T-2

Goal: In this set of problems, we use the replica method to study the equilibrium properties of a prototypical mean-field model of glasses, 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 $p!/(2N^{p-1}),$ and $p\geq 3$ is an integer. We set up the calculation in these two problems, and conclude it next week.

Key concepts: annealed vs quenched averages, fully-connected models, order parameters, the three steps of a replica calculation.

Quenched vs annealed, and the replica method

In Problems 1, 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 annealed one only requires to do the calculation with $n=1$.

Problem 2.1: the annealed free energy

1. Energy correlations. At variance with the REM, in the spherical $p$-spin the energies at different configurations are correlated. Show that ${\overline {E({\vec {\sigma }})E({\vec {\tau }})}}={\frac {N}{2}}q({\vec {\sigma }},{\vec {\tau }})^{p}+o(1),\quad \quad {\text{where}}\quad \quad q({\vec {\sigma }},{\vec {\tau }})={\frac {1}{N}}\sum _{i=1}^{N}\sigma _{i}\tau _{i}$ 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?

2. 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 $\sigma ^{2}$, then ${\overline {e^{\alpha X}}}=e^{\frac {\alpha ^{2}\sigma ^{2}}{2}}$.

3. Entropy contribution. The volume of a sphere $S_{N}$ 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: $f_{\text{a}}=-{\frac {1}{\beta }}\left({\frac {\beta ^{2}}{4}}+{\frac {1}{2}}\log(2\pi e)\right).$ This 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 2.2: the replica trick and the quenched free energy

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 ${\overline {Z^{n}}}=\int _{S_{N}}\prod _{a=1}^{n}d{\vec {\sigma }}^{a}e^{{\frac {\beta ^{2}}{4}}\sum _{a,b=1}^{n}\left[q({{\vec {\sigma }}^{a},{\vec {\sigma }}^{b}})\right]^{p}}$

   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.

References

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