  In this work we generalize and subsequently apply the Effective Field
Renormalization Group technique to the problem of ferro- and
antiferromagnetically coupled Ising spins with local anisotropy axes in
geometrically frustrated geometries (kagome and pyrochlore lattices). In this
framework, we calculate the various ground states of these systems and the
corresponding critical points. Excellent agreement is found with exact and
Monte Carlo results. The effects of frustration are discussed. As pointed out
by other authors, it turns out that the spin ice model can be exactly mapped to
the standard Ising model but with effective interactions of the opposite sign
to those in the original Hamiltonian. Therefore, the ferromagnetic spin ice is
frustrated, and does not order. Antiferromagnetic spin ice (in both 2 and 3
dimensions), is found to undergo a transition to a long range ordered state.
The thermal and magnetic critical exponents for this transition are calculated.
It is found that the thermal exponent is that of the Ising universality class,
whereas the magnetic critical exponent is different, as expected from the fact
that the Zeeman term has a different symmetry in these systems. In addition,
the recently introduced Generalized Constant Coupling method is also applied to
the calculation of the critical points and ground state configurations. Again,
a very good agreement is found with both exact, Monte Carlo, and
renormalization group calculations for the critical points. Incidentally, we
show that the generalized constant coupling approach can be regarded as the
lowest order limit of the EFRG technique, in which correlations outside a
frustrated unit are neglected, and scaling is substituted by strict equality of
the thermodynamic quantities.
