View source for T-4

<strong>Goal: </strong> understand the information encoded in the replica solution, and the difference between Replica Symmetry (RS) and Replica Symmetry Breaking (RSB). 
<br>
<strong>Techniques: </strong> replica method, variational ansatz, saddle point approximation.<br>

== The overlaps, and why they tell us about glassiness==
[[File:paramagnet.png|thumb|right|x160px|]]
[[File:ferromagnet.png|thumb|right|x145px|]]
[[File:glass.png|thumb|right|x140px|]]
<ul>
<li> '''Order parameter, symmetry breaking: the ferromagnet.''' The order parameter for ferromagnets (e.g., Ising model at <math>T<T_c </math>) is the magnetization, that for clean systems is:
<math display="block">
m_{\pm}=\lim_{h \to 0^{\pm}}\; \lim_{N \to \infty}\; \frac{1}{N}\sum_{i=1}^N \, {\langle \sigma_i \rangle_{ h}}
</math>
where <math>\langle \cdot \rangle_{h} </math> is the Boltzmann average in presence of a magnetic field of strength <math>h </math>. Notice the order of limits: first the thermodynamic limit, and then the limit of zero field. If the average is performed over the Boltzmann measure with  <math>h=0 </math>, one gets  <math>m_{\rm eq}=\lim_{N \to \infty}\sum_{i=1}^N \, N^{-1}{\langle \sigma_i \rangle_{h=0}}=0,</math> because the Hamiltonian/energy function is symmetric with respect to <math>\sigma_i \to -\sigma_i </math>. Thus, the non-zero magnetization is a signature of <ins>symmetry breaking</ins>. 
</li><br>

<li> '''Ergodicity breaking, pure states: the ferromagnet.''' The low-T phase of the ferromagnet also corresponds to <ins>ergodicity breaking</ins>. This is a dynamical concept: the system following some equilibrium dynamics (e.g. Langevin) explores only a sub-part of the equilibrium configurations/free-energy shell, which corresponds to the sector of either positive or negative magnetization. Because of this, averages over dynamical trajectories do not converge to Boltzmann averages <sup>[[#Notes|[*] ]]</sup>:
<math display="block">
m_\infty:= \lim_{t \to \infty} \lim_{N \to \infty}\; \frac{1}{N}\sum_{i=1}^N  \sigma_i(t) = m_{\pm} (\neq  m_{\rm eq}=0).
</math>
<br>
What's going on? When ergodicity is broken, the Boltzmann measure clusters into <ins>ergodic components</ins>, called <ins>pure states</ins>. Labelling them by <math>\alpha </math> and introducing the weight <math>\omega_\alpha </math>, one can re-write the Boltzmann averages <math>\langle \cdot \rangle </math> of any observable <math>A </math> as 
<math display="block">
\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.
</math>
Pure states satisfy the clustering property, which for fully-connected systems reads: <math>\langle \sigma_i \sigma_j \rangle_\alpha=\langle \sigma_i \rangle_\alpha \langle \sigma_j \rangle_\alpha </math>. In the ferromagnet there are two pure states, <math>\alpha= \pm 1 </math>, that correspond to positive and negative magnetization. These two states are related by symmetry. They satisfy the clustering property <math>\langle \sigma_i \sigma_j \rangle_\pm= m^2</math>, while the full Boltzmann measure does not, since <math>\langle \sigma_i \sigma_j \rangle_{\rm eq}= (1/2)\, m^2_{+} + (1/2)\, m^2_{-}=  m^2 (\neq  \langle \sigma_i \rangle_{\rm eq}\langle \sigma_j \rangle_{\rm eq}=0).</math>
We know how to select each of the two pure states: by adding a small magnetic field <math> h </math> (symmetry breaking). 
</li><br>

<div style="font-size:89%">
: <small>[*]</small> - Think at Ising mean-field, where magnetization concentrates dynamically and no need to time average.
</div>


<li> '''Order parameter, ergodicity-breaking, states: the glass.''' In Lecture 1, we have introduced the <ins>Edwards-Anderson order parameter</ins> as:
<math display="block">
 q_{EA}= \lim_{t\to \infty} \lim_{N\to \infty} \frac{1}{N}\sum_{i} \sigma_i(0) \sigma_i(t)
</math>
This measures the autocorrelation between the configuration of the same spin at <math>t=0</math> and that at infinitely larger time. A non-zero value of <math>  q_{EA} </math> is again an indication of ergodicity breaking: if there was not ergodicity breaking, the system would be able to visit dynamically all configurations according to the Boltzmann measure, decorrelating to the initial condition. The fact that <math>  q_{EA} >0</math> indicates that the system, even at later times, is constrained to visit configurations that are not too different from the initial ones: this is because it explores only one of the available pure states, reaching ergodicity only within this pure state!  <br>

The quantity <math>q_{EA}</math> measures the overlap between configurations in the <ins>same state</ins>, that one expects to be the same for all states. In a thermodynamics formalism, it can be written as <sup>[[#Notes|[**] ]]</sup>
<math display="block">
q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha
</math>
Notice 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 <math>m^2</math>. What is peculiar of glassy systems is that in mean-field glassy  models like the spherical <math>p</math>-spin, there are not just two but several different states, not related by symmetry. This shows up in another quantity, the overlap distribution, that replica calculations produce as a natural output. </li><br>
 

<li> '''Replica formalism: where is this info encoded?''' The overlap between configurations in different states is accounted for by the  <ins>overlap distribution</ins>:
<math display="block">
q_{\alpha \beta}= \frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\beta, \quad \quad \quad {P}_{\beta, N}(q)= \sum_{\alpha, \beta} \omega_\alpha\, \omega_\beta\, \delta(q- q_{\alpha \beta}).
</math>
The average value of this distribution (when  <math> N \to \infty</math>) can be computed within the replica formalism, and one finds:
<math display="block">
\mathbb{E}[{P}_\beta(q)]:= \lim_{N \to \infty}  \mathbb{E}[P_{\beta, N}(q)]=\lim_{n \to 0} \frac{2}{n(n-1)}\sum_{a<b}\delta \left(q- q_{ab}^{*}\right),\quad \quad \quad q_{EA}= \max_{a<b} \left\{ q_{a b}^* \right\}
</math>
where <math> q_{ab}^{*}</math> are the saddle point values of the overlap matrix. In the Ising case, a low temperature one has  <math> q_{\alpha \alpha}=m^2</math> and <math> q_{\alpha \neq \beta}=-m^2</math>, and thus <math> {P}_\beta(q)</math> has two peaks at <math> \pm m^2</math>. Moreover, the weights in front of each delta peak is the same, which tells us that there are only two pure states.  In spin-glass models, the structure of the distribution can be more complicated. </li>
</ul>


<div style="font-size:89%">
: <small>[**]</small> - To be precise, in analogy with the magnetization, we should write  <math>q_{EA}=\lim_{\epsilon \to 0}\lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i^1 \, \sigma_i^2 \rangle_\epsilon </math>, where <math> \vec{\sigma}^1, \vec{\sigma}^2 </math> are two copies of the system, and the average is with respect to a tilted Boltzmann measure which contains a small coupling <math> \epsilon </math> between them, which plays the same role of the infinitesimal magnetic fields in the ferromagnet. The coupling is so weak that it only forces the configurations to fall in the same pure state, but a part from this it leaves them independent.
</div>
<br>

== Problems ==
We go back to the spherical <math> p</math>-spin model, and complete the calculation of the free energy under different assumptions on the structure of the overlap matrix. 

=== Problem 4.1: the RS (Replica Symmetric) solution===

[[File:RS.png|thumb|right|x160px|]]
We go back to the saddle point equations for the spherical <math>p</math>-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:

<math display="block">
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}
</math>
Under this assumption, there is a unique saddle point variable, that is <math>q_0</math>. We denote with <math>q_0^*</math> its value at the saddle point.


<ol>
<li><em> RS overlap distribution. </em>
Under this assumption, what is the overlap distribution <math>\mathbb{E}[P_\beta(q)]</math> and what is <math>q_{EA}</math>? In which sense the RS ansatz corresponds to assuming the existence of a unique pure state?
</li>
</ol>
<br>

<ol start="2">
<li> <em> The paramagnet. </em> Show that the saddle point equation for <math> q_0 </math> when <math> n \to 0 </math> is 
<math display="block">
\beta^2 \, p q_0^{p-1}-\frac{q_0}{(1-q_0)^2}=0,
</math>
which admits always the solution <math>q_0^{*}= 0</math>: why is this called the <em>paramagnetic</em> solution? Show that within this RS assumption, the quenched free energy coincides with the annealed one. 
</li>
</ol>
<br>

=== Problem 4.2: the 1-RSB (Replica Symmetry Broken) calculation===
[[File:1RSB.png|thumb|right|x160px|]]
In the previous problem, we have chosen a certain parametrization of the overlap matrix <math>Q</math>, which corresponds to assuming that typically all the copies of the systems fall into configurations that are at overlap  <math>q_0</math> 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 <math>n</math> replicas fall into configurations that are organized in <math>n/\mu</math> groups of size <math>\mu </math> (replicas in the same group are equilibrating in the same pure state); pairs of replicas in the same group are more strongly correlated and have overlap <math>q_1</math>, while pairs of replicas belonging to different groups have a smaller overlap <math>q_0<q_1</math>.  This corresponds to the following block structure for the overlap matrix:
<math display="block">
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}
</math>
Here we have three parameters: <math>\mu, q_0, q_1</math> (in the sketch above, <math>\mu=3</math>). We denote with <math>\mu^{*}, q_0^{*}, q_1^{*}</math> their values at the saddle point.



<ol>
<li> <em> 1-RSB overlap distribution. </em> Show that in this case the overlap distribution is
<math display="block">
\mathbb{E}[P_\beta(q)]= \mu^{*} \delta(q-q_0^{*})+ (1-\mu^{*})\delta(q-q_1^{*})
</math>
What is <math> q_{EA}</math>? In which sense the parameter <math> \mu</math> when <math>n \to 0</math> can be interpreted as a probability weight?
</li>
</ol>
<br>

<ol start="2">
<li><em> 1-RSB free energy and the paramagnet. </em> Under the 1RSB assumption, one can show that the expression for the free energy is:
<math display="block">
f_{1RSB}= - \frac{1}{2 \beta} \left[\beta^2 \left(1+ (\mu-1)q_1^p - \mu q_0^p \right)+ \frac{\mu-1}{\mu} \log (1-q_1)+ \frac{1}{\mu} \log [\mu(q_1-q_0) + 1-q_1]+ \frac{q_0}{\mu(q_1-q_0)+ 1-q_1} \right]\Big|_{q_1^*, q_0^*, \mu^*}
</math>
Under which limit one recovers the the replica symmetric ansatz? Taking the derivatives of the free energy, one gets the saddle point equations for <math> q_0, q_1 </math> and <math> \mu </math>: it appears that <math> q_0=0</math> is again a valid solution of these equations, and that for <math> q_0=0</math> the remaining equations reduce to:
<math display="block">
(\mu-1) \left[ \beta^2 \, p q_1^{p-1}-\frac{1}{\mu}\frac{1}{1-q_1}+ \frac{1}{\mu} \,\frac{1}{1+ (\mu-1)q_1} \right]=0, \quad \quad 
\beta^2\, q_1^p + \frac{1}{\mu^2}\log \left( \frac{1-q_1}{1+ (\mu-1)q_1}\right)+ \frac{q_1}{\mu [1+(\mu-1)q_1]}=0.
</math>
Show that <math> \mu^*= 1 </math> and <math> q_0^*= 0=q_1^* </math> is a solution to these equations. When do you expect this solution to be correct?
</li>
</ol>
<br>

<ol start="3">
<li><em> The "random first-order" transition. </em>  At a given critical temperature <math> T_c </math> one finds that the system of equations is still solved by <math> \mu^*=1 </math>, but the equation for <math> q_1 </math> 
<math display="block">
\beta^2\, q_1^p + \log \left(1-q_1\right)+ q_1=0
</math>
develops a new solution at a <math> q_{1}^{*} \neq 0 </math>. When <math> T </math> decreases below   <math> T_c </math>, one sees that <math> \mu^*</math> becomes smaller than one, and <math> q_{1}^{*} </math> increases towards one. What is different and what is similar with respect to the REM? <br>

<code>Short exercise. </code> Try to estimate <math> T_c </math> numerically by plotting the equations for <math> p=3</math> and different values of  <math> \beta</math>, and check that <math> \beta_c \approx 1.2066</math>.
</li>
</ol>
<br>

== Replica Symmetry Breaking: the Parisi scheme ==

[[File:kRSB.png|thumb|right|x160px|]]
For the spherical <math>p</math>-spin model,  the low-T phase is glassy and the description within a <ins>1-RSB ansatz is exact</ins>: the equilibrium in the glassy phase is described by three quantities: the typical overlap  <math> q_1^{*} </math> between configurations belonging to the same state, the typical overlap <math> q_0^{*} </math> between configurations belonging to different states, and the probability  <math> (1-\mu^{*}) </math> that two configurations extracted at equilibrium belong to the same state.<br>

There are models for which the glassy phase is described by a more complicated  pattern of the mutual overlaps between states, which is obtained iterating the RSB scheme to <math>K</math> levels, ending up with a series of overlaps <math>q_K > q_{K-1} > \cdots >q_0</math>. The underlying picture of the free-energy landscape is as follows: configurations in the same pure state have typical overlap <math>q_K = q_{EA}</math>. They are arranged in clusters such that configurations belonging to different states inside a cluster have overlap <math>q_{K-1}</math>, but such clusters are arranged in other clusters at a higher level, at mutual overlap <math>q_{K-2}</math> and so on.<br>
 
The exact solution of the <ins>Sherrington-Kirkpatrick model</ins> introduced in Lecture 1 is obtained iterating this procedure an infinite number of times (<math> K \to \infty </math>): this is the <ins>full-RSB</ins> solution. In this limit, the overlap distribution is continuous. The intuition that the free-energy landscape in mean-field spin glasses has this hierarchical structure is due to Giorgio Parisi. Both 1RSB and fullRSB features are found in mean-field models of <ins>structural glasses</ins>.  <br>

<!--[[Media:The Parisi Scheme.pdf]]-->

== Check out: key concepts and Exercises==
[[File:Suscept-experiment.png|thumb|left|x140px| Experimental measurements of the magnetic Field Cooled (top) and Zero-Field Cooled  (bottom) susceptibility in a CuMn spin glass. Figure taken from C. Djurberg, K. Jonason, P. Nordblad, https://arxiv.org/abs/cond-mat/9810314]]

Order parameters, ergodicity breaking, pure states, overlaps, overlap distribution, replica-symmetric ansatz, replica symmetry breaking.
<br>

After this lecture, you have the tools to solve <code>Exercise 10 </code> about the magnetic susceptibilities in glasses. Solving the exercise, you will also understand how to recover the experimental results shown in the plot on the left.

== To know more ==
* A note on Elisabeth Garder [http://www.baginsky.de/eli/eg_portr.html]
* Castellani, Cavagna. Spin-Glass Theory for Pedestrians [https://arxiv.org/abs/cond-mat/0505032]
* Zamponi. Mean field theory of spin glasses [https://arxiv.org/abs/1008.4844]
* Parisi. Order parameter for spin-glasses [[Media:Parisi - OrderParameter.pdf| [4] ]]
* Urbani. Statistical Physics of glassy systems: tools and applications [https://physics.bme.hu/sites/physics.bme.hu/files/users/BMETE15MF53_kov/Lecture_Notes_Urbani.pdf [5]]














<!-- ******************** OLD VERSION ***********************-->


<!--<strong>Goal: </strong> 
In this set of problems, we compute the free energy of the spherical <math>p</math>-spin model in the Replica Symmetric (RS) framework, and in the 1-step Replica Symmetry Broken (1-RSB) framework. 
<br>
<strong>Techniques: </strong> replica method, variational ansatz, saddle point approximation.
<br>



== The order parameters: overlaps, and their meaning ==


<ul>
<li> '''Thermodynamics and dynamics.''' Recall: a system equilibrates dynamically at temperature <math> T </math> whenever at sufficiently large timescales it visits configurations, during its dynamical trajectories, with the frequency predicted by the Boltzmann distribution at temperature <math> T </math>. </li><br>

<li> '''Order parameter, ergodicity-breaking, pure states: the ferromagnet.''' Let us recall the theory of ferromagnetism. The order parameter for ferromagnets is the magnetization:
<center>
<math>
m=\lim_{h \to 0} \lim_{N \to \infty}\frac{1}{N}\sum_{i=1}^N \overline{\langle S_i \rangle_{ h}}
</math>
</center>
where <math>\langle \cdot \rangle_{h} </math> 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. Notice the order of limits in the definition: first the thermodynamic limit, and then the limit of zero 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.<br>


A non-zero magnetisation is connected to <ins>ergodicity breaking</ins>, which is a dynamical concept: when a small field <math>h </math> is added, the system in its equilibrium dynamics explores only a sub-part of the phase space, which corresponds to a finite magnetization in the direction of the field.  When ergodicity is broken, the Boltzmann measure clusters into <ins>pure states</ins> (labelled by <math>\alpha </math>) with Gibbs weight <math>\omega_\alpha </math>, meaning that one can re-write the thermal averages <math>\langle \cdot \rangle </math> of any observable <math>A </math> as 
<center>
<math>
\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
</math>
</center>
In the ferromagnet there are two pure states, <math>\alpha= \pm 1 </math>, that correspond to positive and negative magnetization. The free energy barrier that one has to overcome to go from one state to the other diverges when <math> N \to \infty </math>, and thus the system is dynamically trapped only in one state. 
</li><br>


<li> '''Order parameter, ergodicity-breaking, pure states: the glass.''' In Lecture 1, we have introduced the <ins>Edwards-Anderson order parameter</ins> as:
<center> <math>
 q_{EA}= \lim_{t\to \infty} \lim_{N\to \infty} \frac{1}{N}\sum_{i} S_i(0) S_i(t)
</math></center> 
This measures the autocorrelation between the configuration of the same spin at <math>t=0</math> and that at infinitely larger time. A non-zero value of <math>  q_{EA} </math> is again an indication of ergodicity breaking: if there was not ergodicity breaking, the system would be able to visit dynamical all configurations according to the Bolzmann measure, decorrelating to the initial condition. The fact that <math>  q_{EA} >0</math> indicates that the system, even at later times, is constrained to visit configurations that are not too different from the initial ones: this is because it explores only one of the available pure states! The difference with the ferromagnets is that in models like the spherical <math>p</math>-spin, there are not just two but many different pure states. <br>

The quantity <math>q_{EA}</math> measures the overlap between configurations belonging to the <ins>same pure state</ins>, that one expects to be the same for all states. In a thermodynamics formalism, it can be re-written as <sup>[[#Notes|[*] ]]</sup>
<center>
<math>
q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha
</math>
</center>
Notice 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 <math>m^2</math>, where <math>m</math> is the magnetization. </li><br>
 

<li> '''Replica formalism: where is this info encoded?''' One can generalize this and consider the overlap between configurations in different pure states, and the <ins>overlap distribution</ins>:
<center>
<math>
q_{\alpha \beta}= \frac{1}{N}\sum_i \langle S_i \rangle_\alpha \langle S_i \rangle_\beta, \quad \quad \quad {P}(q)= \sum_{\alpha, \beta} \omega_\alpha\, \omega_\beta\, \delta(q- q_{\alpha \beta}).
</math>
</center>
The disorder average of quantities can be computed within the replica formalism, and one finds:
<center>
<math>
\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\}
</math>
</center>
where <math> Q_{ab}^{SP}</math> 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 <math> Q</math> thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution <math> \overline{P}(q)</math>. This distribution measures the probability that two copies of the system, equilibrating in the same disordered environment, end up having overlap <math>q</math>. In the Ising case, a low temperature one has  <math> q_{\alpha \alpha}=m^2</math> and <math> q_{\alpha \neq \beta}=-m^2</math>, and thus <math> \overline{P}(q)</math> has two peaks at <math> \pm m^2</math>. </li>
</ul>


<div style="font-size:89%">
: <small>[*]</small> - Notice that to be precise, in analogy with the magnetization, we should write  <math>
q_{EA}=\lim_{\epsilon \to 0}\lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i^1 \, \sigma_i^2 \rangle_\epsilon,
</math>
where <math> \vec{\sigma}^1, \vec{\sigma}^2 </math> are two copies of the system, and the average is with respect to a tilted Boltzmann measure which contains a small coupling <math> \epsilon </math> between them, which plays the same role of the infinitesimal magnetic fields in the ferromagnet  The coupling is so weak that it only forces the configurations to fall in the same pure state, but a part from this it leaves them independent.
</div>
<br>

== Problems ==
=== Problem 3.1: the RS (Replica Symmetric) calculation===

We go back to the saddle point equations for the spherical <math>p</math>-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:

<center>
<math>
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}
</math>
</center>
Under this assumption, there is a unique saddle point variable, that is <math>q_0</math>. We denote with <math>q_0^{SP}</math> its value at the saddle point.


<ol>
<li><em> RS overlap distribution. </em>
Under this assumption, what is the overlap distribution <math>\overline{P}(q)</math> and what is <math>q_{EA}</math>? In which sense the RS ansatz corresponds to assuming the existence of a unique pure state?
</li>
</ol>
<br>

<ol start="2">
<li> <em> Self-consistent equations. </em>
Check that the inverse of the overlap matrix is 
<center>
<math>
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}
</math>
</center>
Compute the saddle point equation for  <math>q_0</math> in the limit  <math>n \to 0</math>, and show that this equation admits always the solution <math>q_0= 0</math>: why is this called the <em>paramagnetic</em> solution?
</li>
</ol>
<br>

<ol start="3">
<li><em> RS free energy. </em>
Compute the free energy corresponding to the solution <math>q_0= 0</math>, and show that it reproduces the annealed free energy. Do you have an interpretation for this?
</li>
</ol>
<br>

=== Problem 3.2: the 1-RSB (Replica Symmetry Broken) calculation===
In the previous problem, we have chosen a certain parametrization of the overlap matrix <math>Q</math>, which corresponds to assuming that typically all the copies of the systems fall into configurations that are at overlap  <math>q_0</math> 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 <math>n</math> replicas fall into configurations that are organized in <math>n/m</math> groups of size <math>m</math>; pairs of replicas in the same group are more strongly correlated and have overlap <math>q_1</math>, while pairs of replicas belonging to different groups have a smaller overlap <math>q_0<q_1</math>.  This corresponds to the following block structure for the overlap matrix:
<center>
<math>
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}
</math>
</center>
Here we have three parameters: <math>m, q_0, q_1</math> (in the sketch above, <math>m=3</math>). We denote with <math>m^{SP}, q_0^{SP}, q_1^{SP}</math> their values at the saddle point.





<ol>
<li> <em> 1-RSB overlap distribution. </em> Show that in this case the overlap distribution is
<center>
<math>\overline{P}(q)= m^{SP} \delta(q-q_0^{SP})+ (1-m^{SP})\delta(q-q_1^{SP})
</math>
</center>
What is <math> q_{EA}</math>? In which sense the parameter <math> m</math> can be interpreted as a probability weight?
</li>
</ol>
<br>

<ol start="2">
<li><em> 1-RSB free energy. </em> 
Using that 
<center>
<math>\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]</math>
</center>
show that the free energy now becomes:
<center>
<math>
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]
</math>
</center>
Under which limit this reduces to the replica symmetric expression?
</li>
</ol>
<br>

<ol start="3">
<li><em> Self-consistent equations. </em> 
Compute the saddle point equations with respect to the parameter <math> q_0, q_1 </math> and <math> m </math> are. Check that <math> q_0=0</math> is again a valid solution of these equations, and that for <math> q_0=0</math> the remaining equations reduce to:
<center>
<math>
(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
</math>
</center>
How does one recover the paramagnetic solution?
</li>
</ol>
<br>

<ol start="4">
<li><em> The transition. </em> 
We now look for a solution different from the paramagnetic one. To begin with, we set  <math> m=1 </math> to satisfy the first equation, and look for a solution of 
<center>
<math>
\frac{\beta^2}{2} q_1^p + \log \left(1-q_1\right)+ q_1=0
</math>
</center>
Plot this function for <math> p=3</math> and different values of  <math> \beta</math>, and show that there is a critical temperature <math> T_c</math> where a solution <math> q_1 \neq 0</math> appears: what is the value of this temperature (determined numerically)?
</li>
</ol>
<br>

== Check out: key concepts ==

Order parameters, ergodicity breaking, pure states, overlaps, overlap distribution, replica-symmetric ansatz, replica symmetry breaking.

== To know more ==
* Castellani, Cavagna. Spin-Glass Theory for Pedestrians [https://arxiv.org/abs/cond-mat/0505032]
* Parisi. Order parameter for spin-glasses [[Media:Parisi - OrderParameter.pdf| [2] ]]
* Zamponi. Mean field theory of spin glasses [https://arxiv.org/abs/1008.4844]-->





<!--<strong>Goal: </strong> Understand some physical properties of mean-field spin glasses in the low-T phase: the structure of the free-energy landscape, the response of the system to applied magnetic fields.
<br>

<strong>Techniques: </strong> linear response, fluctuation-dissipation relation.
<br>



== Replica solutions: a classification ==

[[File:Sketch RSB.png|thumb|right|x500px|Decomposition of configuration space into states]]
<ul>
<li> <strong>1-RSB.</strong> We have seen an example of mean-field model, the spherical <math>p</math>-spin, in which the low-T phase is glassy, described by a <ins>1-RSB ansatz</ins> of the overlap matrix. The equilibrium in the glassy phase is described by three quantities: the typical overlap  <math> q_1^{SP} </math> between configurations belonging to the same pure state, the typical overlap <math> q_0^{SP} </math> between configurations belonging to different pure states, and the probability  <math> (1-m^{SP}) </math> 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 <math> q_0^{SP}=0,  q_1^{SP}=1</math> and  <math> m^{SP}=T/T_c </math>. </li><br>


<li><strong>full-RSB. </strong> The <ins>Sherrington-Kirkpatrick model</ins> 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:
<center>
<math>
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}
</math>
</center>
which assumes that replicas are split into <math>m_1</math> blocks, and that inside each block they are further split into <math>m_2<m_1</math> blocks (in the example above,<math>m_1=4, m_2=2</math>). This is also not the correct structure for the SK model. The correct one obtained iterating this procedure an <ins>infinite number of times</ins>, as understood by Parisi [https://iopscience.iop.org/article/10.1088/0305-4470/13/4/009/meta]. Iterating K times, one ends up with a series of overlaps <math>q_K > q_{K-1} > \cdots >q_0</math>. The underlying picture of the free-energy landscape is as follows: equilibrium states have self-overlap <math>q_K = q_{EA}</math>. They are arranged in clusters such that states inside a cluster have overlap <math>q_{K-1}</math>, but such clusters are arranged in other clusters at a higher level, at mutual overlap <math>q_{K-2}</math> and so on. The Parisi solution of the Sherrington Kirkpatrick model is obtained taking <math>K \to \infty</math>. In this limit, the overlap distribution becomes a <ins>continuous distribution</ins>.</li><br>

<li><strong>The underlying physics. </strong> System having a 1-RSB structure differ from those having a  full-RSB structure by several properties at equilibrium, as well as of their dynamics.
</ul>
<br>

== Problems ==
In this problem, we go back to the susceptibilities that were introduced in Lecture 1, and discuss how to compute them within the replica theory.

=== Problem 4.1: magnetic susceptibilities and linear response ===
In an Ising system, the thermodynamic magnetic susceptibility is defined as
<center>
<math>
\chi_{eq}= \frac{d m( h)}{dh}\Big|_{h=0}
</math>
</center>
where <math> m(h)=\lim_{N \to \infty} m_N(h)</math> is the magnetization at a given inverse temperature <math> \beta</math> and external field <math> h</math>, and <math> m_N(h)</math> its finite-size counterpart. The function <math>\chi_{eq}</math> measures the response to changes of the magnetic field.


<ol>
<li>
<em> Ferromagnetism. </em> Using the self-consistent equation for the magnetization in the mean-field Ising model, <math> m= \text{tanh}[\beta(h+m)]</math>, show that <math>\chi_{eq}</math> diverges exactly at the transition temperature <math>\beta=\beta_c=1</math>.
</li>
</ol>


<ol start="2">
<li>
<em> Spin glass. </em>By the Fluctuation-Dissipation Theorem (FDT), we know that the response and correlations at equilibrium are related by
<center>
<math>
\chi_N= \frac{d m_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 }.
</math>
</center>
In a spin glass, by symmetry with respect to sign flips of the couplings it holds <math>\overline{\langle \sigma_i \sigma_j \rangle_c}= 0</math> for  <math> i \neq j  </math>. Show that one can write
<center>
<math>
\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})}</math>
</center>
and thus that this quantity <em>does not</em> diverge at the transition to the spin-glass phase. 
</li>
</ol>

[[File:Suscept-experiment.png|thumb|left|x150px|The ZFC (lower curve) and FC (upper curve) susceptibility. Experimental results taken from [https://arxiv.org/abs/cond-mat/9810314]]]


<ol start="3">
<li><em> Linear response. </em> The quantity  <math>\chi_{eq} </math> 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. <sup>[[#Notes|[*] ]]</sup> In linear response, one would measure however the response <math>\chi_{LR} </math> of the system which is confined to 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 <math>\chi_{LR}< \chi_{eq}</math>?
</li>
</ol>
<br>

<ol start="4">
<li><em> Experiments. </em>The plot on the left shows experimental measurements of the magnetic susceptibility in a spin-glass. The two curves correspond to two different protocols: (i) ZFC (zero-field cooled) protocol:  cool the system at low T, 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?
</li>
</ol>



<div style="font-size:89%">
: <small>[*]</small> - 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 <math>\omega_\alpha </math>.
</div>

== Check out: key concepts==
an idea of the Parisi solution, magnetic susceptibility, linear response.


== References ==
* Parisi. Order parameter for spin-glasses [[Media:Parisi - OrderParameter.pdf| [2] ]]
* Castellani, Cavagna. Spin-Glass Theory for Pedestrians [https://arxiv.org/abs/cond-mat/0505032]
* Zamponi. Mean field theory of spin glasses [https://arxiv.org/abs/1008.4844]
* Comment: the diverging susceptibility
<div style="font-size:89%">
The magnetic susceptibility does not diverge at the spin-glass critical temperature. One can identify a more complicated object, the <em> spin-glass susceptibility </em>, 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, <math> q_{EA} </math>, is itself a 2-point function, while the magnetization that is a 1-point function..
</div>-->