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 $m(T)=b_{0}T_{c}|t|^{1/2}/c$ from the Landau theory and your results in (c) and (d), show that the Ginzburg criterion derived in the class

${\frac {0.063k_{B}T}{\lambda }}\ll \xi (T)[m(T)]^{2}$

can be expressed as

${\frac {0.016k_{B}}{\xi _{0}^{3}T_{c}\Delta C}}\ll |t|^{1/2}$.

For example, in conventional superconductors $\xi _{0}=10^{-7}$ (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 $T_{c}$ as $|t|\sim 10^{-14}$.

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:

$Z=\mathrm {Tr} \,\mathbf {T} ^{N}=\mathrm {Tr} \,(\mathbf {T} ^{2})^{N/2}=\mathrm {Tr} \,\mathbf {T'} ^{N/2}$

The transfer matrix for two-spin cells, $\mathbf {T} ^{2}$, can be written as:

$Z=\mathrm {Tr} \,\mathbf {T} ^{N}=\mathrm {Tr} \,(\mathbf {T} ^{2})^{N/2}=\mathrm {Tr} \,\mathbf {T'} ^{N/2}$