T-2

Problem 1.2: freezing transition & glassiness

In this Problem we compute the equilibrium phase diagram of the model, and in particular the quenched free energy density $f$ which controls the scaling of the typical value of the partition function, $Z\sim e^{-N\beta \,f+o(N)}$. We show that the free energy equals to

$f={\begin{cases}&-\left(T\log 2+{\frac {1}{2T}}\right)\quad {\text{if}}\quad T\geq T_{f}\\&-{\sqrt {2\,\log 2}}\quad {\text{if}}\quad T<T_{f}\end{cases}}\quad \quad T_{f}={\frac {1}{\sqrt {2\,\log 2}}}.$

At $T_{f}$ a transition occurs, often called freezing transition: in the whole low-temperature phase, the free-energy is “frozen” at the value that it has at the critical temperature $T=T_{f}$.
We also characterize the overlap distribution of the model: the overlap between two configurations ${\vec {\sigma }}^{\alpha },{\vec {\sigma }}^{\beta }$ is $q_{\alpha \,\beta }=N^{-1}\sum _{i=1}^{N}\sigma _{i}^{\alpha }\,\sigma _{i}^{\beta }$, and its distribution is

$P(q)=\sum _{\alpha ,\beta }{\frac {e^{-\beta E({\vec {\sigma }}^{\alpha })}}{Z}}{\frac {e^{-\beta E({\vec {\sigma }}^{\beta })}}{Z}}\delta (q-q_{\alpha \beta }).$

1. The freezing transition. The partition function the REM reads $Z=\sum _{\alpha =1}^{2^{N}}e^{-\beta E_{\alpha }}=\int dE\,{\mathcal {N}}(E)e^{-\beta E}.$ Using the behaviour of the typical value of ${\mathcal {N}}$ determined in Problem 1.1, derive the free energy of the model (hint: perform a saddle point calculation). What is the order of this thermodynamic transition?

2. Fluctuations, and back to average vs typical. Similarly to what we did for the entropy, one can define an annealed free energy $f_{\text{a}}$ from ${\overline {Z}}=e^{-N\beta f_{\text{a}}+o(N)}$: show that in the whole low-temperature phase this is smaller than the quenched free energy obtained above. Putting all the results together, justify why the average of the partition function in the low-T phase is "dominated by rare events".

3. Entropy catastrophe. What happens to the entropy of the model when the critical temperature is reached, and in the low temperature phase? What does this imply for the partition function $Z$?

4. Overlap distribution and glassiness. Justify why in the REM the overlap typically can take only values zero and one, leading to

   $P(q)=I_{2}\,\delta (q-1)+(1-I_{2})\,\delta (q),\quad \quad I_{2}={\frac {\sum _{\alpha }z_{\alpha }^{2}}{\left(\sum _{\alpha }z_{\alpha }\right)^{2}}},\quad \quad z_{\alpha }=e^{-\beta E_{\alpha }}$

   Why $I_{2}$ can be interpreted as a probability? Show that

   ${\overline {I_{2}}}={\begin{cases}0\quad &{\text{if}}\quad T>T_{f}\\1-{\frac {T}{T_{f}}}\quad &{\text{if}}\quad T\leq T_{f}\end{cases}}$

   and interpret the result. In particular, why is this consistent with the entropy catastrophe?
   

   Hint: For $T<T_{f}$, use that ${\overline {I_{2}}}={\frac {T_{f}}{T}}{\frac {d}{d\mu }}\log \int _{0}^{\infty }(1-e^{-u-\mu u^{2}})u^{-{\frac {T}{T_{f}}}-1}du{\Big |}_{\mu =0}$

Comment: the low-T phase of the REM is a frozen phase, characterized by the fact that the free energy is temperature independent, and that the typical value of the partition function is very different from the average value. In fact, the low-T phase is also a glass phase with $q_{EA}=1$. This is also a phase where a peculiar symmetry, the so called replica symmetry, is broken. We go back to this concepts in the next sets of problems.

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 $Z$, 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 $n=1$.

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 $p!/(2N^{p-1}),$ and $p\geq 3$ 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 $p$-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 ${\overline {Z}}$ 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 ${\overline {e^{\alpha X}}}=e^{\frac {\alpha ^{2}v}{2}}$.


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.

To know more

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