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<math>\overline{Z^n}= \int_{S_N} \prod_{a=1}^n d \vec{\sigma}^a e^{\frac{\beta^2}{ | <math>\overline{Z^n}= \int_{S_N} \prod_{a=1}^n d \vec{\sigma}^a e^{\frac{\beta^2}{2} \sum_{a,b=1}^n \left[q({\vec{\sigma}^a, \vec{\sigma}^b})\right]^p}</math></center> | ||
Justify why averaging over the disorder induces a coupling between the replicas. | Justify why averaging over the disorder induces a coupling between the replicas. |
Revision as of 19:51, 30 January 2024
Goal: use the replica method to study the equilibrium properties of a prototypical mean-field model of glasses, the spherical -spin model.
Techniques: replica trick, Gaussian integrals, saddle point calculations.
Quenched vs annealed, and the replica trick
In Problems 1 and 2, we defined the quenched free energy as the quantity controlling the scaling of the typical value of the partition function , which means:
The annealed free energy instead controls the scaling of the average value of . It is defined by
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:
which can be easily shown to be true by Taylor expanding . Applying this to the average of the partition function, we see that
Therefore, to compute the quenched free-energy we need to compute the moments and then take the limit . The calculations in the following Problems rely on this replica trick. The annealed one only requires to do the calculation with .
Problems
In this and the following set of problems, we analyse a mean-field model that is slightly more complicated than the REM: the spherical -spin model. In the spherical -spin model the configurations satisfy the spherical constraint , and the energy associated to each configuration is
where the coupling constants are independent random variables with Gaussian distribution with zero mean and variance and is an integer.
Problem 3.1: correlations, p-spin vs REM
- Energy correlations. At variance with the REM, in the spherical -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{\sigma}')}= N \, q(\vec{\sigma}, \vec{\sigma}')^p + O(1) , \quad \quad \text{where} \quad \quad q(\vec{\sigma}, \vec{\sigma}')= \frac{1}{N}\sum_{i=1}^N \sigma_i \sigma'_i } is the overlap between the two configurations. Why can 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 lecture?
Problem 3.2: the annealed free energy. DO IT YOURSELF!
As a preliminary exercise, we compute the annealed free energy of the spherical -spin model.
- 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}}}}
, which is a Gaussian integral. Compute this average.
Hint: if 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} is a centered Gaussian variable with variance , 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 v}{2} }} . - Entropy contribution. The volume of a sphere 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 \mathcal{S}_N}
of radius 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- asymptotic of this to conclude the calculation of 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_{\text{a}}= - \frac{1}{\beta}\left( \frac{\beta^2}{2}+ \frac{1}{2}\log (2 \pi e)\right). } This result is only slightly different with respect to the annealed free-energy of the REM: can you identify the source of this difference?
Problem 3.3: the replica trick and the quenched free energy
In this Problem, we set up the replica calculation of the quenched free energy. This will be done in 3 steps, that are general in each calculation of this type.
- Step 1: average over the disorder. By using the same Gaussian integration discussed above, show that 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}
-th moment of the partition function is
Justify why averaging over the disorder induces a coupling between the replicas.
- Step 2: identify the order parameter. Using the identity 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 1=\int dq_{ab} \delta \left( q({\vec{\sigma}^a, \vec{\sigma}^b})- q_{ab}\right) }
, 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{Z^n}}
can be rewritten as an integral over variables only, 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 \overline{Z^n}= \int \prod_{a<b} d q_{ab} e^{\frac{N\beta^2}{4} \sum_{a,b=1}^n q_{ab}^p+ \frac{Nn}{2} \log (2 \pi e) + \frac{N}{2} \log \det Q+ o(N)} \equiv \int \prod_{a<b} d q_{ab} e^{N \mathcal{A}[Q]+ o(N)}, \quad \quad Q_{ab} \equiv \begin{cases} &q_{ab} \text{ if } a <b\\ &1 \text{ if } a =b\\ &q_{ba}\text{ if } a >b \end{cases}} In the derivation, you can use the fact 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 \int_{S_N} \prod_{a=1}^n d \vec{\sigma}^a \, \prod_{a<b}\delta \left(q({\vec{\sigma}^a, \vec{\sigma}^b})- q_{ab} \right)= e^{N S[Q]+ o(N)}} , where . The 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} is an order parameter: explain the analogy with the magnetisation in the mean-field solution of the Ising model.
- Step 3: the saddle point (RS). For large N, the integral can be computed with a saddle point approximation for general 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}
. The saddle point variables are the matrix elements 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 a \neq b}
. Show that the saddle point equations read
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{\partial \mathcal{A}[Q]}{\partial q_{ab}}=\frac{\beta^2}{4}p q_{ab}^{p-1}+ \frac{1}{2} \left(Q^{-1}\right)_{ab}=0 \quad \quad \text{for } \quad a \neq b } To solve these equations and get the free energy, one first needs to make an assumption on the structure of the matrix , i.e., on the space where to look for the saddle point solutions. We discuss this in the next set of problems.
Check out: key concepts
Annealed vs quenched averages, replica trick, fully-connected models, order parameters, the three steps of a replica calculation.
To know more
- Castellani, Cavagna. Spin-Glass Theory for Pedestrians [1]