T-3

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

Key concepts: order parameters, ergodicity breaking, pure states, overlap distribution, replica-symmetric ansatz, replica symmetry breaking.

The order parameters: overlaps, and their meaning

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

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

  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 $\langle \cdot \rangle _{h}$ 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 $h$ 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.


• 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 $q_{EA}$ 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 \, \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 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 \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 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} , where $m$ is the magnetization.


• 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:

  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)=\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 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 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)} . 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 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 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 \neq \beta}=-m^2} , and thus ${\overline {P}}(q)$ 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 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 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 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 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}} its value at the saddle point.

1. RS overlap distribution. Under this assumption, what is the overlap distribution ${\overline {P}}(q)$ 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. 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 $q_{0}$ 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 $q_{0}=0$, 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 $n$ 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 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} ; 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 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:

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_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, 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. 1-RSB overlap distribution. Show that in this case the overlap distribution 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 \overline{P}(q)= m^{SP} \delta(q-q_0^{SP})+ (1-m^{SP})\delta(q-q_1^{SP}) }

   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. 1-RSB free energy. 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 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 } are. Check 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 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. 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)?

References

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