T-4

Goal of these problems:

Key concepts:

The free-energy landscape of the Sherrington Kirkpatrick 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 $m^{SP}=T/T_{c}$. The structure of the overlap matrix, and thus of the replica formalism, encodes a lot of information on the structure of the Boltzmann measure and 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 and encoded in a complicated pattern of the mutual overlaps between equilibrium states. 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. Iterating K times, one ends up with a series of overlaps $q_{K}>q_{K-1}>\cdots >q_{0}$. 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. The Parisi solution of the Sherrington Kirkpatrick model is obtained taking $K\to \infty$.

Problem 4.1: magnetic susceptibilities and linear response

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

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

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

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

2. Spin-glass. By the Fluctuation-Dissipation Theorem (FDT), we know that the response and correlations at equilibrium are related by

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

   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 one can write

   $\chi _{eq}=\lim _{N\to \infty }\chi _{N}=\beta \left(1-\int dq\,{\overline {P(q)}}\,q\right)\,\quad \quad \quad {\overline {P(q)}}={\overline {\sum _{\alpha ,\beta }\omega _{\alpha }\omega _{\beta }\delta (q-q_{\alpha \beta })}}$

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

3. Response within a state. The quantity $\chi _{eq}$ is the response that one would measure if the system is prepared at equilibrium, then a small magnetic field is applied and the system is given enough time to reach the new equilibrium state. This is reflected in the formula above: one is averaging over all possible pure states, meaning that the system has enough time to sample them according to their thermal weight $\omega _{\alpha }$. One can also define a susceptibility $\chi _{LR}$ (where LR stands for Linear Response) measuring the response within a given pure state: in analogy with the above, what would you expect to be its expression in terms of the overlaps? Could you explain intuitively why $\chi _{LR}<\chi _{eq}$?

The zero-field cooled (lower curve) and field cooled (upper curve) susceptibility. Experimental results taken from Djurberg et al, European Phys. J. B, 10, 1999.

4. Experiments. The plot on the left shows experimental measurements of the magnetic susceptibility in a spin-glass material. The two curves correspond to two different protocols: (i) ZFC (zero-field cooled) protocol: add a very small magnetic field when the system is already at the final low temperature; (ii) FC (field-cooled): cooling the system in presence of a small magnetic field and comparing the observed magnetization with the one measured without this small magnetic field. In the second protocol, the system has the ability to choose the state that is most appropriate in presence of the applied field. Which of the the two susceptibilities defined above describe these two experimental protocols?

Comment: as we have seen, the magnetic susceptibility does not diverge at the spin-glass critical temperature, at variance with a ferromagnet. One can identify a more complicated object, the spin-glass susceptibility, that does diverge at the transition. At variance with the magnetic susceptibility, that is related to a 2-point function (the correlation, which involves two spins), the spin-glass susceptibility is associated to a 4-point function. This is consistent with the fact that the order parameter of the spin-glass phase, $q_{EA}$, is itself a 2-point function, at variance with the magnetization that is a 1-point function.