T-I

Goal: derive the equilibrium phase diagram of the simplest spin-glass model, the Random Energy Model (REM).
Techniques: saddle point approximation, Legendre transform, probability theory.

Problem 1.1: the energy landscape of the REM

Entropy of the Random Energy Model

The REM has been introduced in ^[1] . In the REM the system can take $2^{N}$ configurations ${\vec {S}}=(S_{1},\cdots ,S_{N})$ with $S_{i}=\pm 1$. To each configuration $\alpha =1,\cdots ,2^{N}$ is assigned a random energy $E_{\alpha }$. The random energies are independent, taken from a Gaussian distribution $p(E)=(\pi N)^{-1/2}e^{-{\frac {E^{2}}{N}}}$. In this problem we study the random variable ${\mathcal {N}}(E)dE$, that is the number of configurations having energy $E_{\alpha }\in [E,E+dE]$. We show that for large $N$ it scales as ${\mathcal {N}}(E)=e^{NS\left(E/N\right)+o(N)}$ . We show that the typical value of $S(\epsilon )$, the quenched entropy of the model (see sketch), is given by:

$S(\epsilon )={\begin{cases}\log 2-\epsilon ^{2}\quad &{\text{ if }}|\epsilon |\leq {\sqrt {\log 2}}\\0\quad &{\text{ if }}|\epsilon |>{\sqrt {\log 2}}\end{cases}}$

The point where the entropy vanishes, $\epsilon =-{\sqrt {\log 2}}$, is the energy density of the ground state. The entropy is maximal at $\epsilon =0$: the highest number of configurations have vanishing energy density.

1. Averages: the annealed entropy. We begin by computing the annealed entropy $S_{\text{a}}$, which is defined by the average ${\overline {{\mathcal {N}}(E)}}={\text{exp}}\left(NS_{\text{a}}\left(E/N\right)+o(N)\right)$. Compute this function using the representation ${\mathcal {N}}(E)dE=\sum _{\alpha =1}^{2^{N}}\chi _{\alpha }(E)dE\;$ [with $\chi _{\alpha }(E)=1$ if $E_{\alpha }\in [E,E+dE]$ and $\chi _{\alpha }(E)=0$ otherwise]. When does $S_{\text{a}}$ coincide with $S$?

2. Self-averaging. For $|\epsilon |\leq {\sqrt {\log 2}}$ the quantity ${\mathcal {N}}$ is self-averaging: its distribution concentrates around the average value ${\overline {\mathcal {N}}}$ when $N\to \infty$. Show this by showing that $\sigma ^{2}={\overline {{\mathcal {N}}^{2}}}-{\overline {\mathcal {N}}}^{2}\sim {\overline {\mathcal {N}}}$. This property of being self-averaging is no longer true in the region where the annealed entropy is negative: why does one expect fluctuations to be relevant in this region?

3. Rare events vs typical values. For $|\epsilon |>{\sqrt {\log 2}}$ the annealed entropy is negative: the average number of configurations with those energy densities is exponentially small in $N$. This implies that the probability to get configurations with those energy is exponentially small in $N$: these configurations are rare. Do you have an idea of how to show this, using the expression for ${\overline {\mathcal {N}}}$? What is the typical value of ${\mathcal {N}}$ in this region? Justify why the point where the entropy vanishes coincides with the ground state energy of the model.

Problem 1.2: the free energy and the freezing transition

We now compute the equilibrium phase diagram of the model, and in particular the quenched free energy density $f$ which controls the scaling of the typical value of the partition function, $Z\sim e^{-N\beta \,f+o(N)}$. We show that the free energy equals to

$f={\begin{cases}&-\left(T\log 2+{\frac {1}{4T}}\right)\quad {\text{if}}\quad T\geq T_{c}\\&-{\sqrt {\log 2}}\quad {\text{if}}\quad T<T_{c}\end{cases}}\quad \quad T_{c}={\frac {1}{2{\sqrt {\log 2}}}}.$

At $T_{c}$ a transition occurs, often called freezing transition: in the whole low-temperature phase, the free-energy is “frozen” at the value that it has at the critical temperature $T=T_{c}$.

1. The thermodynamical transition and the freezing. The partition function the REM reads $Z=\sum _{\alpha =1}^{2^{N}}e^{-\beta E_{\alpha }}=\int dE\,{\mathcal {N}}(E)e^{-\beta E}.$ Using the behaviour of the typical value of ${\mathcal {N}}$ determined in Problem 1.1, derive the free energy of the model (hint: perform a saddle point calculation). What is the order of this thermodynamic transition?

2. Entropy. What happens to the entropy of the model when the critical temperature is reached, and in the low temperature phase? What does this imply for the partition function $Z$?

3. Fluctuations, and back to average vs typical. Similarly to what we did for the entropy, one can define an annealed free energy $f_{\text{a}}$ from ${\overline {Z}}=e^{-N\beta f_{\text{a}}+o(N)}$: show that in the whole low-temperature phase this is smaller than the quenched free energy obtained above. Putting all the results together, justify why the average of the partition function in the low-T phase is "dominated by rare events".

Comment: the low-T phase of the REM is a frozen phase, characterized by the fact that the free energy is temperature independent, and that the typical value of the partition function is very different from the average value. In fact, the low-T phase is also a glass phase : it is a phase where a peculiar symmetry, the so called replica symmetry, is broken. We go back to this concepts in the next sets of problems.

Check out: key concepts

Self-averaging, average value vs typical value, large deviations, rare events, saddle point approximation, freezing transition.

To know more

• Derrida. Random-energy model: limit of a family of disordered models [1]