T-3

Goal: 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.

The order parameters: overlaps, and their meaning

• Order parameter, ergodicity-breaking, pure states: the ferromagnet. The order parameter for ferromagnets is the magnetization. It is defined 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 m=\lim_{h \to 0} \lim_{N \to \infty}\frac{1}{N}\sum_{i=1}^N \overline{\langle S_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. Notice the order of limits in the definition: 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.
  

  
  A non-zero magnetisation is also connected to ergodicity breaking, which is a dynamical concept: 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, following some 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 pure states (labelled by 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 } ) 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 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 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. The free energy barrier that one has to overcome to go from one state to the other diverges when $N\to \infty$, and thus the system is dynamically trapped only in one state.



• Order parameter, ergodicity-breaking, pure states: the spin-glass. In Lecture 1, we have introduced the Edwards-Anderson order parameter as: $q_{EA}=\lim _{t\to \infty }\lim _{N\to \infty }{\frac {1}{N}}\sum _{i}S_{i}(0)S_{i}(t)$

  
  

  
  

  $q_{EA}=\lim _{N\to \infty }{\frac {1}{N}}\sum _{i}{\overline {\langle \sigma _{i}\rangle ^{2}}}$

  which plays the same role as
  

  
  

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

  $\langle A\rangle =\sum _{\alpha }\omega _{\alpha }\langle A\rangle _{\alpha },\quad \quad \quad \omega _{\alpha }={\frac {Z_{\alpha }}{Z}},\quad \quad \quad Z_{\alpha }=\int _{{\vec {\sigma }}\in {\text{ state }}\alpha }d{\vec {\sigma }}e^{-\beta E[{\vec {\sigma }}]}=\langle e^{-\beta E[{\vec {\sigma }}]}\rangle _{\alpha }$

  In a mean-field spin glass, there are more than two pure states. The quantity $q_{EA}$ measures the overlap between configurations belonging to the same pure state, that one expects to be the same for all states:
  

  $q_{EA}=q_{\alpha \alpha }=\lim _{N\to \infty }{\frac {1}{N}}\sum _{i}\langle \sigma _{i}\rangle _{\alpha }\langle \sigma _{i}\rangle _{\alpha }$

  Notice that to be precise, we should write

  $q_{EA}=\lim _{\epsilon \to 0}\lim _{N\to \infty }{\frac {1}{N}}\sum _{i}\langle \sigma _{i}^{1}\,\sigma _{i}^{2}\rangle _{\epsilon },$

  where $\sigma ^{1},\sigma ^{2}$ are two copies of the system, and the average is with respect to a tilted Boltzmann measure which contains a small coupling $\epsilon$ between them. 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. 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 $m$ is the magnetization.


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

  $q_{\alpha \beta }={\frac {1}{N}}\sum _{i}\langle \sigma _{i}\rangle _{\alpha }\langle \sigma _{i}\rangle _{\beta },\quad \quad \quad {P}(q)=\sum _{\alpha ,\beta }\omega _{\alpha }\,\omega _{\beta }\,\delta (q-q_{\alpha \beta }).$

  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 $Q_{ab}^{SP}$ 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 $Q$ 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 same disordered environment, end up having overlap $q$. In the Ising case, a low temperature one has $q_{\alpha \alpha }=m^{2}$ and $q_{\alpha \neq \beta }=-m^{2}$, and thus ${\overline {P}}(q)$ has two peaks at $\pm m^{2}$.

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:

$Q={\begin{pmatrix}1&q_{0}&q_{0}\cdots &q_{0}\\q_{0}&1&q_{0}\cdots &q_{0}\\&\cdots &&\\q_{0}&q_{0}&q_{0}\cdots &1\end{pmatrix}}$

Under this assumption, there is a unique saddle point variable, that is $q_{0}$. We denote with $q_{0}^{SP}$ its value at the saddle point.

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

2. Self-consistent equations. Check that the inverse of the overlap matrix 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^{-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 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} : why is this called the paramagnetic solution?

3. RS free energy. 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 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} 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 $q_{1}$, while pairs of replicas belonging to different groups have a smaller overlap $q_{0}<q_{1}$. 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: $m,q_{0},q_{1}$ (in the sketch above, $m=3$). We denote with $m^{SP},q_{0}^{SP},q_{1}^{SP}$ their values at the saddle point.

1. 1-RSB overlap distribution. Show that in this case the overlap distribution is

   ${\overline {P}}(q)=m^{SP}\delta (q-q_{0}^{SP})+(1-m^{SP})\delta (q-q_{1}^{SP})$

   What is $q_{EA}$? In which sense the parameter $m$ can be interpreted as a probability weight?

2. 1-RSB free energy. Using that

   $\log \det Q=n{\frac {m-1}{m}}\log(1-q_{1})+{\frac {n-m}{m}}\log[m(q_{1}-q_{0})+1-q_{1}]+\log \left[nq_{0}+m(q_{1}-q_{0})+1-q_{1}\right]$

   show that the free energy now becomes:

   $f_{1RSB}=-{\frac {1}{2\beta }}\left[{\frac {\beta ^{2}}{2}}\left(1+(m-1)q_{1}^{p}-mq_{0}^{p}\right)+{\frac {m-1}{m}}\log(1-q_{1})+{\frac {1}{m}}\log[m(q_{1}-q_{0})+1-q_{1}]+{\frac {q_{0}}{m(q_{1}-q_{0})+1-q_{1}}}\right]$

   Under which limit this reduces to the replica symmetric expression?

3. Self-consistent equations. Compute the saddle point equations with respect to the parameter $q_{0},q_{1}$ and $m$ are. Check that $q_{0}=0$ is again a valid solution of these equations, and that for $q_{0}=0$ the remaining equations reduce to:

   $(m-1)\left[{\frac {\beta ^{2}}{q}}pq_{1}^{p-1}-{\frac {1}{m}}{\frac {1}{1-q_{1}}}+{\frac {1}{m}}{\frac {1}{1+(m-1)q_{1}}}\right]=0,\quad \quad {\frac {\beta ^{2}}{2}}q_{1}^{p}+{\frac {1}{m^{2}}}\log \left({\frac {1-q_{1}}{1+(m-1)q_{1}}}\right)+{\frac {q_{1}}{m[1+(m-1)q_{1}]}}=0$

   How does one recover the paramagnetic solution?

4. The transition. We now look for a solution different from the paramagnetic one. To begin with, we set $m=1$ to satisfy the first equation, and look for a solution of

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

   Plot this function for $p=3$ and different values of $\beta$, and show that there is a critical temperature $T_{c}$ where a solution $q_{1}\neq 0$ appears: what is the value of this temperature (determined numerically)?

Check out: key concepts of this TD

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

References

• Castellani, Cavagna. Spin-Glass Theory for Pedestrians [1]
• Zamponi. Mean field theory of spin glasses [2]