L-1

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_M(E) = e^{M \log(1 - P^<(E))} \sim \exp\left(-M P^<(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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2} . Using integration by parts, we can write :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E \to -\infty}  :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(E)} defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , the typical value of the minimum energy is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min} = a_M + b_M z}

The constants Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_M} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_M} absorb the dependence on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , while the random variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} is distributed according to a probability distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(z)} that does not depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} .

In the Gaussian case, we start from the third relation introduced earlier and expand Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(E)} around Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_M} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_M(E) \sim \exp\left(-M P^<(E)\right) = \exp\left(-M e^{A(E)}\right) \sim \exp\left(-M e^{A(a_M) +A'(a_M) (E - a_M) + \ldots}\right)}

By setting

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E)} decay faster than any power law.
    • Example: the Gaussian case discussed here or exponential distributions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E) = \exp(E) \quad \text{with} \quad E\in (-\infty, 0)} .
  • Scaling Form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(z) = \exp(z) \exp(-e^{z})}

• Weibull Distribution:
  • Characteristics:
    • Applies to distributions with finite lower bounds Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_0 } .
    • Example: Uniform distribution in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (E_0, E_1)} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E) = \exp(-(E-E_0)) \quad \text{with} \quad E\in (E_0,\infty)} .
  • Scaling Form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(z) = \begin{cases} k z^{k-1} \exp(-z^k), & z \geq 0, \\ 0, & z < 0. \end{cases}}

here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_M=E_0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k } controls the behavior of the distribution close to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_0 }  : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(E) \sim (E-E_0)^k} .

• Fréchet Distribution:
  • Characteristics:
    • Applies when the tails of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E)} exhibit a power-law decay Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim E^{-\alpha}} .
    • Example: Pareto or Lévy distributions.
  • Scaling Form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E)} .

Density above the minimum

Definition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n(x) } :

Given a realization, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n(x) } is defined as the number of random variables above the minimum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min} } such that their value is smaller than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min} +x} . This quantity is a random variable, and we are interested in its average value:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)} = \sum_k k \, \text{Prob}[n(x) = k] }

The key relation for this quantity is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)} = M \int_{-\infty}^\infty dE \; \left[P^>(E) - P^>(E+x)\right] \frac{d P^>(E)^{M-1}}{dE} = M \int_{-\infty}^\infty dE \; \left[P^<(E+x) - P^<(E)\right] \frac{d Q_{M-1}(E)}{dE} }

Using asymptotic forms: So far, no approximations have been made. To proceed, we use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{M-1}(E)\approx Q_M(E)} and its asymptotics:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d Q_{M-1}(E)}{dE} \; dE \sim \exp(\frac{E-a_M}{b_M}) \exp(-\exp(\frac{E-a_M}{b_M})) dE = P(z) dz }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=(E-a_M)/b_M } . The contribution to the integral comes then form the region near Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_M} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^<(E) \sim e^{A(a_M) +A'(a_M) (E-a_M)} } . We can then arrive to:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=2^N} Gaussian random variables with zero mean and variance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2_M = \frac{\log M}{\log 2} = N} . The minimum energy is a random variable belonging to the Gumbel universality class. From the results for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_M } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_M} derived in the previous section, we have:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min} =a_M +b_M z = - \sqrt{2 \log 2}\; N +\frac{1}{2} \frac{ \log (4 \pi \log2 N)}{\sqrt{2 \log 2}} + \frac{z}{\sqrt{2 \log 2}}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} is a random variable distributed according to the Gumbel distribution.

• Key Observations:

The leading term of the non-stochastic part, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\sqrt{2 \log 2} \; N} , is extensive, scaling linearly with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} .

The fluctuations, represented by the term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z/ \sqrt{2 \log 2} } , are independent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim 1/M} .

• Low-Temperature Phase:

Below a critical freezing temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} -independent probabilities.

Calculating the Freezing Temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f}

Thanks to the computation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)}} , we can identify the fingerprints of the glassy phase and calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} . Let's compare the weight of the ground state against the weight of all other states:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sum_\alpha z_\alpha}{z_{\alpha_{\min}}} = 1 + \sum_{\alpha \ne \alpha_{\min}} \frac{z_\alpha}{z_{\alpha_{\min}}} = 1 + \sum_{\alpha \ne \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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T > T_f= b_M = 1/\sqrt{2 \log2}} ):

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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T < T_f= b_M = 1/\sqrt{2 \log2}} ):

In this regime, the integral is finite:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E)} determines the dependence of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_M} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} describes the fluctuations of the ground state energy:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} :

• For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = 0} : A freezing transition is guaranteed. For the Random Energy Model discussed earlier, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\left(E_{\min} - \overline{E_{\min}}\right)^2} \sim L^{2\theta}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} is the linear size of the system and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = E - T S} . We will explore systems where the fluctuations of the ground state exhibit a positive and temperature-independent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} . In such cases, only the glassy phase exists, aligning with the $\omega >0$ scenario in REMs.

On the other hand, in some systems, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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).