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

Edwards Anderson order parameter

Since 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 \bar{ J}=0 } , the model does not display spatial magnetic order, such as ferro/antiferro order. The idea is to distinguish:

• a paramagnet that explores configurations with all possible orientations
• a glass where the orientation are random, but frozen (i.e.immobile).

The glass phase is then characterized by long range correlation in time without any long range correlation in space. The order parameters 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 q_{EA}= \lim_{t\to \infty} \lim_{N\to \infty} \frac{1}{N}\sum_{i} \sigma_i(0) \sigma_i(t) }

In the paramagnetic phase $q_{EA}=0$, in the glass 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 q_{EA}>0 } . One can wonder is this transition is thermodynamic. For example, the magnetic susceptibility does not diverge at the freezing temperature, but the magnetization is not the order parameter 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= \sum_i \sigma_i =0 } . Here the order parameter is $q_{EA}$ and it can be proved that its susceptibility is the non-linear susceptibility.

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{M}{H}= \chi + a_3 H^2 +a_5 H^4+ \ldots }

Experiments showed that 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_3, a_5} are indeed singular. This means that agreement the existence of a thermodynamic transition at $T_{f}$ is an experimental fact.

The SK model

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

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= - \sum_{i,j} \frac{J_{ij}}{ \sqrt{N}} \sigma_i \sigma_j }

At the inverse 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 \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 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_\alpha } is the energy associated to the configuration 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 \alpha } . This model presents a thermodynamic transition.

Random energy model

The solution of the SK is difficult. To make progress we first study the radnom energy model (REM) introduced by B. Derrida. This model neglects the correlations between the $M=2^{N}$ configurations and assumes 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 E_{\alpha} } as iid variables.

• Show that the energy distribution 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 p(E_\alpha) =\frac{1}{\sqrt{2 \pi \sigma_M^2}}e^{-\frac{E_{\alpha}^2}{2 \sigma_M^2}}}

and determine $\sigma _{M}^{2}=N=\log M/\log 2$

We provide different solutions of the Random Energy Model (REM). The first one focus on the statistics of the smallest energies among the ones associated to 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 M=2^N} configurations. For this, we need to become familiar with the main results of extreme value statistic of iid variables.

Extreme value statistics

Consider 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 M=2^N} energies: $E_{1},...,E_{M}$ as iid variables, drawn from 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)} (Gaussian for the REM). It is useful to introduce the probability to find an energy smaller than E:

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 dx p(x) }

.

Hence the probability to find an energy larger than E is $P^{>}(E)=\int _{E}^{+\infty }dxp(x)=1-P^{<}(E)$. We denote

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}=\min(E_1,...,E_M)}

Our goal is to compute the cumulative 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 Q_M(\epsilon)\equiv\text{Prob}(E_{\min}> \epsilon)} for large M. To achieve this we need three key relations:

• first relation (exact):

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

• Second relation (estimation): typical value of the minimum, namely 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}} } :

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_{\min}^{\text{typ}}) = \frac1 M }

.

• Third relation (approximation) valid 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 M\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 Q_M(\epsilon) = e^{M\log\left(1-P^<(\epsilon)\right)} \sim \exp\left(-M P^<(\epsilon)\right) }

Gaussian case and beyond

The asymptotic tail ofFailed 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)} 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 P^<(E)=\int_{-\infty}^E dx p(x) \sim \frac{\sigma}{\sqrt{2 \pi}|E|}e^{-\frac{E^2}{2 \sigma_M^2}} \; } 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}

Hence, the typical value of the minimum is

$E_{\min }^{\text{typ}}=-\sigma {\sqrt {2\log M}}+{\frac {1}{2}}{\sqrt {\log(\log M)}}+O(1)$

Let us to be more general and consider tails

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^{-\frac{|E|^\gamma}{2 \sigma_M^2}} \; . }

In the spirit of the central limit theorem we are looking 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, b_M} are M-dependent while 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 of order one drawa from the M-independent 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)} . Shows that

• at the leading order 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= -\left[2 \sigma^2_M \log M\right]^{1/\gamma}}
• 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= \frac{2 \sigma^2_M}{\gamma |a_M|^{\gamma-1}} }
• 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) = e^{z} e^{-e^{z}}} which is the Gumbel distribution

Ground state fluctuations

Depending on 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)} we have a different dependence of M of both 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, b_M } . It is convenient to emphasize the N dependence we define

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 \equiv 1/y_N \propto N^{\omega} }

Note that the typical fluctuations of 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 \sim 1/y_N} . In particular 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 \overline{ \left(E_{\min} - \overline {E_{\min}}\right)^2 }\propto N^{2\omega}}

We will see that three different scenarios occur 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 } .

Density of states above the minimum

For a given disorder realization, we compute 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) } , the number of configurations above the minimum with an energy 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} . 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}[d(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} }

Taking the average, we get 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}[d(x) = k] } . We use the following identity

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}{d A} \sum_{k=0}^{M-1} \binom{M-1}{k} (A-B)^k B^{M-1-k}= (M-1)(A-B)A^{M-2} }

we arrive to the final 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 dE \; p(E) \left[P^>(E) - P^>(E+x) \right] P^>(E)^{M-2} }

Replace 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=a_M + b_M z } and 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 \overline{n(x)} = \left(e^{y_N x}-1\right) \int_{-\infty}^{\infty} dz e^{2 z -e^{z} } = \left(e^{y_N x}-1\right)\quad \text{with} \; y_N \sim N^{-\omega} }

The Glass phase

In the Glass phase the measure is concentrated in few configurations which has a finite occupation probability while in the paramagnetic phase the occupation probability 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 \sim 1/M } . As a consequence the entropy is extensive in the paramagnetic phase and sub-extensive in the glass phase. It is useful to evaluate the occupation probability of the ground state in the infinite system:

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{z_{\alpha_{\min}}}{\sum_{\alpha=1}^M z_\alpha}= \frac{1}{1+\sum_{\alpha\ne \alpha_{\min}} z_\alpha}\sim \frac{1}{1+\int_0^\infty dx\, e^{-\beta x} \left(e^{y_N x}-1\right) } }

• In the high temperature phase, for $\beta <y_{N}$, the occupation probability is close to zero, meaning that the ground state is not deep enough to make the system glassy
• In the low temperature phase, 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 \beta>y_N } , the above integral is finite. Hence, 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 \beta=1/T, T_f=1/y_N} you can find

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{z_{\alpha_{\min}}}{\sum_{\alpha=1}^M z_\alpha}= \frac{1}{1+ \frac{T^2}{T_f-T} } }

This means that below the freezing temperature, the ground state is occupied with a finite probability as in Bose-Einstein Condensation.

Take home message

Let us recall 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 y_N \sim N^{-\omega}} , so that three situations can occur

• 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} , there is no freezing transition as there are too many states just above the minimum. This is the situation of many low-dimensional systems such as the Edwards Anderson model is 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} there are two important features: (i) there is only the glass phase, (ii) the system condensate only in the ground state because the excited states have very high energy. We will see that in real systems (i) is not always the case and that the 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} can change with temperature. This situation can be realistic (there is a very deep groud sate), but it is not revolutionary as the following one.
• 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} there is for sure a freezing transition. For the Random Energy Model defined above 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}} One important feature of this transition that we will see in the next tutorial is that the condensation does not occur only in the ground state but in a large (yet not extensive) number of low energy exctitations.

References

• "Spin glasses: Experimental signatures and salient outcomes", 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
• "Spin glass i-vii" P.W. Anderson, Physics Today, 1988