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Problem 1: Ginzburg criterion

This problem explores Ginzburg criterion for the validity of mean-field theory in arbitrary spatial dimension and its expression in terms of experimentally measurable quantities.

(a) The general solution for the correlation function in arbitrary spatial dimension within the mean-field theory can be written as:

assuming that distance is much larger than the lattice spacing . Generalize the Ginzburg criterion

for the validity of the mean-field theory to arbitrary spatial dimension to show that it is satisfied if

.

where and are critical exponents for describing vanishing of the order parameter and divergence of the correlation length , respectively.

(b) Using your result in (a), find the upper critical dimension for the Ising model above which its critical behavior near temperature is well-described by the mean-field theory.


(c) The Ginzburg criterion can be expressed in terms of the measurable quantities, such as critical temperature , the correlation length at , and the jump (predicted by the mean-field theory) in the specific heat at .

Problem 2: Predictions of the Landau theory for the critical exponents and

(a) Starting from the Gibbs free energy density in Landau theory:

show that isothermal susceptibility is given by for and for , so that critical exponent according to Landau mean-field theory. HINT: Find the value of the order parameter which minimizes and use .

(b) Show that at the critical point, and hence critical exponent , where is defined by .

Problem 3: Renormalization group for 1D Ising model using transfer matrix method

The partition function for the N-spin Ising chain can be written as the trace of the N-th power of the transfer matrix T. Another way to reduce the number of degrees of freedom is the describe the system in terms of two-spin cells, where the partition function is written as:

The transfer matrix for two-spin cells, , can be written as: