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: magnetic susceptibilities

In this problem, we consider the magnetic susceptibilities. In an Ising system, the thermodynamic magnetic susceptibility is

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

where $m(\beta ,h)=\lim _{N\to \infty }m_{N}(\beta ,h)$ is the magnetization at inverse temperature $\beta$ and external field $h$, and $m_{N}(\beta ,h)$ its finite-size counterpart. The function $\chi (\beta )$ measures the response to changes of the magnetic field.

1. Using the self-consistent equation for the magnetization in the mean-field Ising model, $m=tanh[\beta (h+m)]$, show that $\chi (\beta )$ diverges exactly at the transition temperature $\beta =\beta _{c}=1$.

2. By the Fluctuation Dissipation relation, we know that the response and correlations at equilibrium are related by

   $\chi _{N}(\beta )={\frac {dm_{N}(\beta ,h)}{dh}}{\Big |}_{h=0}={\frac {\beta }{N}}\sum _{ij}{\overline {\langle \sigma _{i}\sigma _{j}\rangle _{c}}}={\frac {\beta }{N}}\sum _{ij}{\overline {\left(\langle \sigma _{i}\sigma _{j}\rangle -\langle \sigma _{i}\rangle \langle \sigma _{j}\rangle \right)}}.$

   In a spin glass, by symmetry with respect to sign flips of the couplings it holds ${\overline {\langle \sigma _{i}\sigma _{j}\rangle _{c}}}=0$ for $i\neq j$. Show that

   $\chi (\beta )=\lim _{N\to \infty }\chi _{N}(\beta )=\int dq\,{\overline {\sum _{\alpha ,\beta }\omega _{\alpha }\omega _{\beta }\delta (q-q_{\alpha \beta })}}\,q\,=\int dq\,{\overline {P(q)}}\,q\,$

   and thus that this quantity does not diverge at the transition to the spin-glass phase.

3. The quantity $\chi (\beta )$ is the response one would measure if the system is prepared at equilibrium, then a small magnetic field is applied and the new equilibrium state is reached. In this protocol, one is assuming that the system has time to reach the new equilibrium in presence of the field. In analogy with the above, what would be the susceptibility $\chi _{LR}$ that measures the response of the system within a given pure state? Could you explain intuitively why $\chi _{LR}<\chi$?

Comment: diverging susceptibility is another one

• • • • • • • In the Ising case, the magnetization $m_{N}(\beta ,h)$ behaves as in Fig. for $T<T_{c}$. Using the plot, show that the limits $N\to \infty$ and $h\to 0$ do not commute: if $N\to \infty$ is taken first as in the definition of the thermodynamics susceptibility, one gets a finite value for $\chi (\beta )$; if instead $h\to 0$ is taken before, $\chi _{N}(\beta )$ diverges. Confirm the last observation using the FDT an the fact that ${\overline {\langle \sigma _{i}\sigma _{j}\rangle _{c}}}=m^{2}$.