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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_0^{SP} } between configurations belonging to different pure states, and the probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2<m_1} blocks (in the example above,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_K > q_{K-1} > \cdots >q_0} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \to \infty} . The underlying picture of the free-energy landscape is as follows: equilibrium states have self-overlap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_K = q_{EA}} . They are arranged in clusters such that states inside a cluster have overlap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{K-1}} , but such clusters are arranged in other clusters at a higher level, at mutual overlap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{K-2}} and so on. In the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \to \infty} , the overlap distribution becomes a continuous function.

Problem 4.1: the susceptibilities

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(\beta)= \frac{d m(\beta, h)}{dh}\Big|_{h=0} }

where $m(\beta ,h)=\lim _{N\to \infty }m_{N}(\beta ,h)$ is the magnetization at inverse temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} and external field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_N(\beta,h)} its finite-size counterpart. By the Fluctuation Dissipation relation, we also know that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_N(\beta)= \frac{d m_N(\beta, h)}{dh} \Big|_{h=0}= \frac{\beta}{N}\sum_{ij} \langle \sigma_i \sigma_j \rangle_{c} =\frac{\beta}{N}\sum_{ij} \left(\langle \sigma_i \sigma_j \rangle- \langle \sigma_i \rangle \langle \sigma_j \rangle \right). }

Consider the mean field Ising model. Let , which satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m= tanh[\beta(h+m)]} . For finite N, behaves as [plot].

Define the

where assumed that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \to \infty} before Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \to 0} .

Show that 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(\beta)= \frac{\beta (1-m^2)}{1-\beta [1-m^2]} }

Finite for $T>T_{c}=1$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T \to T_c} , this quantity diverges. Consider now Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \to 0} that one is taking to get the susceptibility do not commute: if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \to \infty} is taken first, the system remains trapped in the pure state selected by h, and the susceptibility when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \to 0} remains finite; if instead Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \to 0} before, the susceptibility is proportional to N and so it diverges.

The susceptibility at finite N is given by FDT

In the ferromagnet, using that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\langle \sigma_i \sigma_j \rangle_c}= m^2} show divergence. 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \neq j } . Show that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N \to \infty}\overline{\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 \, = } </math>

Show that this quantity does not diverge at the spin glass transition.

The interpretation of the susceptibility is one would measure if the system is prepared at equilibrium, then a small magnetic field is applied and the new equilibrium state is reached. you let the system reach the best free energy states in presence of the field: This is called field-cooled.

What would be the susceptibility that measures the response of the system within a given state? Could you explain why