I've recently made myself more familiar with Laughlin and Pines' "quantum protectorate" arguments (1, 2), which seems built on Anderson's famous More is Different article (3). This does seem in disagreement with Dirac's 1929 sentiments, (4):
The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.
(though I know not everyone agrees with this, e.g. this chemistry SE thread) and perhaps these arguments are still considered speculative. However, there are a few mentions of the failures of ab initio methods (which seems quite reductionist to me), and it seems implied that they cannot be enough to predict some effects in condensed matter, especially superconductivity.
My question is, to what extent is this true, and how would one typically try to model matter that is in some "protectorate," e.g. superconductivity?
In these cases, are ab initio methods not suitable (and say, even with incredible computing power, still not suitable?)? Would (for example) renormalisation group methods be more reliable than ab initio methods? If they fail, is this a failure of our reductionist model ("theory of everything"), or is it a failure of our approximations in solving the reductionist model? Is there reason to believe that we can't, for example, recover the Josephson effect (which relies on spontaneous symmetry breaking) by direct solution of the Schrödinger equation (with our technology, but also in principle, i.e. if we imagine a world where we can solve such a large system with FCI/QMC)? Similarly, on the more extreme "patently absurd" end, protein functionality and the human brain: these are obviously inaccessible just in terms of computing power, though I suppose asking questions about ab initio neuroscience quickly turns into poppycock.