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:

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

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

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

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

At variance with the REM, in the spherical $p$-spin the energies at different configurations are correlated. Show that

${\overline {E({\vec {\sigma }})E({\vec {\sigma }}')}}=N\,q({\vec {\sigma }},{\vec {\sigma }}')^{p}+o(N),\quad \quad {\text{where}}\quad \quad q({\vec {\sigma }},{\vec {\sigma }}')={\frac {1}{N}}\sum _{i=1}^{N}\sigma _{i}\sigma '_{i}$

is the overlap between the two configurations. Why can we say that for $p\to \infty$ this model converges with the REM?

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: ${\overline {Z^{n}}}=\int \prod _{a<b}dq_{ab}e^{{\frac {N\beta ^{2}}{2}}\sum _{a,b=1}^{n}q_{ab}^{p}+{\frac {Nn}{2}}\log(2\pi e)+{\frac {N}{2}}\log \det Q+o(N)}\equiv \int \prod _{a<b}dq_{ab}e^{N{\mathcal {A}}[Q]+o(N)},\quad \quad Q_{ab}\equiv {\begin{cases}&q_{ab}{\text{ if }}a<b\\&1{\text{ if }}a=b\\&q_{ba}{\text{ if }}a>b\end{cases}}$

   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 ${\frac {\partial {\mathcal {A}}[Q]}{\partial q_{ab}}}=\beta ^{2}\,pq_{ab}^{p-1}+\left(Q^{-1}\right)_{ab}{\Big |}_{Q=Q^{*}}=0\quad \quad {\text{for }}\quad a<b,$

   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:

$P_{N,\beta }(q)=\int _{{\mathcal {S}}_{N}}d{\vec {\sigma }}\,\int _{{\mathcal {S}}_{N}}d{\vec {\sigma }}'{\frac {e^{-\beta E({\vec {\sigma }})}}{Z}}{\frac {e^{-\beta E({\vec {\sigma }}')}}{Z}}\delta \left(q-q({\vec {\sigma }},{\vec {\sigma }}')\right)$

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,

${\overline {P_{\beta }(q)}}=\lim _{N\to \infty }{\overline {P_{N,\beta }(q)}}=\lim _{n\to 0}{\frac {2}{n(n-1)}}\sum _{a>b}\delta \left(q-q_{ab}^{*}\right),$

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 ${\overline {P_{\beta }(q)}}$. Moreover, the Edwards-Anderson order parameter introduced in Lecture 1 can also be read out from the formalism:

$q_{EA}=\max \left\{q_{a\neq b}^{*}\right\}.$

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]