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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Z}} boils down to computing the average Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a centered Gaussian variable with variance $v$, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{e^{\alpha X}}=e^{\frac{\alpha^2 v}{2} }} .


2. Entropy contribution. The volume of a sphere Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S}_N} of radius ${\sqrt {N}}$ in dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1=\int dq_{ab} \delta \left( q({\vec{\sigma}^a, \vec{\sigma}^b})- q_{ab}\right) } , show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Z^n}} can be rewritten as an integral over $n(n-1)/2$ variables only, as: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Z^n}= \int \prod_{a<b} d q_{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} d q_{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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^{N S[Q]+ o(N)}} , where $S[Q]=n\log(2\pi e)/2+(1/2)\log \det Q$. The matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . The saddle point variables are the matrix elements $q_{ab}$ with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \neq b} . Show that the saddle point equations read Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathcal{A}[Q]}{\partial q_{ab}}=\frac{\beta^2}{4}p q_{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]