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.

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 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!/ (2 N^{p-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 p \geq 3} is an integer.

Quenched vs annealed

In TD1, we defined the quenched free energy density as the quantity controlling 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 } , which means:

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= -\lim_{N \to \infty} \frac{1}{\beta N} \overline{ \log Z}. }

The 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_{\rm ann} } instead controls the scaling of the average value 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 Z } . It is defined 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 f_{\rm ann} = -\lim_{N \to \infty} \frac{1}{\beta N} \log \overline{Z}. }

These formulas differ by the order in which the logarithm and the average over disorder are taken. Computing the average of the logarithm is in general a hard problem, which one can address by using a smart representation of the logarithm, that goes under the name of replica trick:

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 x= \lim_{n \to 0} \frac{x^n-1}{n} }

which can be easily shown to be true by Taylor expanding 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 x^n= e^{n \log(x)}= 1+ n\log x+ O(n^2) } . Applying this to the average of the partition function, we see 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 f= -\lim_{N \to \infty} \lim_{n \to 0}\frac{1}{\beta N n} \frac{\overline{Z^n}-1}{n}. }

Therefore, to compute the free-energy we need to compute the moments 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{Z^n}}} and then take 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} .

Problem 1: the annealed free energy

Let us compute this quantity.

1. Energy correlations. At variance with the REM, in 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 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 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(\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 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 \to \infty } this model converges with the REM discussed in the previous TD?

2. Energy contribution. Show that computing 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{Z}} boils down to computing the average 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^{-\beta J_{i_1 \, \cdots i_p} \sigma_{i_1} \cdots \sigma_{i_p}}}} . Compute this average. Hint: if X is a centered Gaussian variable with variance 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 \sigma^2} , then 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^{\alpha X}}=e^{\frac{\alpha^2 \sigma^2}{2} }} .

3. Entropy contribution. The volume of a sphere of radius ${\sqrt {N}}$ in dimension 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} is given 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 N^{\frac{N}{2}} \pi^{\frac{N}{2}}/\left(\frac{N}{2}\right)!} . Use the large-N asymptotic of this to conclude the calculation of the annealed free energy: the final result is only slightly different with respect to the free-energy of the REM in the high-temperature phase: can you identify the source of this difference?

Problem 2: the replica trick and the quenched free energy

The quenched free energy density is the quantity controlling the scaling of the typical value of the partition function $Z$. This means that

This calculation can be done in 3 main steps.

1. Step 1: average over the disorder. ccc

2. Step 2: identify the order parameter. cccc

3. Step 3: the saddle point. cccc