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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

Experiments

Spin glass behviour was first reported in non-magnetic metals (Cu, Fe, Au,...) doped with a few percent of a magnetic impurities, typically Mn. At low doping, Mn magnetic moments feel the Ruderman–Kittel–Kasuya–Yosida (RKKY) inetraction which has a random sign because of the random location of Mn atoms in the non-magnetic metal. A freezing temparature $T_{f}$ seprates the high-temperature paramagnetic phase from the low temeprature spin glass phase:

• Above $T_{f}$ the susceptibility obeys to the standard Curie law $\chi (T)\sim 1/T$.
• Below $T_{f}$, a strong metastability is observed: at the origin of the difference between the field cooled (FC) and the zero field cooled (ZFC) protocols. In zero field cooled ZFC, the susceptibility decays with $T$. In FC, the susceptibility freezes at $T_{f}$: $\chi _{FC}(T<T_{f})=\chi (T_{f})$

Edwards Anderson model

The first important theoretical attempt for spin glasses in the Edwards Anderson model. We consider for simplicity the Ising version of this model.

Ising spins takes two values $\sigma =\pm 1$ and live on a lattice of $N$ sites $i=1,2,\ldots ,N$. The enregy is written as a sum between the nearest neighbours <i,j>:

$E=-\sum _{<i,j>}J_{ij}\sigma _{i}\sigma _{j}$

Edwards and Anderson proposed to study this model for couplings $J$ that are i.i.d. random variables with zero mean. We set $\pi (J)$ the coupling distribution and we indicate the average over the couplings, called disorder average, with an overline:

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

We will discuss two distributions:

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

Edwards Anderson order parameter

Since ${\bar {J}}=0$, 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

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

In the paramagnetic phase $q_{EA}=0$, in the glass phase $q_{EA}>0$. 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 $M=\sum _{i}\sigma _{i}=0$. Here the order parameter is $q_{EA}$ and it can be proved that its susceptibility is the non-linear susceptibility.

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

Experiments showed that $a_{3},a_{5}$ 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:

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

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 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 $E_{\alpha }$ as iid variables.

• Show that the energy distribution is

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

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 $M=2^{N}$ configurations. For this, we need to become familiar with the main results of extreme value statistic of iid variables.

Extreme value statistics

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

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

.

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

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

Our goal is to compute the cumulative distribution $Q_{M}(\epsilon )\equiv {\text{Prob}}(E_{\min }>\epsilon )$ 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 $E_{\min }^{\text{typ}}$:

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

.

• Third relation (approximation) valid for $M\to \infty$

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

Gaussian case and beyond

The asymptotic tail of$P^{<}(E)$ is

$P^{<}(E)=\int _{-\infty }^{E}dxp(x)\sim {\frac {\sigma }{{\sqrt {2\pi }}|E|}}e^{-{\frac {E^{2}}{2\sigma _{M}^{2}}}}\;$ for $E\to -\infty$

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

$P^{<}(E)\sim e^{-{\frac {|E|^{\gamma }}{2\sigma _{M}^{2}}}}\;.$

In the spirit of the central limit theorem we are looking for a scaling form $E_{\min }=a_{M}+b_{M}z$. The constants $a_{M},b_{M}$ are M-dependent while $z$ is a random variable of order one drawa from the M-independent distribution $P(z)$. Shows that

• at the leading order $a_{M}=-\left[2\sigma _{M}^{2}\log M\right]^{1/\gamma }$
• $b_{M}={\frac {2\sigma _{M}^{2}}{\gamma |a_{M}|^{\gamma -1}}}$
• $P(z)=e^{z}e^{-e^{z}}$ which is the Gumbel distribution

Ground state fluctuations

Depending on the distribution $p(E)$ we have a different dependence of M of both $a_{M},b_{M}$. It is convenient to emphasize the N dependence we define

$b_{M}\equiv 1/y_{N}\propto N^{\omega }$

Note that the typical fluctuations of the minimum $\sim 1/y_{N}$. In particular we can write:

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

We will see that three different scenarios occur depending on the sign of $\omega$.

Density of states above the minimum

For a given disorder realization, we compute $n(x)$, the number of configurations above the minimum with an energy smaller than $E_{\min }+x$. The key relation for this quantity is:

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

Taking the average, we get ${\overline {n(x)}}=\sum _{k}k\,{\text{Prob}}[d(x)=k]$. We use the following identity

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

we arrive to the final form

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

Replace $E=a_{M}+b_{M}z$ and obtain

${\overline {n(x)}}=\left(e^{y_{N}x}-1\right)\int _{-\infty }^{\infty }dze^{2z-e^{z}}=\left(e^{y_{N}x}-1\right)\quad {\text{with}}\;y_{N}\sim N^{-\omega }$

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 $\sim 1/M$. 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:

${\frac {z_{\alpha _{\min }}}{\sum _{\alpha =1}^{M}z_{\alpha }}}={\frac {1}{1+\sum _{\alpha \neq \alpha _{\min }}z_{\alpha }}}\sim {\frac {1}{1+\int _{0}^{\infty }dx\,e^{-\beta x}\left(e^{y_{N}x}-1\right)}}$

• 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 $\beta >y_{N}$, the above integral is finite. Hence, setting $\beta =1/T,T_{f}=1/y_{N}$ you can find

${\frac {z_{\alpha _{\min }}}{\sum _{\alpha =1}^{M}z_{\alpha }}}={\frac {1}{1+{\frac {T^{2}}{T_{f}-T}}}}$

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 $y_{N}\sim N^{-\omega }$, so that three situations can occur

• For $\omega <0$, 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 $\omega >0$ 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 $\omega$ 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 $\omega =0$ there is for sure a freezing transition. For the Random Energy Model defined above $T_{f}=1/{\sqrt {2\log 2}}$ 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