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: getting acquainted with the model

In the spherical $p$-spin model the configurations ${\vec {\sigma }}=(\sigma _{1},\cdots ,\sigma _{N})$ that the system can take satisfy the spherical constraint $\sum _{i=1}^{N}\sigma _{i}^{2}=N$, and the energy associated to each configuration is

$E({\vec {\sigma }})=\sum _{1\leq i_{1}\leq i_{2}\leq \cdots i_{p}\leq N}J_{i_{1}\,i_{2}\cdots i_{p}}\sigma _{i_{1}}\sigma _{i_{2}}\cdots \sigma _{i_{p}},$

where the coupling constants $J_{i_{1}\,i_{2}\cdots i_{p}}$ are independent random variables with Gaussian distribution with zero mean and variance $p!/(2N^{p-1}),$ and $p\geq 3$ is an integer.

1. Energy correlations. At variance with the REM, in the spherical $p$-spin the energies at different configurations are correlated. Show that ${\overline {E({\vec {\sigma }})E({\vec {\tau }})}}=Nq({\vec {\sigma }},{\vec {\tau }})^{p}/2+o(1)$, where $q({\vec {\sigma }},{\vec {\tau }})={\frac {1}{N}}\sum _{i=1}^{N}\sigma _{i}\tau _{i}$ is the overlap between the two configurations. Why an we say that for $p\to \infty$ this model converges with the REM discussed in the previous TD?

Problem 2: the annealed free energy

In TD1, we defined the quenched free energy density as the quantity controlling the scaling of the typical value of the partition function $Z$. The annealed free energy $f_{\rm {ann}}$ instead controls the scaling of the average value of $Z$. It is defined by

$f_{\rm {ann}}=-\lim _{N\to \infty }{\frac {1}{\beta N}}\log {\overline {Z}}.$

Let us compute this quantity.

1. Show that computing the average free energy boils down to computing the average ${\overline {e^{-\beta J_{i_{1}\,\cdots i_{p}}\sigma _{i_{1}}\cdots \sigma _{i_{p}}}}}$ 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: the quenched free energy

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.