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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 $i=1,2,\ldots ,N$. The energy of the system is expressed as a sum over nearest neighbors $\langle i,j\rangle$:

$E=-\sum _{\langle i,j\rangle }J_{ij}\sigma _{i}\sigma _{j}.$

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:

${\overline {J}}\equiv \int dJ\,J\,\pi (J)=0.$

We will consider two specific coupling distributions:

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

Edwards Anderson order parameter

Since ${\overline {J}}=0$, 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:

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

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:

${\frac {M}{H}}=\chi +a_{3}H^{2}+a_{5}H^{4}+\ldots$

where $\chi$ is the linear susceptibility, and $a_{3},a_{5},\ldots$ 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 $T_{f}$.

The SK model

Sherrington and Kirkpatrik considered the fully connected version of the model with Gaussian couplings:

$E=-\sum _{i,j}{\frac {J_{ij}}{\sqrt {N}}}\sigma _{i}\sigma _{j}$

At the inverse temperature $\beta$, the partion function of the model is

$Z=\sum _{\alpha =1}^{2^{N}}z_{\alpha },\quad {\text{with}}\;z_{\alpha }=e^{-\beta E_{\alpha }}$

Here $E_{\alpha }$ is the energy associated to the configuration $\alpha$. This model presents a thermodynamic transition.

Random energy model

The solution of the Sherrington-Kirkpatrick (SK) model is challenging. To make progress, we first study the Random Energy Model (REM), introduced by B. Derrida. This model simplifies the problem by neglecting correlations between the $M=2^{N}$ configurations and assuming that the energies $E_{\alpha }$ are independent and identically distributed (i.i.d.) random variables. Here, "independent" means that the energy of one configuration does not influence the energy of another, e.g., a configuration identical to the previous one except for a spin flip. "Identically distributed" indicates that all configurations follow the same probability distribution.

Energy Distribution: Show that the energy distribution is given by:

$p(E_{\alpha })={\frac {1}{\sqrt {2\pi \sigma _{M}^{2}}}}\exp \left(-{\frac {E_{\alpha }^{2}}{2\sigma _{M}^{2}}}\right)$

and determine that:

$\sigma _{M}^{2}=N={\frac {\log M}{\log 2}}$

.

In the following, we present the original solution of the model. Here, we characterize the glassy phase by analyzing the statistical properties of the smallest energy values among the $M=2^{N}$ configurations. To address this, it is necessary to make a brief detour into the theory of extreme value statistics for i.i.d. random variables.

Detour: Extreme Value Statistics

Consider the $M$ energies $E_{1},\dots ,E_{M}$ as independent and identically distributed (i.i.d.) random variables drawn from a distribution $p(E)$. It is useful to introduce the cumulative probability of finding an energy smaller than E:

$P^{<}(E)=\int _{-\infty }^{E}dx\,p(x)$

The complementary probability of finding an energy larger than E is:

$P^{>}(E)=\int _{E}^{+\infty }dx\,p(x)=1-P^{<}(E)$

We define:

$E_{\min }=\min(E_{1},\dots ,E_{M})$

Our goal is to compute the cumulative distribution:

$Q_{M}(E)\equiv {\text{Prob}}(E_{\min }>E)$

for large $M$. To achieve this, we rely on three key relations:

• First relation:

$Q_{M}(E)=\left(P^{>}(E)\right)^{M}$

This relation is exact but depends on $M$ and the precise form of $p(E)$. However, in the large $M$ limit, a universal behavior emerges.

• Second relation: The typical value of the minimum energy, $E_{\min }^{\text{typ}}$, satisfies:

$P^{<}(E_{\min }^{\text{typ}})={\frac {1}{M}}$

This is an estimation of the typical value of the minimum. It is a crucial relation that will be used frequently in this context.

• Third relation: For $M\to \infty$, we have:

$Q_{M}(E)=e^{M\log(1-P^{<}(E))}\sim \exp \left(-MP^{<}(E)\right)$

This is an approximation valid around the typical value of the minimum energy.

A Concrete Example: The Gaussian Case

To understand how a universal scaling form emerges, let us analyze in detail the case of a Gaussian distribution with zero mean and variance $\sigma ^{2}$. Using integration by parts, we can write :

$P^{<}(E)=\int _{-\infty }^{E}{\frac {dx}{\sqrt {2\pi \sigma ^{2}}}}\,e^{-{\frac {x^{2}}{2\sigma ^{2}}}}={\frac {1}{2{\sqrt {\pi }}}}\int _{\frac {E^{2}}{2\sigma ^{2}}}^{\infty }{\frac {dt}{\sqrt {t}}}e^{-t}={\frac {\sigma }{{\sqrt {2\pi }}|E|}}e^{-{\frac {E^{2}}{2\sigma ^{2}}}}-{\frac {1}{4{\sqrt {\pi }}}}\int _{\frac {E^{2}}{2\sigma ^{2}}}^{\infty }{\frac {dt}{t}}e^{-t}$

Hence we derive the following asymptotic expansion for $E\to -\infty$ :

$P^{<}(E)\approx {\frac {\sigma }{{\sqrt {2\pi }}|E|}}e^{-{\frac {E^{2}}{2\sigma ^{2}}}}+O({\frac {e^{-{\frac {E^{2}}{2\sigma ^{2}}}}}{|E|^{2}}})$

It is convenient to introduce the function $A(E)$ defined as

$P^{<}(E)=\exp(A(E))\quad A_{\text{gauss}}(E)=-{\frac {E^{2}}{2\sigma ^{2}}}-\log({\frac {{\sqrt {2\pi }}|E|}{\sigma }})+\ldots$

Using this expansion and the second relation introduced earlier, show that for large $M$, the typical value of the minimum energy is:

$E_{\min }^{\text{typ}}=-\sigma {\sqrt {2\log M}}\left(1-{\frac {1}{4}}{\frac {\log(4\pi \log M)}{\log M}}+\ldots \right)$

The Scaling Form in the Large M Limit

In the spirit of the central limit theorem, we look for a scaling form:

$E_{\min }=a_{M}+b_{M}z$

The constants $a_{M}$ and $b_{M}$ absorb the dependence on $M$, while the random variable $z$ is distributed according to a probability distribution $P(z)$ that does not depend on $M$.

In the Gaussian case, we start from the third relation introduced earlier and expand $A(E)$ around $a_{M}$:

$Q_{M}(E)\sim \exp \left(-MP^{<}(E)\right)=\exp \left(-Me^{A(E)}\right)\sim \exp \left(-Me^{A(a_{M})+A'(a_{M})(E-a_{M})+\ldots }\right)$

By setting

$a_{M}=E_{\min }^{\text{typ}}=-\sigma {\sqrt {2\log M}}+\ldots \quad {\text{and}}\quad b_{M}={\frac {1}{A'(a_{M})}}={\frac {\sigma }{\sqrt {2\log M}}}$

we have

$\exp(A(a_{M}))={\frac {1}{M}}\quad {\text{and}}\quad Q_{M}(E)\equiv {\text{Prob}}(E_{\min }>E)\sim \exp \left(-\exp({\frac {E-a_{M}}{b_{M}}})\right)$

Therefore, the variable $z=(E-a_{M})/b_{M}$ is distributed according an M independent distribution. It is possible to generalize the result and classify the scaling forms into three distinct universality classes:

• Gumbel Distribution:
  • Characteristics:
    • Applies when the tails of $p(E)$ decay faster than any power law.
    • Example: the Gaussian case discussed here or exponential distributions $p(E)=\exp(E)\quad {\text{with}}\quad E\in (-\infty ,0)$.
  • Scaling Form:

$P(z)=\exp(z)\exp(-e^{z})$

• Weibull Distribution:
  • Characteristics:
    • Applies to distributions with finite lower bounds $E_{0}$.
    • Example: Uniform distribution in $(E_{0},E_{1})$ or $p(E)=\exp(-(E-E_{0}))\quad {\text{with}}\quad E\in (E_{0},\infty )$.
  • Scaling Form:

$P(z)={\begin{cases}kz^{k-1}\exp(-z^{k}),&z\geq 0,\\0,&z<0.\end{cases}}$

here $a_{M}=E_{0}$ and $k$ controls the behavior of the distribution close to $E_{0}$ : $P(E)\sim (E-E_{0})^{k}$.

• Fréchet Distribution:
  • Characteristics:
    • Applies when the tails of $p(E)$ exhibit a power-law decay $\sim E^{-\alpha }$ .
    • Example: Pareto or Lévy distributions.
  • Scaling Form:

$P(z)={\begin{cases}\alpha |z|^{-(\alpha +1)}\exp(-|z|^{-\alpha }),&z\leq 0,\\0,&z>0.\end{cases}}$

These three classes, known as the Gumbel, Weibull, and Fréchet distributions, represent the universality of extreme value statistics and cover all possible asymptotic behaviors of $p(E)$.

Density above the minimum

Definition of $n(x)$:

Given a realization, $n(x)$ is defined as the number of random variables above the minimum $E_{\min }$ such that their value is smaller than $E_{\min }+x$. This quantity is a random variable, and we are interested in its average value:

${\overline {n(x)}}=\sum _{k}k\,{\text{Prob}}[n(x)=k]$

The key relation for this quantity is:

${\text{Prob}}[n(x)=k]=M{\binom {M-1}{k}}\int _{-\infty }^{\infty }dE\;p(E)[P^{>}(E)-P^{>}(E+x)]^{k}P^{>}(E+x)^{M-k-1}$

We use the following identity to sum over $k$:

$\sum _{k=0}^{M-1}k{\binom {M-1}{k}}(A-B)^{k}B^{M-1-k}=(A-B){\frac {d}{dA}}\sum _{k=0}^{M-1}{\binom {M-1}{k}}(A-B)^{k}B^{M-1-k}=(M-1)(A-B)A^{M-2}$

to arrive at the form:

${\overline {n(x)}}=M(M-1)\int _{-\infty }^{\infty }dE\;p(E)\left[P^{>}(E)-P^{>}(E+x)\right]P^{>}(E)^{M-2}$

which simplifies further to:

${\overline {n(x)}}=M\int _{-\infty }^{\infty }dE\;\left[P^{>}(E)-P^{>}(E+x)\right]{\frac {dP^{>}(E)^{M-1}}{dE}}=M\int _{-\infty }^{\infty }dE\;\left[P^{<}(E+x)-P^{<}(E)\right]{\frac {dQ_{M-1}(E)}{dE}}$

Using asymptotic forms: So far, no approximations have been made. To proceed, we use $Q_{M-1}(E)\approx Q_{M}(E)$ and its asymptotics:

${\frac {dQ_{M-1}(E)}{dE}}\;dE\sim \exp({\frac {E-a_{M}}{b_{M}}})\exp(-\exp({\frac {E-a_{M}}{b_{M}}})){\frac {dE}{b_{M}}}=P(z)dz$

where $z=(E-a_{M})/b_{M}$. The contribution to the integral comes then form the region near $a_{M}$ where $P^{<}(E)\sim e^{A(a_{M})+A'(a_{M})(E-a_{M})}$. We can then arrive to:

${\overline {n(x)}}=\left(e^{x/b_{M}}-1\right)\int _{-\infty }^{\infty }dz\;e^{2z-e^{z}}=\left(e^{x/b_{M}}-1\right)$

Back to the REM

Returning now to the Random Energy Model (REM), recall that for each realization of disorder, we obtain $M=2^{N}$ Gaussian random variables with zero mean and variance $\sigma _{M}^{2}={\frac {\log M}{\log 2}}=N$. The minimum energy is a random variable belonging to the Gumbel universality class. From the results for $a_{M}$ and $b_{M}$ derived in the previous section, we have:

$E_{\min }=a_{M}+b_{M}z=-{\sqrt {2\log 2}}\;N+{\frac {1}{2}}{\frac {\log(4\pi \log 2N)}{\sqrt {2\log 2}}}+{\frac {z}{\sqrt {2\log 2}}}$

where $z$ is a random variable distributed according to the Gumbel distribution.

• Key Observations:

The leading term of the non-stochastic part, $-{\sqrt {2\log 2}}\;N$, is extensive, scaling linearly with $N$.

The fluctuations, represented by the term $z/{\sqrt {2\log 2}}$, are independent of $N$.

Phase Transition in the Random Energy Model

The Random Energy Model (REM) exhibits two distinct phases:

• High-Temperature Phase:

At high temperatures, the system is in a paramagnetic phase where the entropy is extensive, and the occupation probability of a configuration is approximately $\sim 1/M$.

• Low-Temperature Phase:

Below a critical freezing temperature $T_{f}$, the system transitions into a glassy phase. In this phase, the entropy becomes subextensive (i.e., the extensive contribution vanishes), and only a few configurations are visited with finite, $M$-independent probabilities.

Calculating the Freezing Temperature $T_{f}$

Thanks to the computation of ${\overline {n(x)}}$, we can identify the fingerprints of the glassy phase and calculate $T_{f}$. Let's compare the weight of the ground state against the weight of all other states:

${\frac {\sum _{\alpha }z_{\alpha }}{z_{\alpha _{\min }}}}=1+\sum _{\alpha \neq \alpha _{\min }}{\frac {z_{\alpha }}{z_{\alpha _{\min }}}}=1+\sum _{\alpha \neq \alpha _{\min }}e^{-\beta (E_{\alpha }-E_{\min })}\sim 1+\int _{0}^{\infty }dx\,{\frac {d{\overline {n(x)}}}{dx}}\,e^{-\beta x}=1+\int _{0}^{\infty }dx\,{\frac {e^{x/b_{M}}}{b_{M}}}\,e^{-\beta x}$

Behavior in Different Phases:

• High-Temperature Phase ($T>T_{f}=b_{M}=1/{\sqrt {2\log 2}}$):

In this regime, the weight of the excited states diverges. This indicates that the ground state is not deep enough to render the system glassy.

• Low-Temperature Phase ($T<T_{f}=b_{M}=1/{\sqrt {2\log 2}}$):

In this regime, the integral is finite:

$\int _{0}^{\infty }dx\,e^{(1/b_{M}-\beta )x}/b_{M}={\frac {1}{\beta b_{M}-1}}={\frac {T}{T_{f}-T}}$

This result implies that below the freezing temperature $T_{f}$, the weight of all excited states is of the same order as the weight of the ground state. Consequently, the ground state is occupied with a finite probability, reminiscent of Bose-Einstein condensation.

More general REM and systems in Finite dimensions

In random energy models with i.i.d. random variables, the distribution $p(E)$ determines the dependence of $a_{M}$ and $b_{M}$ on M, and consequently their scaling with N, the number of degrees of freedom. It is insightful to consider a more general case where an exponent $\omega$ describes the fluctuations of the ground state energy:

${\overline {\left(E_{\min }-{\overline {E_{\min }}}\right)^{2}}}\sim b_{M}^{2}\propto N^{2\omega }$

Three distinct scenarios emerge depending on the sign of $\omega$:

• For $\omega <0$: The freezing temperature vanishes with increasing system size, leading to the absence of a freezing transition. This scenario occurs in many low-dimensional systems, such as the Edwards-Anderson model in two dimensions.

• For $\omega =0$: A freezing transition is guaranteed. For the Random Energy Model discussed earlier, $T_{f}=1/{\sqrt {2\log 2}}$. An important feature of this transition, as will be explored in the next section, is that condensation does not occur solely in the ground state but also involves a large (albeit not extensive) number of low-energy excitations.

• For $\omega >0$: The freezing temperature grows with the system size, resulting in only the glassy phase. The system condenses entirely into the ground state since the excited states are characterized by prohibitively high energies. This scenario, while less intricate than the $\omega =0$ case, corresponds to a glassy phase with a single deep ground state.

The extent to which these scenarios persist in real systems in finite dimensions remains an open question. In these systems, the fluctuations of the ground state energy are characterized by an exponent $\theta$:

${\overline {\left(E_{\min }-{\overline {E_{\min }}}\right)^{2}}}\sim L^{2\theta }$

where $L$ is the linear size of the system and $N=L^{D}$ is the number of degrees of freedom.

At finite temperatures, an analogous exponent can be defined by studying the fluctuations of the free energy, $F=E-TS$. We will explore systems where the fluctuations of the ground state exhibit a positive and temperature-independent $\theta$. In such cases, only the glassy phase exists, aligning with the $\omega >0$ scenario in REMs.

On the other hand, in some systems, $\theta$ is positive at low temperatures but vanishes at high temperatures. This behavior defines a glass transition mechanism entirely distinct from those observed in mean-field models.

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).