Temporary HW

Problem 1: Ginzburg criterion in arbitrary spatial dimension and upper critical dimension

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 $a$.

(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.

Problem 2: 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 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:

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