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 $Z$, 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_{ ann} } instead controls the scaling of the average value of $Z$. 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_{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 $x^{n}=e^{n\log(x)}=1+n\log x+O(n^{2})$. 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 quenched 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 $n\to 0$. The annealed one only requires to do the calculation 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 n=1} .

Problem 1: the annealed free energy

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 ${\overline {E({\vec {\sigma }})E({\vec {\tau }})}}=Nq({\vec {\sigma }},{\vec {\tau }})^{p}/2+o(1)$, 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 ${\overline {Z}}$ 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 ${\overline {e^{\alpha X}}}=e^{\frac {\alpha ^{2}\sigma ^{2}}{2}}$.

3. Entropy contribution. The volume of a sphere of radius 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 \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 $N^{\frac {N}{2}}\pi ^{\frac {N}{2}}/\left({\frac {N}{2}}\right)!$. 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

1. Step 1: average over the disorder. ccc

2. Step 2: identify the order parameter. cccc

3. Step 3: the saddle point. cccc