T-2

In this set of problems, we use the replica method to study the equilibrium properties of a prototypical toy model of glasses, the spherical $p$-spin model.

Problem 1: the energy landscape of the REM

In this exercise we study the number ${\mathcal {N}}(E)dE$ of configurations having energy $E_{\alpha }\in [E,E+dE]$. This quantity is a random variable. For large $N$, we will show that its typical value scales as

${\mathcal {N}}(E)=e^{N\Sigma \left({\frac {E}{N}}\right)+o(N)},\quad \quad \Sigma (\epsilon )={\begin{cases}\log 2-\epsilon ^{2}\quad &{\text{ if }}|\epsilon |\leq {\sqrt {\log 2}}\\0\quad &{\text{ if }}|\epsilon |>{\sqrt {\log 2}}\end{cases}}$

The function $\Sigma (\epsilon )$ is the entropy of the model, and it is sketched in Fig. X. The point where the entropy vanishes, $\epsilon =-{\sqrt {\log 2}}$, is the energy density of the ground state, consistently with what we obtained in the lecture. 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 $\Sigma ^{A}$, which is the function that controls the behaviour of the average number of configurations at a given energy, ${\overline {{\mathcal {N}}(E)}}={\text{exp}}\left(N\Sigma ^{A}\left({\frac {E}{N}}\right)+o(N)\right)$. 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], together with the distribution $p(E)$ of the energies of the REM configurations. When does $\Sigma ^{A}$ coincide with the entropy defined above (which we define as the “quenched” entropy in the following)?

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 that this is the case by computing the second moment ${\overline {{\mathcal {N}}^{2}}}$ and using the central limit theorem. Show that this 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.

Comment: this analysis of the landscape suggests that in the large $N$ limit, the fluctuations due to the randomness become relevant when one looks at the bottom of their energy landscape, close to the ground state energy density. We show below that this intuition is correct, and corresponds to the fact that the partition function $Z$ has an interesting behaviour at low temperature.

Problem 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, 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 $f^{A}$ from ${\overline {Z}}=e^{-N\beta f^{A}+o(N)}$: 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 $aglassphase$ in the sense discussed in the lecture. It is characterized by the fact that Replica Symmetry is broken, as one sees explicitly by re-deriving the free energy through the replica method. We go back to this in the next lectures/TDs.

Problem 3: freezing as a localization/condensation transition

In this final exercise, we show how the freezing transition can be understood in terms of extreme valued statistics (discussed in the lecture) and localization. We consider the energies of the configurations and define $E_{\alpha }=-N{\sqrt {\log 2}}+\delta E_{\alpha }$, so that

${Z}=e^{\beta N{\sqrt {\log 2}}}\sum _{\alpha =1}^{2^{N}}e^{-\beta \delta E_{\alpha }}=e^{\beta N{\sqrt {\log 2}}}\sum _{\alpha =1}^{2^{N}}z_{\alpha }$

We show that $Z$ is a sum of random variables that become heavy tailed for $T<T_{c}$, implying that the central limit theorem is violated and this sum is dominated by few terms, the largest ones. This can be interpreted as the occurrence of localization.

1. Heavy tails and concentration. Compute the distribution of the variables $\delta E_{\alpha }$ and show that for $(\delta E)^{2}/N\ll 1$ this is an exponential. Using this, compute the distribution of the $z_{\alpha }$ and show that it is a power law, $p(z)={\frac {c}{z^{1+\mu }}}\quad \quad \mu ={\frac {2{\sqrt {\log 2}}}{\beta }}$ When $T<T_{c}$, one has $\mu <1$: the distribution of $z$ becomes heavy tailed. What does this imply for the sum $Z$? Is this consistent with the behaviour of the partition function and of the entropy discussed in Problem 2? Why can one talk about a localization or condensation transition?

2. Inverse participation ratio. The low temperature behaviour of the partition function an be characterized in terms of a standard measure of localization (or condensation), the Inverse Participation Ratio (IPR) defined as: $IPR={\frac {\sum _{\alpha =1}^{2^{N}}z_{\alpha }^{2}}{[\sum _{\alpha =1}^{2^{N}}z_{\alpha }]^{2}}}=\sum _{\alpha =1}^{2^{N}}\omega _{\alpha }^{2}\quad \quad \omega _{\alpha }={\frac {z_{\alpha }}{\sum _{\alpha =1}^{2^{N}}z_{\alpha }}}.$

   When $z$ is power law distributed with exponent $\mu$, one can show (HOMEWORK!) that

   $IPR={\frac {\Gamma (2-\mu )}{\Gamma (\mu )\Gamma (1-\mu )}}.$

   Discuss how this quantity changes across the transition at $\mu =1$, and how this fits with what you expect in general in a localized phase.