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.

Quenched vs annealed, and the replica trick

• Spherical p-spin model. In the spherical $p$-spin model the configurations ${\vec {S}}=(S_{1},\cdots ,S_{N})$ satisfy the spherical constraint $\sum _{i=1}^{N}S_{i}^{2}=N$, and the energy associated to each configuration is $E({\vec {S}})=\sum _{1\leq i_{1}\leq i_{2}\leq \cdots i_{p}\leq N}J_{i_{1}\,i_{2}\cdots i_{p}}S_{i_{1}}S_{i_{2}}\cdots S_{i_{p}},$

  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.



• 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 $Z$, 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 calculation of these TDs relies on this replica trick. The annealed one only requires to do the calculation with $n=1$.
  
  

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

  
  

  1. Energy correlations. At variance with the REM, in the spherical $p$-spin the energies at different configurations are correlated. Show that ${\overline {E({\vec {S}})E({\vec {S}}')}}={\frac {N}{2}}q({\vec {S}},{\vec {S}}')^{p}+O(1),\quad \quad {\text{where}}\quad \quad q({\vec {S}},{\vec {S}}')={\frac {1}{N}}\sum _{i=1}^{N}S_{i}S'_{i}$ 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 ${\overline {e^{-\beta J_{i_{1}\,\cdots i_{p}}S_{i_{1}}\cdots S_{i_{p}}}}}$, 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 ${\mathcal {S}}_{N}$ 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: $f_{\text{a}}=-{\frac {1}{\beta }}\left({\frac {\beta ^{2}}{4}}+{\frac {1}{2}}\log(2\pi e)\right).$ 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 ${\overline {Z^{n}}}=\int _{S_{N}}\prod _{a=1}^{n}d{\vec {\sigma }}^{a}e^{{\frac {\beta ^{2}}{4}}\sum _{a,b=1}^{n}\left[q({{\vec {\sigma }}^{a},{\vec {\sigma }}^{b}})\right]^{p}}$

     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.

  
  

  Check out: key concepts

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

  References

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