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| == Problem 1: Predictions of the Landau theory for the critical exponents <math> \gamma </math> and <math> \delta </math> ==
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| (a) Starting from the Gibbs free energy density in Landau theory:
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| <math> g(T,m) = a(T) + \frac{b(T)}{2}m^2 + \frac{c(T)}{4} m^4 - hm </math>
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| show that isothermal susceptibility <math> \chi = (\partial m/\partial h)_T </math> is given by <math> \chi = 1/b_0(T-T_c) </math> for <math> T>T_c </math> and <math> \chi = 1/b_0(T-T_c) </math> for <math> T>T_c </math>, so that critical exponent <math> \gamma =1 </math> according to Landau mean-field theory. HINT: Find the value of the order parameter <math> m </math> which minimizes <math> g(T,m) </math> and use <math> b=b_0 (T-T_c) </math>.
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| (b) Show that <math> cm^3 = h </math> at the critical point, and hence critical exponent <math> \delta = 3 </math>, where <math> \delta </math> is defined by <math> m \sim H^{1/\delta} </math>.
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| ==Problem 2: Ginzburg criterion for the range of validity of mean-field theory ==
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| 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.
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| (a) The general solution for the correlation function in arbitrary spatial dimension <math> d </math> within the mean-field theory can be written as:
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| <math> G(r) \sim \frac{e^{-r/\xi}}{r^{d-2}} </math>
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| assuming that distance <math> r \gg a </math> is much larger than the lattice spacing <math> a </math>. Generalize the Ginzburg criterion
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| <math> \frac{G(r)}{\int m^2 d\mathbf{r}} \ll 1 </math>
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| for the validity of the mean-field theory to arbitrary spatial dimension <math> d </math> to show that it is satisfied if
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| <math> d>2+2\beta/\nu </math>.
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| where <math> \beta </math> and <math> \nu </math> are critical exponents for describing vanishing of the order parameter <math> m </math> and divergence of the correlation length <math> \xi </math>, respectively.
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| (b) Using your result in (a), find the ''upper critical dimension'' for the Ising model above which its critical behavior near temperature <math> T_c </math> is well-described by the mean-field theory.
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| (c) The Ginzburg criterion can be expressed in terms of the measurable quantities, such as critical temperature <math> T_c </math>, the correlation length <math> \xi_0 </math> at <math> T=0 </math>, and
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| the jump (predicted by the mean-field theory) in the specific heat <math> \Delta C </math> at <math> T=T_c </math>. Use Landau-Ginzburg solution for the correlation length <math> \xi^2 = - \lambda/2b = - lambda/[2b_0(T-T_c)] </math> at <math> T<T_c </math> to express it as
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| <math> \xi(T) = \xi_0 |t|^{-1/2} </math>
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| and show that the correlation length extrapolated to <math> T=0 </math> is given by:
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| <math> \xi_0^2 = \frac{\lambda}{2b_0 T_c} </math>.
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| Thus, the parameter <math> \lambda </math>, which measures the strength of fluctuations in the Landau-Ginzburg form of the free energy, can be eliminated in favor of measurable quantity <math> \xi_0 </math> and the parameter <math> b_0 </math>.
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| (d) Use Landau mean-field theory to show that <math> b_0 </math> can be expressed in terms of the jump <math> \Delta C </math> in the specific heat (see page 84 in Plischke and Bergersen textbook)
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| at <math> T_c </math>:
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| <math> b_0^2 = (2 c/T_c) \Delta C </math>.
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| (e) Using <math> m(T) = b_0 T_c |t|^{1/2}/c </math> from the Landau theory and your results in (c) and (d), show that the Ginzburg criterion derived in the class
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| <math> \frac{0.063 k_B T}{\lambda} \ll \xi(T) [m(T)]^2 </math>
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| can be expressed as
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| <math> \frac{0.016 k_B}{\xi_0^3 T_c \Delta C} \ll |t|^{1/2} </math>.
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| For example, in conventional superconductors <math> \xi_0 = 10^{-7} </math> (radius of the Cooper pair formed by two electrons), so that the Ginzburg criterion tells us that mean-field theory description is valid even for temperatures as close to <math> T_c </math> as <math> |t| \sim 10^{-14} </math>.
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| ==Problem 3: Renormalization group for 1D Ising model using transfer matrix method ==
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| In this problem you will apply renormalization group (for which [http://nobelprize.org/nobel_prizes/physics/laureates/1982/ the Nobel Prize in Physics 1982] was awarded to Kenneth G. Wilson) to 1D Ising model ''in the external magnetic field'' <math> h_\mathrm{ext} </math>. 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''' (page 78-79 in the Plischke & Bergersen textbook). 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:
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| <math> Z = \mathrm{Tr}\, \mathbf{T}^N = \mathrm{Tr}\, (\mathbf{T}^2)^{N/2} = \mathrm{Tr}\, \mathbf{T'}^{N/2} </math>
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| The transfer matrix for two-spin cells, <math> \mathbf{T}^2 </math>, can be written as:
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| <math> \mathbf{T}^2 = \mathbf{T} \mathbf{T} =
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| \begin{pmatrix}
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| e^{2K+2h} + e^{-2K} & e^{h} + e^{-h} \\
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| e^{-h} + e^{h} & e^{2K - 2h} + e^{-2K}
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| \end{pmatrix}
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| </math>.
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| where <math> K = \beta J </math> and <math> h = \beta h_\mathrm{ext} </math>.
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| We require that <math> \mathbf{T'} </math> has the same form as <math> \mathbf{T} </math>:
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| <math> \mathbf{T}' =
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| \begin{pmatrix}
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| e^{K'+h'} & e^{-K'} \\
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| e^{-K'} & e^{K'-h'}
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| \end{pmatrix}
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| </math>
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| where a parameter <math> C </math> must be introduced because matching of <math> \mathbf{T}' </math> and <math> \mathbf{T}^2 </math> requires to match three matrix elements of such symmetric <math> 2 \times 2 </math> matrices, which is impossible with only two variables <math> K' </math> and <math> h' </math>.
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| (a) Show that the three unknowns satisfy the three conditions:
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| <math>
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| \begin{align}
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| Ce^{K'}e^{h'}& = e^{2K + 2h} + e^{-2K},\\
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| Ce^{-K'} & = e^{h} + e^{-h} , \\
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| Ce^{K'}e^{-h'} & = e^{2K-2h} + e^{-2K}.
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| \end{align}
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| </math>
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| (b) Show that the solution of equations in (a) can be written as:
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| <math>
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| \begin{align}
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| e^{-2h'} & = \frac{e^{2K - 2h} + e^{-2K}}{e^{2K + 2h} + e^{-2K}}, \\
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| e^{4K'} & = \frac{e^{4K} + e^{-2h} + e^{2h} + e^{-4K}}{e^{h} + e^{-h}}, \\
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| C^4 & = [ e^{4K} + e^{-2h} + e^{2h} + e^{-4K}][e^h + e^{-h}]^2.
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| \end{align}
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| </math>
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| (c) Show that the recursion relations in (b) reduce to:
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| <math> K' = R(K) = \frac{1}{2} \ln [\cosh(2K)] </math> for <math> h=0 </math>. For <math> h \neq 0 </math>
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