Desperately seeking a way to find order parameters

Much of condensed matter physics is concerned with finding the relevant order parameter for new phases of matter. Indeed this is a good way to win a Nobel Prize! This is much of what was done by Ginzburg, Neel, Leggett, de Gennes,....

A fundamental and controversial question is whether one can a priori predict new order parameters. Historically, the progression has always been:

1. Experimental discovery of a new phase of matter.
2. Proposal of an order parameter and a phenomenological (Ginzburg-Landau) theory to explain a range of experiments.
3. Proposal of an effective Hamiltonian which has a ground state with the desired spontaneous symmetry breaking and associated order parameter.
4. Justification of the effective Hamiltonian from so-called "ab initio" electronic structure calculations starting with Schrodinger's equation and the actual chemical composition of specific materials. 

The grand challenge is to invert this process, even just one step.

Laughlin and Pines seem to claim that this is essentially impossible.

There are some interesting fundamental philosophical questions as to whether the obstacles are ones of practical difficulty versus fundamental physics.

There is really interesting 2006 PRL, Systematic Derivation of Order Parameters through Reduced Density Matrices, by Shunsuke Furukawa, Grégoire Misguich, and Masaki Oshikawa.
Essentially they claim to have found a way to go from 3. to 2. above. In particular, given the results of an exact diagonalisation calculation of the low lying states of a lattice model they give a procedure to find the order parameter from looking at two nearly degenerate ground states.
They then apply the method to two concrete examples: a Heisenberg spin model on a ladder with ring exchange, and a quantum dimer model on the Kagome lattice. The method gives the correct order parameters in the first case and for the second shows there is no order parameter. I found this quite impressive and promising.

The PRL also promises future work generalising the method to more than two degenerate ground states and suggests application to frustrated two-dimensional quantum antiferromagnets. Unfortunately, I have not been able to find such work.