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. We set up the calculation in these two problems, and conclude it next week.
Techniques: replica trick, Gaussian integrals, saddle point calculations.

Quenched vs annealed, and the replica trick

In Problems 1, we defined the quenched free energy as the quantity controlling the scaling of the typical value of the partition function 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 Z } , which means:

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= -\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 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 Z } . It is defined 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 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 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^n= e^{n \log x}= 1+ n\log x+ O(n^2) } . Applying this to the average of the partition function, we see 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 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 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 \to 0} . The calculation of these TDs relies on this replica trick. The annealed one only requires to do the calculation 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 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 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} -spin model the configurations 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 \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

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 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 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 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 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 \geq 3} is an integer.

Problem 2.1: the annealed free energy of the spherical p-spin

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


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


3. 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 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 \sqrt{N}} in dimension $N$ 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}{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

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 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_{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 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 ${\overline {Z^{n}}}$ can be rewritten as an integral over 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(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 $\int _{S_{N}}\prod _{a=1}^{n}d{\vec {\sigma }}^{a}\,\prod _{a<b}\delta \left(q({{\vec {\sigma }}^{a},{\vec {\sigma }}^{b}})-q_{ab}\right)=e^{NS[Q]+o(N)}$, where 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 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 $n$. The saddle point variables are the matrix elements 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_{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 ${\frac {\partial {\mathcal {A}}[Q]}{\partial q_{ab}}}={\frac {\beta ^{2}}{4}}pq_{ab}^{p-1}+{\frac {1}{2}}\left(Q^{-1}\right)_{ab}=0\quad \quad {\text{for }}\quad a\neq b$

   To solve these equations and get the free energy, one first needs to make an assumption on the structure of the matrix 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} , 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]