T-3

Goal of these problems: In this set of problems, we compute the free energy of the spherical 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 p} -spin model in the Replica Symmetric (RS) framework, and in the 1-step Replica Symmetry Broken (1-RSB) framework.

Key concepts:

The order parameters: overlaps, and their meaning

• Order parameter. In the lectures, we have introduced the Edwards-Anderson order parameter

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_{EA}= \lim_{N \to \infty}\frac{1}{N}\sum_i \overline{\langle \sigma_i \rangle^2} }

which plays the same role as the magnetization in a ferromagnet. Let us recap what happens for a ferromagnet. The magnetization is defined as:

$m=\lim _{h\to 0}\lim _{N\to \infty }{\frac {1}{N}}\sum _{i=1}^{N}{\overline {\langle \sigma _{i}\rangle _{h}}}$

where 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 \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. This quantity is non-zero in the low-T ferromagnetic phase; it measures ergodicity breaking: when a small 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 } is added, the system at equilibrium explores only a sub-part of the phase space which corresponds to a magnetization in the direction of the 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.

• Pure states. When ergodicity is broken, the Boltzmann measure clusters into pure states (labelled by $\alpha$) with Gibbs weight 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 \omega_\alpha } , meaning that one can re-write the thermal averages 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 \langle \cdot \rangle } of any observable $A$ as

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 \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, 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 \alpha= \pm 1 } , that correspond to positive and negative magnetization. In a mean-field spin glass, there are more than two pure states. The quantity 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_{EA}} measures the overlap between configurations belonging to the same pure state, that one expects to be the same for all states:

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_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha }

Notice that to be precise, we should write

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_{EA}=\lim_{\epsilon \to 0}\lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i^1 \rangle_\epsilon \langle \sigma_i^2 \rangle_\epsilon, }

where the average is with respect to a tilted Boltzmann measure which contains a small coupling $\epsilon$ between two configurations 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 \sigma^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 \sigma^2} of the same system, that forces them to fall in the same pure state but a part from this it leaves them independent. This small coupling 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 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} is the magnetization.

• Overlap distribution. One can generalize this and consider also the overlap between configurations in different pure states, and the overlap distribution:

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_{\alpha \beta}= \frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_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 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_{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 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} 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 sae disordered environment, end up having 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} . In the Ising case, a low temperature one has 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_{\alpha \alpha}=m^2} and $q_{\alpha \neq \beta }=-m^{2}$, and thus 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{P}(q)} has two peaks at 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 \pm m^2} .

Problem 3.1: the RS (Replica Symmetric) calculation

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:

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_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 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} . We denote with $q_{0}^{SP}$ its value at the saddle point.

1. Under this assumption, what is the overlap distribution 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{P}(q)} and what 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 q_{EA}} ? In which sense the RS ansatz corresponds to assuming the existence of a unique pure state?

2. Check that the inverse of the overlap matrix is

   $Q^{-1}={\begin{pmatrix}\alpha &\beta &\beta \cdots &\beta \\\beta &\alpha &\beta \cdots &\beta \\&\cdots &&\\\beta &\beta &\beta \cdots &\alpha \end{pmatrix}}\quad \quad {\text{with}}\quad \alpha ={\frac {1+(n-2)q_{0}}{1+(n-2)q_{0}-(n-1)q_{0}^{2}}}\quad {\text{and}}\quad \beta ={\frac {-q_{0}}{1+(n-2)q_{0}-(n-1)q_{0}^{2}}}$

   Compute the saddle point equation 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 q_0} 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 n \to 0} , and show that this equation admits always the solution $q_{0}=0$: why is this called the paramagnetic solution?

3. Compute the free energy corresponding to the solution 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= 0} , and show that it reproduces the annealed free energy. Do you have an interpretation for this?

Problem 3.2: the 1-RSB (Replica Symmetry Broken) calculation

In the previous problem, we have chosen a certain parametrization of 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} , 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 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} replicas fall into configurations that are organized in 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/m} groups of size $m$; pairs of replicas in the same group are more strongly correlated and 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_1} , while pairs of replicas belonging to different groups have a smaller 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<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: 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, q_0, q_1} (in the sketch 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=3} ). We denote 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 m^{SP}, q_0^{SP}, q_1^{SP}} their values at the saddle point.

1. What is the overlap distribution in this case? What 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 q_{EA}} ? In which sense the parameter 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} can be interpreted as a probability weight?

2. 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 \log \det Q=n\frac{m-1}{m} \log (1-q_1)+ \frac{n-m}{m} \log [m(q_1-q_0) + 1-q_1]+ \log \left[nq_0+ m(q_1-q_0)+ 1-q_1 \right]}

   show that the free energy now becomes:

   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 f_{1RSB}= - \frac{1}{2 \beta} \left[ \frac{\beta^2}{2} \left(1+ (m-1)q_1^p - m q_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?

3. Compute the saddle point equations with respect to the parameter 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, q_1 } and $m$ are. Check that $q_{0}=0$ is again a valid solution of these equations, and that 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 q_0=0} the remaining equations reduce to:

   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) \left[ \frac{\beta^2}{q}p q_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?

4. We now look for a solution different from the paramagnetic one. To begin with, we set 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 } to satisfy the first equation, and look for a solution of

   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 \frac{\beta^2}{2} q_1^p + \log \left(1-q_1\right)+ q_1=0 }

   Plot this function 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 p=3} and different values of 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 show that there is a critical 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 T_c} where a solution 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_1 \neq 0} appears: what is the value of this temperature (determined numerically)?