T-3: Difference between revisions

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<!--<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
<center>
<math>
q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle S_i \rangle_\alpha \langle S_i \rangle_\alpha
</math>
</center>
Notice that to be precise, in analogy with the magnetization, we should write
<center>
<math>
q_{EA}=\lim_{\epsilon \to 0}\lim_{N \to \infty}\frac{1}{N}\sum_i \langle S_i^1 \, S_i^2 \rangle_\epsilon,
</math>
</center>
where <math> \vec{S}^1, \vec{S}^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 <sup>[[#Notes|[*] ]]</sup>. 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 <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> - 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]-->

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