T-II-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

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}=\frac{1}{N}\sum_i \overline{\langle \sigma_i \rangle^2} }

This quantity is a measure of ergodicity breaking: when ergodicity is broken, the Boltzmann measure clusters into pure states $\alpha$, meaning that one can re-write the sum over all configurations as a sum over 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 Z= \int d \vec{\sigma} e^{-\beta E[\vec{\sigma}]}=\sum_{\alpha} Z_\alpha, \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 Ising model at low temperature there are two pure states, 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:

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

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

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

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), }

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 $Q$ thus encodes a lot of information on the system; in particular, they contain the information on whether spin glass order emerges:

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 \overline{P}(q)} is the probability that two copies of the system, equilibrating in the same disordered environment, end up having overlap q with each others. 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)} is an indicator of ergodicity breaking in the systems: it indicates that the Boltzmann measure is partitioned into several pure states (like the $+m$ 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} states in the Ising case, 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), and it measures the typical overlaps between configurations belonging to the same, or different states. In Ising, ${\overline {P}}(q)=\delta (q-m^{2})$.

Problem 3.1: the RS (Replica Symmetric) calculation

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} .

1. Check that the inverse of the overlap matrix is

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

   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?

2. 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?

1. Overlpa interpretation

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 formula 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} ).

1. 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:

   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?

2. 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 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 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:

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

   How does one recover the paramagnetic solution?

3. 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 $p=3$ 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 $q_{1}\neq 0$ appears: what is the value of this temperature (determined numerically)?