T-2

In this set of problems, we use the replica method to study the equilibrium properties of a prototypical toy model of glasses, the spherical $p$-spin model.

Problem 1: getting acquainted with the model

In the spherical $p$-spin model the configurations ${\vec {\sigma }}=(\sigma _{1},\cdots ,\sigma _{N})$ that the system can take satisfy the spherical constraint $\sum _{i=1}^{N}\sigma _{i}^{2}=N$, and the energy associated to each configuration is

$E({\vec {\sigma }})=\sum _{1\leq i_{1}\leq i_{2}\leq \cdots i_{p}\leq N}J_{i_{1}\,i_{2}\cdots i_{p}}\sigma _{i_{1}}\sigma _{i_{2}}\cdots \sigma _{i_{p}},$

where the coupling constants $J_{i_{1}\,i_{2}\cdots i_{p}}$ are independent random variables with Gaussian distribution with zero mean and variance $p!/(2N^{p-1}),$ and $p\geq 3$ is an integer.

1. Energy correlations. At variance with the REM, in the spherical $p$-spin the energies at different configurations are correlated. Show 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 \overline{E(\vec{\sigma}) E(\vec{\tau})}= N q(\vec{\sigma}, \vec{\tau})^p/2 + o(1) } , where $q({\vec {\sigma }},{\vec {\tau }})={\frac {1}{N}}\sum _{i=1}^{N}\sigma _{i}\tau _{i}$ is the overlap between the two configurations. Why an we say that for $p\to \infty$ this model converges with the REM discussed in the previous TD?

2. Self-averaging. For $|\epsilon |\leq {\sqrt {\log 2}}$ the quantity ${\mathcal {N}}$ is self-averaging: its distribution concentrates around the average value ${\overline {\mathcal {N}}}$ when 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 \infty } . Show that this is the case by computing the second moment 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{\mathcal{N}^2} } and using the central limit theorem. Show that this is no longer true in the region where the annealed entropy is negative: why does one expect fluctuations to be relevant in this region?

3. Rare events vs typical values. 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 |\epsilon| > \sqrt{\log 2} } the annealed entropy is negative: the average number of configurations with those energy densities is exponentially small in $N$. This implies that the probability to get configurations with those energy is exponentially small 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 } : these configurations are rare. Do you have an idea of how to show this, using the expression 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 \overline{\mathcal{N}}} ? What is the typical value of ${\mathcal {N}}$ in this region? Justify why the point where the entropy vanishes coincides with the ground state energy of the model.

Comment: this analysis of the landscape suggests that in the large 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 } limit, the fluctuations due to the randomness become relevant when one looks at the bottom of their energy landscape, close to the ground state energy density. We show below that this intuition is correct, and corresponds to the fact that the partition function 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 } has an interesting behaviour at low temperature.

Problem 2: the annealed free energy

We now compute the equilibrium phase diagram of the model, and in particular the quenched free energy density $f$ which controls the scaling of the typical value of the partition function, 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 \sim e^{-N \beta \, f +o(N) } } . We show that the free energy 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 f = \begin{cases} &- \left( T \log 2 + \frac{1}{4 T}\right) \quad \text{if} \quad T \geq T_c\\ & - \sqrt{\log 2} \quad \text{if} \quad T <T_c \end{cases} \quad \quad T_c= \frac{1}{2 \sqrt{\log 2}}. }

At $T_{c}$ a transition occurs, often called freezing transition: in the whole low-temperature phase, the free-energy is “frozen” at the value that it has at the 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= T_c } .

1. The thermodynamical transition and the freezing. The partition function the REM reads 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 = \sum_{\alpha=1}^{2^N} e^{-\beta E_\alpha}= \int dE \, \mathcal{N}(E) e^{-\beta E}. } Using the behaviour of the typical value of ${\mathcal {N}}$ determined in Problem 1, derive the free energy of the model (hint: perform a saddle point calculation). What is the order of this thermodynamic transition?

2. Entropy. What happens to the entropy of the model when the critical temperature is reached, and in the low temperature phase? What does this imply for the partition function 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} ?

3. Fluctuations, and back to average vs typical. Similarly to what we did for the entropy, one can define an annealed free energy 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^A } from ${\overline {Z}}=e^{-N\beta f^{A}+o(N)}$: show that in the whole low-temperature phase this is smaller than the quenched free energy obtained above. Putting all the results together, justify why the average of the partition function in the low-T phase is "dominated by rare events".

Comment: the low-T phase of the REM is a frozen phase, characterized by the fact that the free energy is temperature independent, and that the typical value of the partition function is very different from the average value. In fact, the low-T phase is also 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 glass phase } in the sense discussed in the lecture. It is characterized by the fact that Replica Symmetry is broken, as one sees explicitly by re-deriving the free energy through the replica method. We go back to this in the next lectures/TDs.

Problem 3: the quenched free energy

In this final exercise, we show how the freezing transition can be understood in terms of extreme valued statistics (discussed in the lecture) and localization. We consider the energies of the configurations and define 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 E_\alpha= - N \sqrt{\log 2} + \delta E_\alpha } , so that

${Z}=e^{\beta N{\sqrt {\log 2}}}\sum _{\alpha =1}^{2^{N}}e^{-\beta \delta E_{\alpha }}=e^{\beta N{\sqrt {\log 2}}}\sum _{\alpha =1}^{2^{N}}z_{\alpha }$

We show 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 Z } is a sum of random variables that become heavy tailed 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 T < T_c } , implying that the central limit theorem is violated and this sum is dominated by few terms, the largest ones. This can be interpreted as the occurrence of localization.

1. Heavy tails and concentration. Compute the distribution of the variables $\delta E_{\alpha }$ and show 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 (\delta E)^2/N \ll 1 } this is an exponential. Using this, compute the distribution of the $z_{\alpha }$ and show that it is a power law, 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(z)= \frac{c}{z^{1+\mu}} \quad \quad \mu= \frac{2 \sqrt{\log 2}}{\beta} } When 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 < T_c } , 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 \mu<1 } : the distribution of $z$ becomes heavy tailed. What does this imply for the sum 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 } ? Is this consistent with the behaviour of the partition function and of the entropy discussed in Problem 2? Why can one talk about a localization or condensation transition?

2. Inverse participation ratio. The low temperature behaviour of the partition function an be characterized in terms of a standard measure of localization (or condensation), the Inverse Participation Ratio (IPR) 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 IPR= \frac{\sum_{\alpha=1}^{2^N} z_\alpha^2}{[\sum_{\alpha=1}^{2^N} z_\alpha]^2}= \sum_{\alpha=1}^{2^N} \omega_\alpha^2 \quad \quad \omega_\alpha=\frac{ z_\alpha}{\sum_{\alpha=1}^{2^N} z_\alpha}. }

   When $z$ is power law distributed with exponent 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 \mu } , one can show (HOMEWORK!) 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 IPR= \frac{\Gamma(2-\mu)}{\Gamma(\mu) \Gamma(1-\mu)}. }

   Discuss how this quantity changes across the transition 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 \mu=1 } , and how this fits with what you expect in general in a localized phase.