T-3

Goal of these problems: In this set of problems, we compute the free energy of the spherical $p$-spin model in the Replica Symmetric (RS) framework, and in the 1-step Replica Symmetry Broken (1-RSB) framework.

Key concepts:

The order parameters: overlaps, and their meaning

• Order parameter. In the lectures, we have introduced the Edwards-Anderson order parameter

$q_{EA}=\lim _{N\to \infty }{\frac {1}{N}}\sum _{i}{\overline {\langle \sigma _{i}\rangle ^{2}}}$

which plays the same role as the magnetization in a ferromagnet. Let us recap what happens for a ferromagnet. The magnetization is defined as:

$m=\lim _{h\to 0}\lim _{N\to \infty }{\frac {1}{N}}\sum _{i=1}^{N}{\overline {\langle \sigma _{i}\rangle _{h}}}$

where $\langle \cdot \rangle _{h}$ is the Boltzmann average in presence of a small magnetic field, and the average over the disorder can be neglected because this quantity is self-averaging. This quantity is non-zero in the low-T ferromagnetic phase; it measures ergodicity breaking: when a small field $h$ is added, the system at equilibrium explores only a sub-part of the phase space which corresponds to a magnetization in the direction of the field; in a finite system, the magnetization vanishes when the field is switched off. If the infinite size limit is taken before, though, the magnetization persists.

• Pure states. When ergodicity is broken, the Boltzmann measure clusters into pure states (labelled by $\alpha$) with Gibbs weight $\omega _{\alpha }$, meaning that one can re-write the thermal averages $\langle \cdot \rangle$ of any observable $A$ as

$\langle A\rangle =\sum _{\alpha }\omega _{\alpha }\langle A\rangle _{\alpha },\quad \quad \quad \omega _{\alpha }={\frac {Z_{\alpha }}{Z}},\quad \quad \quad Z_{\alpha }=\int _{{\vec {\sigma }}\in {\text{ state }}\alpha }d{\vec {\sigma }}e^{-\beta E[{\vec {\sigma }}]}=\langle e^{-\beta E[{\vec {\sigma }}]}\rangle _{\alpha }$

In the ferromagnet there are two pure states, $\alpha =\pm 1$, that correspond to positive and negative magnetization. In a mean-field spin glass, there are more than two pure states. The quantity $q_{EA}$ measures the overlap between configurations belonging to the same pure state, that one expects to be the same for all states:

$q_{EA}=q_{\alpha \alpha }=\lim _{N\to \infty }{\frac {1}{N}}\sum _{i}\langle \sigma _{i}\rangle _{\alpha }\langle \sigma _{i}\rangle _{\alpha }$

Notice that to be precise, we should write

$q_{EA}=\lim _{\epsilon \to 0}\lim _{N\to \infty }{\frac {1}{N}}\sum _{i}\langle \sigma _{i}^{1}\rangle _{\epsilon }\langle \sigma _{i}^{2}\rangle _{\epsilon },$

where the average is with respect to a tilted Boltzmann measure which contains a small coupling $\epsilon$ between two configurations $\sigma ^{1}$ and $\sigma ^{2}$ of the same system, that forces them to fall in the same pure state but a part from this it leaves them independent. This small coupling plays the same role of the infinitesimal magnetic fields in the ferromagnet. Notice also that this non-zero in a spin-glass phase, but it is also different from zero in unfrustrated ferromagnets: in the low-T phase of an Ising model, it equals to $m^{2}$, where $m$ is the magnetization.

• Overlap distribution. One can generalize this and consider also the overlap between configurations in different pure states, and the overlap distribution:

$q_{\alpha \beta }={\frac {1}{N}}\sum _{i}\langle \sigma _{i}\rangle _{\alpha }\langle \sigma _{i}\rangle _{\beta },\quad \quad \quad {P}(q)=\sum _{\alpha ,\beta }\omega _{\alpha }\,\omega _{\beta }\,\delta (q-q_{\alpha \beta }).$

The disorder average of quantities can be computed within the replica formalism, and one finds:

${\overline {P}}(q)=\lim _{n\to 0}{\frac {2}{n(n-1)}}\sum _{a>b}\delta \left(q-Q_{ab}^{SP}\right),\quad \quad \quad q_{EA}=\max \left\{Q_{a\neq b}^{SP}\right\}$

where $Q_{ab}^{SP}$ are the saddle point values of the overlap matrix introduced in Problem 2.2. The solution of the saddle point equations for the overlap matrix $Q$ thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution ${\overline {P}}(q)$. This distribution measures the probability that two copies of the system, equilibrating in the sae disordered environment, end up having overlap $q$. In the Ising case, a low temperature one has $q_{\alpha \alpha }=m^{2}$ and $q_{\alpha \neq \beta }=-m^{2}$, and thus ${\overline {P}}(q)$ has two peaks at $\pm m^{2}$.

Problem 3.1: the RS (Replica Symmetric) calculation

We go back to the saddle point equations for the spherical $p$-spin model derived in the previous problems. Let us consider the simplest possible ansatz for the structure of the matrix Q, that is the Replica Symmetric (RS) ansatz:

$Q={\begin{pmatrix}1&q_{0}&q_{0}\cdots &q_{0}\\q_{0}&1&q_{0}\cdots &q_{0}\\&\cdots &&\\q_{0}&q_{0}&q_{0}\cdots &1\end{pmatrix}}$

Under this assumption, there is a unique saddle point variable, that is $q_{0}$. We denote with $q_{0}^{SP}$ its value at the saddle point.

1. Under this assumption, what is the overlap distribution ${\overline {P}}(q)$ and what is $q_{EA}$? In which sense the RS ansatz corresponds to assuming the existence of a unique pure state?

2. Check that the inverse of the overlap matrix is

   $Q^{-1}={\begin{pmatrix}\alpha &\beta &\beta \cdots &\beta \\\beta &\alpha &\beta \cdots &\beta \\&\cdots &&\\\beta &\beta &\beta \cdots &\alpha \end{pmatrix}}\quad \quad {\text{with}}\quad \alpha ={\frac {1+(n-2)q_{0}}{1+(n-2)q_{0}-(n-1)q_{0}^{2}}}\quad {\text{and}}\quad \beta ={\frac {-q_{0}}{1+(n-2)q_{0}-(n-1)q_{0}^{2}}}$

   Compute the saddle point equation for $q_{0}$ in the limit $n\to 0$, and show that this equation admits always the solution $q_{0}=0$: why is this called the paramagnetic solution?

3. Compute the free energy corresponding to the solution $q_{0}=0$, and show that it reproduces the annealed free energy. Do you have an interpretation for this?

Problem 3.2: the 1-RSB (Replica Symmetry Broken) calculation

In the previous problem, we have chosen a certain parametrization of the overlap matrix $Q$, which corresponds to assuming that typically all the copies of the systems fall into configurations that are at overlap $q_{0}$ with each others, no matter what is the pair of replicas considered. This assumption is however not the good one at low temperature. We now assume a different parametrisation, that corresponds to breaking the symmetry between replicas: in particular, we assume that typically the $n$ replicas fall into configurations that are organized in $n/m$ groups of size $m$; pairs of replicas in the same group are more strongly correlated and have overlap $q_{1}$, while pairs of replicas belonging to different groups have a smaller overlap $q_{0}<q_{1}$. This corresponds to the following block structure for the overlap matrix:

$Q={\begin{pmatrix}1&q_{1}&q_{1}&q_{0}&q_{0}\cdots &q_{0}\\q_{1}&1&q_{1}&q_{0}&q_{0}\cdots &q_{0}\\q_{1}&q_{1}&1&q_{0}&q_{0}\cdots &q_{0}\\\cdots \\\cdots \\\cdots \\q_{0}&q_{0}\cdots &q_{0}&1&q_{1}&q_{1}\\q_{0}&q_{0}\cdots &q_{0}&q_{1}&1&q_{1}\\q_{0}&q_{0}\cdots &q_{0}&q_{1}&q_{1}&1\\\end{pmatrix}}$

Here we have three parameters: $m,q_{0},q_{1}$ (in the sketch above, $m=3$). We denote with $m^{SP},q_{0}^{SP},q_{1}^{SP}$ their values at the saddle point.

1. What is the overlap distribution in this case? What is $q_{EA}$? In which sense the parameter $m$ can be interpreted as a probability weight?

2. Using that

   $\log \det Q=n{\frac {m-1}{m}}\log(1-q_{1})+{\frac {n-m}{m}}\log[m(q_{1}-q_{0})+1-q_{1}]+\log \left[nq_{0}+m(q_{1}-q_{0})+1-q_{1}\right]$

   show that the free energy now becomes:

   $f_{1RSB}=-{\frac {1}{2\beta }}\left[{\frac {\beta ^{2}}{2}}\left(1+(m-1)q_{1}^{p}-mq_{0}^{p}\right)+{\frac {m-1}{m}}\log(1-q_{1})+{\frac {1}{m}}\log[m(q_{1}-q_{0})+1-q_{1}]+{\frac {q_{0}}{m(q_{1}-q_{0})+1-q_{1}}}\right]$

   Under which limit this reduces to the replica symmetric expression?

3. Compute the saddle point equations with respect to the parameter $q_{0},q_{1}$ and $m$ are. Check that $q_{0}=0$ is again a valid solution of these equations, and that for $q_{0}=0$ the remaining equations reduce to:

   $(m-1)\left[{\frac {\beta ^{2}}{q}}pq_{1}^{p-1}-{\frac {1}{m}}{\frac {1}{1-q_{1}}}+{\frac {1}{m}}{\frac {1}{1+(m-1)q_{1}}}\right]=0,\quad \quad {\frac {\beta ^{2}}{2}}q_{1}^{p}+{\frac {1}{m^{2}}}\log \left({\frac {1-q_{1}}{1+(m-1)q_{1}}}\right)+{\frac {q_{1}}{m[1+(m-1)q_{1}]}}=0$

   How does one recover the paramagnetic solution?

4. We now look for a solution different from the paramagnetic one. To begin with, we set $m=1$ to satisfy the first equation, and look for a solution of

   ${\frac {\beta ^{2}}{2}}q_{1}^{p}+\log \left(1-q_{1}\right)+q_{1}=0$

   Plot this function for $p=3$ and different values of $\beta$, and show that there is a critical temperature $T_{c}$ where a solution $q_{1}\neq 0$ appears: what is the value of this temperature (determined numerically)?