Temporary HW

Problem 1: Predictions of the Landau theory for the critical exponents $\gamma$ and $\delta$

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

$g(T,m)=a(T)+{\frac {b(T)}{2}}m^{2}+{\frac {c(T)}{4}}m^{4}-hm$

show that isothermal susceptibility $\chi =(\partial m/\partial h)_{T}$ is given by $\chi =1/b_{0}(T-T_{c})$ for $T>T_{c}$ and $\chi =1/b_{0}(T-T_{c})$ for $T>T_{c}$, so that critical exponent $\gamma =1$ according to Landau mean-field theory. HINT: Find the value of the order parameter $m$ which minimizes $g(T,m)$ and use $b=b_{0}(T-T_{c})$.

(b) Show that $cm^{3}=h$ at the critical point, and hence critical exponent $\delta =3$, where $\delta$ is defined by $m\sim H^{1/\delta }$.

Problem 2: Ginzburg criterion for the range of validity of mean-field theory

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 $d$ within the mean-field theory can be written as:

$G(r)\sim {\frac {e^{-r/\xi }}{r^{d-2}}}$

assuming that distance $r\gg a$ is much larger than the lattice spacing $a$. Generalize the Ginzburg criterion

${\frac {G(r)}{\int m^{2}d\mathbf {r} }}\ll 1$

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

$d>2+2\beta /\nu$.

where $\beta$ and $\nu$ are critical exponents for describing vanishing of the order parameter $m$ and divergence of the correlation length $\xi$, respectively.

(b) Using your result in (a), find the upper critical dimension for the Ising model above which its critical behavior near temperature $T_{c}$ 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 $T_{c}$, the correlation length $\xi _{0}$ at $T=0$, and the jump (predicted by the mean-field theory) in the specific heat $\Delta C$ at $T=T_{c}$. Use Landau-Ginzburg solution for the correlation length $\xi ^{2}=-\lambda /2b=-lambda/[2b_{0}(T-T_{c})]$ at $T<T_{c}$ to express it as

$\xi (T)=\xi _{0}|t|^{-1/2}$

and show that the correlation length extrapolated to $T=0$ is given by:

$\xi _{0}^{2}={\frac {\lambda }{2b_{0}T_{c}}}$.

Thus, the parameter $\lambda$, which measures the strength of fluctuations in the Landau-Ginzburg form of the free energy, can be eliminated in favor of measurable quantity $\xi _{0}$ and the parameter $b_{0}$.

(d) Use Landau mean-field theory to show that $b_{0}$ can be expressed in terms of the jump $\Delta C$ in the specific heat (see page 84 in Plischke and Bergersen textbook) at $T_{c}$:

$b_{0}^{2}=(2c/T_{c})\Delta C$.

(e) Using Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m(T) = b_0 T_c |t|^{1/2}/c } from the Landau theory and your results in (c) and (d), show that the Ginzburg criterion derived in the class

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{0.063 k_B T}{\lambda} \ll \xi(T) [m(T)]^2 }

can be expressed as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{0.016 k_B}{\xi_0^3 T_c \Delta C} \ll |t|^{1/2} } .

For example, in conventional superconductors Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \xi_0 = 10^{-7} } (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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_c } as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |t| \sim 10^{-14} } .

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

In this problem you will apply renormalization group (for which the Nobel Prize in Physics 1982 was awarded to Kenneth G. Wilson) to 1D Ising model. 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:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z = \mathrm{Tr}\, \mathbf{T}^N = \mathrm{Tr}\, (\mathbf{T}^2)^{N/2} = \mathrm{Tr}\, \mathbf{T'}^{N/2} }

The transfer matrix for two-spin cells, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{T}^2 } , can be written as:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{T}^2 = \mathbf{T} \mathbf{T} = \begin{pmatrix} e^{2K+2h} + e^{-2K} & e^{h} + e^{-h} \\ e^{-h} + e^{h} & e^{2K - 2h} + e^{-2K} \end{pmatrix} } .

We require that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{T'} } has the same form as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{T} } :

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{T}' = \begin{pmatrix} e^{K'+h'} & e^{-K}' \\ e^{-K'} & e^{K'-h'} \end{pmatrix} }

where a parameter Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C } must be introduced because matching of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{T}' } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{T}^2 } requires to match three matrix elements, which is impossible with only two variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle K' } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h' } .