Homework Set 6

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

(a) Starting from the Gibbs free energy density in the Landau phenomenological formulation of the mean-field 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 $\chi ^{-1}=2b_{0}(T_{c}-T)$ 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 $m\sim H^{1/\delta }$. NOTE: Exponent $\delta$ is defined for $T=T_{c}$ or ($t=0$).

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 $m^{2}(T)=b_{0}T_{c}|t|/c$ 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

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

where $T=T_{c}$ was set since the Ginzburg criterion is valid only near the critical temperature.

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 $|t|\sim 10^{-14}$.

NOTE: Chapter 9 from Gould & Tobochnik textbook has simple introduction to basic concepts used in this problem.

Problem 3: Renormalization group for 1D Ising model in external magnetic field using transfer matrix method

In this problem you will apply the renormalization group (for which the Nobel Prize in Physics 1982 was awarded to Kenneth G. Wilson) to 1D Ising model in the external magnetic field $h_{\mathrm {ext} }$. 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:

$\mathbf {T} ^{2}=\mathbf {T} \mathbf {T} ={\begin{pmatrix}e^{2K+2h}+e^{-2K}&e^{h}+e^{-h}\\e^{-h}+e^{h}&e^{2K-2h}+e^{-2K}\end{pmatrix}}$.

where $K=\beta J$ and $h=\beta h_{\mathrm {ext} }$.

We require that $\mathbf {T'}$ has the same form as $\mathbf {T}$:

$\mathbf {T} '=C{\begin{pmatrix}e^{K'+h'}&e^{-K'}\\e^{-K'}&e^{K'-h'}\end{pmatrix}}$

where a parameter $C$ must be introduced because matching of $\mathbf {T} '$ and $\mathbf {T} ^{2}$ requires to match three matrix elements of such symmetric $2\times 2$ matrices, which is impossible with only two variables $K'$ and $h'$.

(a) Show that the three unknowns satisfy the three conditions:

${\begin{aligned}Ce^{K'}e^{h'}&=e^{2K+2h}+e^{-2K},\\Ce^{-K'}&=e^{h}+e^{-h},\\Ce^{K'}e^{-h'}&=e^{2K-2h}+e^{-2K}.\end{aligned}}$

(b) Show that the solution of equations in (a) can be written as:

${\begin{aligned}e^{-2h'}&={\frac {e^{2K-2h}+e^{-2K}}{e^{2K+2h}+e^{-2K}}},\\e^{4K'}&={\frac {e^{4K}+e^{-2h}+e^{2h}+e^{-4K}}{(e^{h}+e^{-h})^{2}}},\\C^{4}&=[e^{4K}+e^{-2h}+e^{2h}+e^{-4K}][e^{h}+e^{-h}]^{2}.\end{aligned}}$

(c) Show that the recursion relations in (b) reduce to:

$K'=R(K)={\frac {1}{2}}\ln[\cosh(2K)]$

for $h=0$. For $h\neq 0$, start from some initial state $(K_{0},h_{0})=(1,0.1)$ and calculate a typical renormalization group trajectory. To what phase (paramagnetic or ferromagnetic) does the fixed point correspond? NOTE: In general, RG for 1D Ising model has a line of trivial fixed points satisfying $K^{*}=0$ and arbitrary $h^{*}$, which corresponds to the paramagnetic phase, and an unstable ferromagnetic fixed point at $K^{*}=\infty$ and $h^{*}=0$.