T-4

Goal: understand the information encoded in the replica solution, and the difference between Replica Symmetry (RS) and Replica Symmetry Breaking (RSB).
Techniques: replica method, variational ansatz, saddle point approximation.

The overlaps, and why they tell us about glassiness

Order parameter, ergodicity-breaking, states: the ferromagnet. The order parameter for ferromagnets is the magnetization:
$m=\lim _{h\to 0}\lim _{N\to \infty }{\frac {1}{N}}\sum _{i=1}^{N}{\overline {\langle \sigma _{i}\rangle _{h}}}$

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. Notice the order of limits in the definition: first the thermodynamic limit, and then the limit of zero 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.

A non-zero magnetisation is connected to ergodicity breaking, which is a dynamical concept: when there is a small field $h$ and the system follows some equilibrium dynamics (e.g. Langevin), it explores only a sub-part of the phase space, which corresponds to the sector of positive/negative magnetization. When ergodicity is broken, the Boltzmann measure clusters into states (labelled by $\alpha$ ) with 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.

Order parameter, ergodicity-breaking, states: the glass. In Lecture 1, we have introduced the Edwards-Anderson order parameter as: $q_{EA}=\lim _{t\to \infty }\lim _{N\to \infty }{\frac {1}{N}}\sum _{i}\sigma _{i}(0)\sigma _{i}(t)$
This measures the autocorrelation between the configuration of the same spin at $t=0$ and that at infinitely larger time. A non-zero value of $q_{EA}$ is again an indication of ergodicity breaking: if there was not ergodicity breaking, the system would be able to visit dynamically all configurations according to the Boltzmann measure, decorrelating to the initial condition. The fact that $q_{EA}>0$ indicates that the system, even at later times, is constrained to visit configurations that are not too different from the initial ones: this is because it explores only one of the available states! The difference with the ferromagnets is that in mean-field glassy models like the spherical $p$ -spin, there are not just two but many different states.

The quantity $q_{EA}$ measures the overlap between configurations belonging to the same state, that one expects to be the same for all states. In a thermodynamics formalism, it can be re-written as

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

Notice that to be precise, in analogy with the magnetization, we should write

$q_{EA}=\lim _{\epsilon \to 0}\lim _{N\to \infty }{\frac {1}{N}}\sum _{i}\langle \sigma _{i}^{1}\,\sigma _{i}^{2}\rangle _{\epsilon },$
where ${\vec {\sigma }}^{1},{\vec {\sigma }}^{2}$ are two copies of the system, and the average is with respect to a tilted Boltzmann measure which contains a small coupling $\epsilon$ between them, which 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.

Replica formalism: where is this info encoded? One can generalize this and consider the overlap between configurations in different states, and the overlap distribution:
$q_{\alpha \beta }={\frac {1}{N}}\sum _{i}\langle S_{i}\rangle _{\alpha }\langle S_{i}\rangle _{\beta },\quad \quad \quad {P}(q)=\sum _{\alpha ,\beta }\omega _{\alpha }\,\omega _{\beta }\,\delta (q-q_{\alpha \beta }).$

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 same 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 $\pm m^{2}$ .

[*] - The coupling is so weak that it only forces the configurations to fall in the same pure state, but a part from this it leaves them independent.

Problems

Problem 4.1: the RS (Replica Symmetric) solution

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.

RS overlap distribution. 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?

RS free energy. The saddle point equation for $q_{0}$ when $n\to 0$ is
${\frac {\beta ^{2}}{2}}pq_{0}^{p-1}-{\frac {1}{2}}{\frac {q_{0}}{(1-q_{0})^{2}}}=0,$

which admits always the solution $q_{0}^{SP}=0$ : why is this called the paramagnetic solution? Show that within this RS assumption, the quenched free energy coincides with the annealed one. Do you have an interpretation for this in terms of replicas?

Problem 4.2: 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-RSB overlap distribution. Show that in this case the overlap distribution is
${\overline {P}}(q)=m^{SP}\delta (q-q_{0}^{SP})+(1-m^{SP})\delta (q-q_{1}^{SP})$

What is $q_{EA}$ ? In which sense the parameter $m$ can be interpreted as a probability weight?

1-RSB free energy and saddle point equations. Under the 1RSB assumption, the expression for the free energy is:
$f_{1RSB}=-{\frac {1}{2\beta }}\left[{\frac {\beta ^{2}}{2}}\left(1+(m-1)q_{1}^{p}-mq_{0}^{p}\right)+{\frac {m-1}{m}}\log(1-q_{1})+{\frac {1}{m}}\log[m(q_{1}-q_{0})+1-q_{1}]+{\frac {q_{0}}{m(q_{1}-q_{0})+1-q_{1}}}\right]$

Under which limit this reduces to the replica symmetric expression? Compute the saddle point equations with respect to the parameter $q_{0},q_{1}$ and $m$ . 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?

The "random first-order" transition. One solution to the saddle point equations is $m^{SP}=1$ and $q_{0}^{SP}=0=q_{1}^{SP}$ . This is the only good solution one finds at high temperature. However, at a given critical temperature $T_{c}$ one finds that the system of equations is still solved by $m^{SP}=1$ , but the equation for $q_{1}$
${\frac {\beta ^{2}}{2}}q_{1}^{p}+\log \left(1-q_{1}\right)+q_{1}=0$

develops a new solution at a $q_{1}^{SP}\neq 0$ : try to estimate this temperature numerically by plotting this function for $p=3$ and different values of $\beta$ . When $T$ decreases below $T_{c}$ , one sees that $m^{SP}$ becomes smaller than one, and $q_{1}^{SP}$ increases towards one. What is different and what is similar with respect to the REM?