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.

The model In the spherical $p$-spin model the configurations $\overrightarrow{\sigma }=({\sigma }_1,\cdots ,{\sigma }_N)$ 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

$E(\overrightarrow{\sigma })=\sum _{1\le i_1\le i_2\le \cdots i_p\le N}{J}_{i_1{i}_2\cdots i_p}{\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_p},$

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 $p\ge 3$ is an integer.

Quenched vs annealed

In TD1, we defined the quenched free energy density as the quantity controlling the scaling of the typical value of the partition function $Z$, which means:

$f=-\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{\beta N}\overline{\log Z}.$

The annealed free energy $f_{\mathrm{a}\mathrm{n}\mathrm{n}}$ instead controls the scaling of the average value of  $Z$. It is defined by

$f_{\mathrm{a}\mathrm{n}\mathrm{n}}=-\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{\beta N}\log \bar{Z}.$

These formulas differ by the order in which the logarithm and the average over disorder are taken. Computing the average of the logarithm is in general a hard problem, which one can address by using a smart representation of the logarithm, that goes under the name of replica trick:

$\log x=\underset{n\to 0}{\lim }\frac{x^n-1}{n}$

which can be easily shown to be true by Taylor expanding $x^n=e^{n\log (x)}=1+n\log x+O(n^2)$. Applying this to the average of the partition function, we see that

$f=-\underset{N\to \mathrm{\infty }}{\lim }\underset{n\to 0}{\lim }\frac{1}{\beta Nn}\frac{\overline{Z^n}-1}{n}.$

Therefore, to compute the free-energy we need to compute the moments $\overline{Z^n}$ and then take the limit $n\to 0$.

Problem 1: the annealed free energy

Let us compute this quantity.

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

2. Energy contribution. Show that computing $\bar{Z}$ boils down to computing the average $\stackrel{‾}{e^{-\beta J_{i_1\cdots i_p}{\sigma }_{i_1}\cdots {\sigma }_{i_p}}}$. Compute this average. Hint: if X is a centered Gaussian variable with variance ${\sigma }^2$, then $\overline{e^{\alpha X}}=e^{\frac{{\alpha }^2{\sigma }^2}{2}}$.

3. Entropy contribution. The volume of a sphere of radius $\sqrt{N}$ in dimension $N$ is given by $N^{\frac{N}{2}}{\pi }^{\frac{N}{2}}/\left(\frac{N}{2}\right)!$. Use the large-N asymptotic of this to conclude the calculation of the annealed free energy: the final result is only slightly different with respect to the free-energy of the REM in the high-temperature phase: can you identify the source of this difference?

Problem 2: the replica trick and the quenched free energy

The quenched free energy density is the quantity controlling the scaling of the typical value of the partition function $Z$. This means that

This calculation can be done in 3 main steps.

1. Step 1: average over the disorder. ccc

2. Step 2: identify the order parameter. cccc

3. Step 3: the saddle point. cccc