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Goal: Spin glass transition. From the experiments with the anomaly on the magnetic susceptibility to order parameter of the transition. We will discuss the arguments linked to extreme value statistics

Spin glass Transition

Spin glass behavior was first observed in experiments with non-magnetic metals (such as Cu, Fe, Au, etc.) doped with a small percentage of magnetic impurities, typically Mn. At low doping levels, the magnetic moments of Mn atoms interact via the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. This interaction has a random sign due to the random spatial distribution of Mn atoms within the non-magnetic metal. A freezing temperature, $T_f$, separates the high-temperature paramagnetic phase from the low-temperature spin glass phase:

• Above $T_f$: The magnetic susceptibility follows the standard Curie law, $\chi (T)\sim 1/T$.
• Below $T_f$: Strong metastability emerges, leading to differences between the field-cooled (FC) and zero-field-cooled (ZFC) protocols:

(i) In the ZFC protocol, the susceptibility decreases with decreasing temperature, $T$.

(ii)In the FC protocol, the susceptibility freezes at $T_f$, remaining constant at ${\chi }_{FC}(T<T_f)=\chi (T_f)$.

Understanding whether these data reveal a true thermodynamic transition and determining the nature of this new "glassy" phase remains an open challenge to this day. However, in the early 1980s, spin glass models were successfully solved within the mean-field approximation. In this limit, it is possible to determine the phase diagram and demonstrate the existence of a glassy phase where the entropy vanishes at a finite temperature. Furthermore, a condensation of the Gibbs measure onto a few configurations is observed.

Edwards Anderson model

The first significant theoretical attempt to describe spin glasses is the Edwards-Anderson model. For simplicity, we will consider the Ising version of this model. Ising spins take two values, ${\sigma }_i=\pm 1$, and are located on a lattice with $N$ sites, indexed by $i=1,2,\dots ,N$. The energy of the system is expressed as a sum over nearest neighbors $⟨i,j⟩$:

$E=-\sum _{⟨i,j⟩}{J}_{ij}{\sigma }_i{\sigma }_j.$

Edwards and Anderson proposed studying this model with couplings $J_{ij}$ that are independent and identically distributed (i.i.d.) random variables with a zero mean. The coupling distribution is denoted by $\pi (J)$, and the average over the couplings, referred to as the disorder average, is indicated by an overline:

$\bar{J}\equiv \int dJJ\pi (J)=0.$

We will consider two specific coupling distributions:

• Gaussian couplings: $\pi (J)=\exp (-J^2/2)/\sqrt{2\pi }$.
• Coin-toss couplings: $J=\pm 1$, chosen with equal probability $1/2$.

Edwards Anderson order parameter

Since $\bar{J}=0$, the model does not exhibit spatial magnetic order, such as ferromagnetic or antiferromagnetic order. Instead, the idea is to distinguish between two phases:

• Paramagnetic phase: Configurations are explored with all possible spin orientations.
• Spin glass phase: Spin orientations are random but frozen (i.e., immobile).

The glass phase is characterized by long-range correlations in time, despite the absence of long-range correlations in space. The order parameter for this phase is:

$q_{EA}=\underset{t\to \mathrm{\infty }}{\lim }\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\sum _i{\sigma }_i(0){\sigma }_i(t),$

where

$q_{EA}$

measures the overlap of the spin configuration with itself after a long time.

In the paramagnetic phase, $q_{EA}=0$, while in the spin glass phase, $q_{EA}>0$.

This raises the question of whether the transition at $T_f$ is truly thermodynamic in nature. Indeed, in the definition of the Edwards-Anderson (EA) parameter, time seems to play a role, and the magnetic susceptibility does not diverge at the freezing temperature $T_f$. The divergence of the magnetic susceptibility in ferromagnets is due to the fact that the magnetization $M=\sum _i{\sigma }_i$ serves as the order parameter, distinguishing the ordered and disordered phases. However, in the spin glass model, magnetization is zero in both phases and the order parameter is $q_{EA}$. It can be shown that the associated susceptibility corresponds to the nonlinear susceptibility:

$\frac{M}{H}=\chi +a_3{H}^2+a_5{H}^4+\dots$

where

$\chi$

is the linear susceptibility, and

$a_3,a_5,\dots$

are higher-order coefficients. Experiments have demonstrated that

$a_3$

and

$a_5$

exhibit singular behavior, providing experimental evidence for the existence of a thermodynamic transition at

$T_f$

.

References

• Spin glass i-vii, P.W. Anderson, Physics Today, 1988
• Spin glasses: Experimental signatures and salient outcome, E. Vincent and V. Dupuis, Frustrated Materials and Ferroic Glasses 31 (2018).
• Theory of spin glasses, S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5 965 (1975).
• Non-linear susceptibility in spin glasses and disordered systems, H. Bouchiat, Journal of Physics: Condensed Matter, 9, 1811 (1997).
• Solvable Model of a Spin-Glass, D. Sherrington and S. Kirkpatrick, Physical Review Letters, 35, 1792 (1975).
• Random-Energy Model: An Exactly Solvable Model of Disordered Systems, B.Derrida,Physical Review B, 24, 2613 (1980).
• Extreme value statistics of correlated random variables: a pedagogical review, S. N. Majumdar, A. Pal, and G. Schehr, Physics Reports 840, 1-32, (2020).