium properties of ferromagnetic...”

Statistical mechanics of spin glasses and neural networks 8\3\16 no sound :(

Spin Glasses and Complexity

Spin Glasses and Computational Complexity. Which types of interactions give us computationally hard problems and spin glasses? I will survey what is known as we close in on finding the simplest complex spin systems.

Random Matrices and complexity of Spin Glasses, used for the analysis of some deep learning networks, see Deep learning theory (Loss surface of multilayer networks paper).