L-1

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 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} seprates the high-temperature paramagnetic phase from the low temeprature spin glass phase:

• 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} the susceptibility obeys to the standard Curie law 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 \chi(T) \sim 1/T} .
• Below 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} , 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 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} . In FC, the susceptibility freezes at 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} : 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 \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 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=\pm 1} and live on a lattice 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 } sites 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 i=1,2,\ldots,N } . The enregy is written as a sum between the nearest neighbours <i,j>:

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> } J_{ij} \sigma_i \sigma_j }

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

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{J} \equiv \int d J \, J \, \pi(J)=0 }

We will discuss two 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, 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 J= \pm 1 } , selected with 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 } .

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}}{2 \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

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= \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 at 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_c=?? } .

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 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 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^2}}e^{-\frac{E_{\alpha}^2}{2 \sigma^2}}}

and determine 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}

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: 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_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, but we remain general in this section). It is useful to use the following notations:

• 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) } , it is 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_E^{+\infty} dx p(x) = 1- P^<(E) } , it is the probability to find an energy larger than 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 two key relations:

• The first relation is exact:

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

• The second relation identifies the 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 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 P^<(a_M) = \frac1 M }

.

Let us consider the limit, 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 \lim_{M\to \infty} (1-\frac{k}{M})^M =\exp(-k)} , which allow to re-write the first relation:

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) \sim \exp\left(-M P^<(\epsilon)\right)}

This relation holds only when 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 \epsilon \approx a_M } and one hase to expand around this value. However, a direct Taylor expansion does not ensures that probabilities remain positive. Hence, 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 P^<(\epsilon)=\exp(-A(\epsilon)) } and remark 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(\epsilon) } is a decreasing function. We propose the following Taylor expansion

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(\epsilon) =a_M + A'(a_M)(\epsilon -a_M) = a_M - y_N(\epsilon -a_M) }

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 on N or 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, y_N } . It is convenient to 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 y_N \propto N^{-\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 } . Using this expansion we derive:

• The famous Gumbel 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) \sim \exp\left(-e^{ y_N (\epsilon-a_M)}\right) }

• 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:

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

Density of states above the minimum

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

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

Taking the average ${\overline {d(x)}}=\sum _{k}k{\text{Prob}}(d(x)=k)$, we derive

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

In the above integral, $E$ is the energy of the minimum. Hence, we can use the Taylor expansion $A(E)=a_{M}-y_{N}(E-a_{M})$. In particular we can write

• $p(E)={\frac {d}{dE}}P^{<}(E)=-A'(E)e^{-A(E)}\sim y_{N}e^{y_{N}(E-a_{M})}/M$
• $P^{>}(E)-P^{>}(E+x)=e^{-A(E+x)}-e^{-A(E)}\sim e^{y_{N}(E-a_{M})}\left(e^{y_{N}x}-1\right)/M$
• $P^{>}(E)^{M-2}=Q_{M-2}(E)\sim \exp \left(-e^{y_{N}(E-a_{M})}\right)$

Calling $u=y_{N}(E-a_{M})$ we obtain

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

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.

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. 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 (but not extensive) number of low energy exctitations.

Exercise L1-A: the Gaussian case

Specify these results to the Guassian case and find $P^{<}(E)=\int _{-\infty }^{E}dxp(x)\sim {\frac {\sigma }{{\sqrt {2\pi }}|E|}}e^{-{\frac {E^{2}}{2\sigma ^{2}}}}\;$ for $x\to -\infty$

• the typical value of the minimum

%

$a_{M}=\sigma {\sqrt {2\log M}}-{\frac {1}{2}}{\sqrt {\log(\log M)}}+O(1)$

• The expression $A(\epsilon )={\frac {\epsilon ^{2}}{2\sigma ^{2}}}-{\frac {\sqrt {2\pi }}{\sigma }}\log |\epsilon |+\ldots$
• The expression of the Gumbel distribution for the Gaussian case

$Q_{M}(\epsilon )\sim \exp \left(-e^{-{\frac {\sqrt {2\log M}}{\sigma }}(\epsilon -a_{M})}\right)$

Bibliography

• Theory of spin glasses, S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5 965, 1975