T-3

Goal: use the replica method to compute the quenched free energy of a prototypical mean-field model of glasses, the spherical $p$-spin model.
Techniques: replica trick, Gaussian integrals, saddle point calculations.

Quenched vs annealed, and the replica trick

In Problems 1 and 2, we defined the quenched free energy as the physically relevant quantity controlling the scaling of the typical value of the partition function $Z$, which means:

$$f=-\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{\beta N}\mathbb{𝔼}[\log Z].$$

The annealed free energy $f_{\text{a}}$ instead controls the scaling of the average value of $Z$, and it is defined by $$f_{\text{a}}=-\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{\beta N}\log \mathbb{𝔼}[Z].$$ These formulas differ by the order in which the logarithm and the average over disorder are taken.

For the REM, because of the simplicity of the model we could compute the quenched free-energy by computing the typical value of the number of configurations with a given energy. This can not be done in general, and one has to resort to the computation of the average of the logarithm, which is a hard problem. To address it, one can use a smart representation of the logarithm, that goes under the name of replica trick: $$\log x=\underset{n\to 0}{\lim }\frac{x^n-1}{n},x>0$$ 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 $$f=-\underset{N\to \mathrm{\infty }}{\lim }\underset{n\to 0}{\lim }\frac{1}{\beta N}\frac{\mathbb{𝔼}[Z^n]-1}{n}.$$ Therefore, to compute the quenched free-energy we need to compute the moments $\mathbb{𝔼}[Z^n]$ and then take the limit $n\to 0$. The calculations in the following Problems rely on this replica trick. The annealed one only requires to do the calculation with $n=1$.

The spherical p-spin model

We discuss the Replica formalism for a mean-field model that is slightly more complicated than the REM: the spherical $p$-spin model. In the spherical $p$-spin model the configurations $\overrightarrow{\sigma }=({\sigma }_1,\cdots ,{\sigma }_N)$ satisfy the spherical constraint $\sum _{i=1}^N{\sigma }_i^2=N$, and the energy associated to each configuration is $$E(\overrightarrow{\sigma })=\sum _{1\le i_1<i_2<\cdots i_p\le N}{J}_{i_1{i}_2\cdots i_p}{\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_p},$$ where the coupling constants $J_{i_1{i}_2\cdots i_p}$ are independent random variables with Gaussian distribution with zero mean and variance $p!/N^{p-1},$ and $p\ge 3$ is an integer.

Problems

Problem 3: the replica trick and the quenched free energy

In this Problem, we set up the replica calculation of the quenched free energy. This will be done in 3 steps, that are general in each calculation of this type.

1. Step 1: average over the disorder. By using the same Gaussian integration discussed above, show that the $n$-th moment of the partition function is $$\mathbb{𝔼}[Z^n]=\underset{S_N}{\int }\prod _{a=1}^{n}d{\overrightarrow{\sigma }}^a\text{exp}\left(\frac{{\beta }^2}{2}\sum _{a,b=1}^n{[q({\overrightarrow{\sigma }}^a,{\overrightarrow{\sigma }}^b)]}^p\right)$$ Justify why averaging over the disorder induces a coupling between the replicas.

2. Step 2: identify the order parameter. Using the identity $1=\int d{q}_{ab}\delta (q({\overrightarrow{\sigma }}^a,{\overrightarrow{\sigma }}^b)-q_{ab})$, show that $\mathbb{𝔼}[Z^n]$ can be rewritten as an integral over $n(n-1)/2$ variables only, as: $$\mathbb{𝔼}[Z^n]=\int \prod _{a<b}d{q}_{ab}{e}^{\frac{N{\beta }^2}{2}\sum _{a,b=1}^n{q}_{ab}^p+\frac{Nn}{2}\log (2\pi e)+\frac{N}{2}\log \det Q+o(N)}\equiv \int \prod _{a<b}d{q}_{ab}{e}^{N{𝒜}[Q]+o(N)},Q_{ab}\equiv \{\begin{array}{ll}& q_{ab}\text{ if }a<b\\ & 1\text{ if }a=b\\ & q_{ba}\text{ if }a>b\end{array}$$ In the derivation, you can use the fact that $\underset{S_N}{\int }\prod _{a=1}^{n}d{\overrightarrow{\sigma }}^a\prod _{a<b}\delta (q({\overrightarrow{\sigma }}^a,{\overrightarrow{\sigma }}^b)-q_{ab})=e^{NS[Q]+o(N)}$, where $S[Q]=n\log (2\pi e)/2+(1/2)\log \det Q$. The matrix $Q$ is an order parameter: explain the analogy with the magnetisation in the mean-field solution of the Ising model.

3. Step 3: the saddle point (RS). For large N, the integral can be computed with a saddle point approximation for general $n$. The saddle point variables are the matrix elements $q_{ab}$ with $a\ne b$. Show that the saddle point equations read $$\frac{\partial {𝒜}[Q]}{\partial q_{ab}}={\beta }^{2}p{q}_{ab}^{p-1}+{\left(Q^{-1}\right)}_{ab}{|}_{Q=Q^{*}}=0\text{for }a<b,$$ where we called $Q^{*}=(Q_{ab}^{*})$ the value at which the saddle point is attained. To solve these equations and get the free energy, one first needs to make an assumption on the structure of the matrix $Q$, i.e., on the space where to look for the saddle point solutions. We discuss this in the next set of problems.

The physics within the replica formalism: a first hint

In Problem 2, we have introduced the overlap distribution, that for the spherical $p$-spin model takes the form:

$$P_{N,\beta }(q)=\underset{{{𝒮}}_N}{\int }d\overrightarrow{\sigma }\underset{{{𝒮}}_N}{\int }d\overrightarrow{\tau }\frac{e^{-\beta E(\overrightarrow{\sigma })}}{Z}\frac{e^{-\beta E(\overrightarrow{\tau })}}{Z}\delta (q-q(\overrightarrow{\sigma },\overrightarrow{\tau }))$$ This is the distribution of overlaps between configurations extracted at equilibrium, in a fixed given realisation of disorder. On the other hand, the problems above show that replicas play exactly this game: they are different configurations of the system, extracted with a Boltzmann weight at a given realisation of the random energy landscape. They serve as probes for capturing correlations between equilibrium configurations. The saddle point solution $q_{ab}^{*}$ capture precisely the information on the average distribution of overlaps in the system. More precisely, one can show that $$\mathbb{𝔼}[P_{\beta }(q)]=\underset{N\to \mathrm{\infty }}{\lim }\mathbb{𝔼}[P_{N,\beta }(q)]=\underset{n\to 0}{\lim }\frac{2}{n(n-1)}\sum _{a>b}\delta (q-q_{ab}^{*}),$$ The solution of the saddle point equations for the overlap matrix $Q$ thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution $\mathbb{𝔼}[P_{\beta }(q)]$. Moreover, the Edwards-Anderson order parameter introduced by Alberto in Lecture 1 can also be read out from the formalism: $$q_{EA}=\max \left\{q_{a\ne b}^{*}\right\}.$$

Check out: key concepts and exercises

Annealed vs quenched averages, replica trick, fully-connected models, order parameters, the three steps of a replica calculation.
To practice on the calculations, you can do Exercise 9

To know more

• Castellani, Cavagna. Spin-Glass Theory for Pedestrians [1]