T-3

Goal: use the replica method to study the equilibrium properties 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 quantity controlling the scaling of the typical value of the partition function $Z$, which means:

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

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

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

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:

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

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

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

Therefore, to compute the quenched free-energy we need to compute the moments ${\overline {Z^{n}}}$ 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 $n=1$.

Problems

In this and the following set of problems, we analyse 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 ${\vec {\sigma }}=(\sigma _{1},\cdots ,\sigma _{N})$ satisfy the spherical constraint $\sum _{i=1}^{N}\sigma _{i}^{2}=N$, and the energy associated to each configuration is

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

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

Problem 3.1: correlations, p-spin vs REM

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

Problem 3.2: the annealed free energy. DO IT YOURSELF!

As a preliminary exercise, we compute the annealed free energy of the spherical $p$-spin model.

1. Energy contribution. Show that computing ${\overline {Z}}$ boils down to computing the average ${\overline {e^{-\beta J_{i_{1}\,\cdots i_{p}}\sigma _{i_{1}}\cdots \sigma _{i_{p}}}}}$, which is a Gaussian integral. Compute this average.
   Hint: if $X$ is a centered Gaussian variable with variance $v$, then ${\overline {e^{\alpha X}}}=e^{\frac {\alpha ^{2}v}{2}}$.


2. Entropy contribution. The volume of a sphere ${\mathcal {S}}_{N}$ 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: $f_{\text{a}}=-{\frac {1}{\beta }}\left({\frac {\beta ^{2}}{2}}+{\frac {1}{2}}\log(2\pi e)\right).$ This result is only slightly different with respect to the annealed free-energy of the REM: can you identify the source of this difference?

Problem 3.3: 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 ${\overline {Z^{n}}}=\int _{S_{N}}\prod _{a=1}^{n}d{\vec {\sigma }}^{a}e^{{\frac {\beta ^{2}}{2}}\sum _{a,b=1}^{n}\left[q({{\vec {\sigma }}^{a},{\vec {\sigma }}^{b}})\right]^{p}}$

   Justify why averaging over the disorder induces a coupling between the replicas.

2. Step 2: identify the order parameter. Using the identity $1=\int dq_{ab}\delta \left(q({{\vec {\sigma }}^{a},{\vec {\sigma }}^{b}})-q_{ab}\right)$, show that ${\overline {Z^{n}}}$ can be rewritten as an integral over $n(n-1)/2$ variables only, as: ${\overline {Z^{n}}}=\int \prod _{a<b}dq_{ab}e^{{\frac {N\beta ^{2}}{4}}\sum _{a,b=1}^{n}q_{ab}^{p}+{\frac {Nn}{2}}\log(2\pi e)+{\frac {N}{2}}\log \det Q+o(N)}\equiv \int \prod _{a<b}dq_{ab}e^{N{\mathcal {A}}[Q]+o(N)},\quad \quad Q_{ab}\equiv {\begin{cases}&q_{ab}{\text{ if }}a<b\\&1{\text{ if }}a=b\\&q_{ba}{\text{ if }}a>b\end{cases}}$

   In the derivation, you can use the fact that $\int _{S_{N}}\prod _{a=1}^{n}d{\vec {\sigma }}^{a}\,\prod _{a<b}\delta \left(q({{\vec {\sigma }}^{a},{\vec {\sigma }}^{b}})-q_{ab}\right)=e^{NS[Q]+o(N)}$, 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 $a\neq b$. Show that the saddle point equations read ${\frac {\partial {\mathcal {A}}[Q]}{\partial q_{ab}}}={\frac {\beta ^{2}}{4}}pq_{ab}^{p-1}+{\frac {1}{2}}\left(Q^{-1}\right)_{ab}=0\quad \quad {\text{for }}\quad a\neq b$

   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.

Check out: key concepts

Annealed vs quenched averages, replica trick, fully-connected models, order parameters, the three steps of a replica calculation.

To know more

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