Concerning the understanding of criticality one can proceed purely mathematically. Consider first 2-dimensional systems and 4-D conformal invariance of Yang-Mills theories.
Basic building bricks of TGD vision
The details of this generalization are not understood but the building bricks have been identified.
Hierarchy of criticalities and hierarchy breakings of conformal invariance
The TGD picture about quantum criticality connects it to the failure of classical non-determinism for Kähler action defining the space-time dynamics. A connection with the hierarchy of Planck constants and therefore dark matter in TGD sense emerges: the number n of conformal equivalence classes for space-time surfaces with fixed ends at the boundaries of causal diamond corresponds to the integer n appearing in the definition of Planck constant heff = n× h.
A more detailed description for the breaking of conformal invariance is as follows. The statement that sub-algebra Vn of full conformal algebra annihilates physical states means that the generators Lkn, k>0, n>0 fixed, annihilate physical states. The generators L-kn, k>0, create zero norm states. Virasoro generators can be of course replaced with generators of Kac-Moody algebra and even those of the symplectic algebra defined above.
Since the action of generators Lm on the algebra spanned by generators Ln+m, m>0, does not lead out from this algebra (ideal is in question), one can pose a stronger condition that all generators with conformal weight k≥ n annihilate the physical states and the space of physical states would be generated by generators Lk, 0<k<n. Similar picture would hold for also for Kac-Moody algebras and symplectic algebra of δ M4+× CP2 with light-like radial coordinate of δ M4+ taking the role of z. Since conformal charge comes as n-multiples of hbar, one could say that one has heff=n× h.
The breaking of conformal invariance would transform finite number of gauge degrees to discrete physical degrees of freedom at criticality. The long range fluctuations associated with criticality are potentially present as gauge degrees of freedom, and at criticality the breaking of conformal invariance takes place and these gauge degrees of freedom are transformed to genuine degrees of freedom inducing the long range correlations at criticality.
Changes of symmetry are assigned with criticality since Landau. Could one say that the conformal subalgebra defining the genuine conformal symmetries changes at criticality and this makes the gauge degrees of freedom visible at criticality?
Emergence of covering spaces associated with the hierarchy of Planck constants
Another picture about hierarchy of Planck constants is based on n-fold covering space bringing in n discrete degrees of freedom. How does this picture relate to the breaking of conformal symmetry? The idea is simple.
One goes to n-fold covering space by replacing z coordinate by w=z1/n. With respect to the new variable w one has just the ordinary conformal algebra with integer conformal weights but in n-fold singular covering of complex plane or sphere. Singularity of the generators explains why Lk(w), k<n, do not annihilate physical states anymore. Sub-algebra would consist of non-singular generators and would act as symmetries and also the stronger condition that Lk, k≥ n, annihilates the physical states could be satisfied. Classically this would mean that the corresponding classical Noether charges for Kähler action are non-vanishing.
Another manner to look the same situation is to use z coordinate. Now conformal weight is fractionized as integer multiples of 1/n and since the generators with fractional conformal weight are singular at origin, one cannot assume that they annihilate the physical states: fractional conformal invariance is broken. Quantally the above conditions on physical states would be satisfied. Sphere - perhaps the sphere assigned with the light-cone boundary or geodesic sphere of CP2 - would be effectively replaced with its n-fold covering space, and due to conformal invariance one would have n additional discrete degrees of freedom.
These discrete degrees of freedom would define n-dimensional Hilbert space space by the n fractional conformal generators. One can also second quantize by assigning oscillator operators to these discrete degrees of freedom. In this picture the effective quantization of Planck constant would result from the condition that conformal weights for the physical states are integers.
Negentropic entanglement and hierarchy of Planck constants
Also a connection with negentropic entanglement associated with density matrix, which is proportional to unit matrix or direct sum of matrices proportional to unit matrices of various dimensions is natural in dark matter phase, emerges. Negentropic entanglement would occur in the new discrete degrees of freedom most naturally. In special 2-particle case negentropic entanglement corresponds to unitary entanglement encountered in quantum computation: large heff of course makes possible long-lived entanglement and its negentropic character implies that Negentropy Maximization Principle favors its generation. An interesting hypothesis to be killed is that the p-adic prime characterizing the space-time sheet divides n.