T-4

Goal of these problems:

Key concepts:

The free-energy landscape of the SK model

We have seen an example of mean-field model, the spherical p-spin, in which the low-T phase is glassy, described by a 1-RSB ansatz of the overlap matrix. The thermodynamics in the glassy phase is described by three quantities: the typical overlap $q_{1}^{SP}$ between configurations belonging to the same pure state, the typical overlap $q_{0}^{SP}$ between configurations belonging to different pure states, and the probability $(1-m^{SP})$ that two configurations extracted at equilibrium belong to the same state. It can be shown that the low-T, frozen phase of the REM is also described by this 1-RSB ansatz with $q_{0}^{SP}=0,q_{1}^{SP}=1$ and $m^{SP}=T/T_{c}$. Replicas are a way to explore the structure of the free-energy landscape.

The Sherrington-Kirkpatrick model introduced in Lecture 1 also has a low-T phase that is glassy. However, the structure of the free-energy landscape is more complicated the mutual overlaps between equilibrium states are organized in a complicated pattern. To understand it, let us consider a 2-RSB ansatz for the overlap matrix:

$Q={\begin{pmatrix}1&q_{2}&q_{1}&q_{1}&q_{0}\cdots &q_{0}\\q_{2}&1&q_{1}&q_{1}&q_{0}\cdots &q_{0}\\q_{1}&q_{1}&1&q_{2}&q_{0}\cdots &q_{0}\\q_{1}&q_{1}&q_{2}&1&q_{0}\cdots &q_{0}\\\cdots \\\cdots \\\cdots \\q_{0}&q_{0}\cdots &q_{2}&1&q_{1}&q_{1}\\q_{0}&q_{0}\cdots &q_{1}&q_{1}&1&q_{2}\\q_{0}&q_{0}\cdots &q_{1}&q_{1}&q_{2}&1\\\end{pmatrix}}$

which assumes that replicas are split into $m_{1}$ blocks, and that inside each block they are further split into $m_{2}<m_{1}$ blocks (in the example above,$m_{1}=4,m_{2}=2$). This is also not the correct structure for the SK model. The correct one obtained iterating this procedure an infinite number of times, as understood by Parisi in his seminal paper of XX. One ends up with a series of overlaps $q_{K}>q_{K-1}>\cdots >q_{0}$ with $K\to \infty$. The underlying picture of the free-energy landscape is as follows: equilibrium states have self-overlap $q_{K}=q_{EA}$. They are arranged in clusters such that states inside a cluster have overlap $q_{K-1}$, but such clusters are arranged in other clusters at a higher level, at mutual overlap $q_{K-2}$ and so on. In the limit $K\to \infty$, the overlap distribution becomes a continuous function.

Problem 4.1: the susceptibilities

Consider the mean field Ising model. Let $m(\beta ,h)$ be the magnetization at inverse temperature $\beta$ and field $h$, which satisfies $m=tanh[\beta (h+m)]$. For finite N, behaves as [plot].

Define the susceptibility

$\chi (\beta ,h)={\frac {dm(\beta ,h)}{dh}}$

Show that 

$\chi (\beta )={\frac {\beta (1-m^{2})}{1-\beta [1-m^{2}]}}$

Finite for $T>T_{c}=1$, $T\to T_{c}$, this quantity diverges. Consider now $T<T_{c}$. The quantity above assumes that N is infinity before h to zero. looking at the graph, Show that the limits $N\to \infty$ and $h\to 0$ that one is taking to get the susceptibility do not commute: if $N\to \infty$ is taken first, the system remains trapped in the pure state selected by h, and the susceptibility when $h\to 0$ remains finite; if instead $h\to 0$ before, the susceptibility is proportional to N and so it diverges.

Show the last point using the FDT

$\chi _{N}(\beta )={\frac {dm_{N}(\beta ,h)}{dh}}{\Big |}_{h=0}$