for the Heavy-Fermions that the qcp is not of the Gaussian kind. The fluctuations appear to be local in space and decay as power laws in time. In other words χ ”( q, ω, p, T ) at p = p$_{c}$ has no singularity as a function of q but has a singularity as ω and T → 0. The upper cut-off in the fluctuations is at and energy ω$_{c}$ , which is only about an order of magnitude smaller than T$_{c}$ . Critical fluctuations proposed on phenomenological grounds for the Cuprates have the form χ ”( q, ω, p$_{c}$, T ) = χ$_{0}$ tanh( ω/T ), with a cut-off at ω$_{c}$ . One could formally put a dynamical exponent z → ∞ and put this in the form in Eq.(20). But this obscures that quite different physical ideas are required to derive such fluctuations than those of the Gaussian kind. For example, the derivation of this class of fluctuations ([9], [10]) for Cuprates rests on finding two classes of orthogonal topological variables one of which has spatially local and temporally logarithmic interactions while the other has spatially logarithmic interactions and temporally local interactions. Classical statistical models are known where the singular fluctuations are determined by topological excitations, for example the Kosterlitz-Thouless transition in the 2d x-y model and transitions in several vertex models [32], which do not fall under the purview of the classical model of phase transitions which are the inspiration for the theory of Gaussian qcp’s. The class of microscopic models where such quantum-criticality occurs is not known. I will refer to such qcp’s as topological qcp’s. 

The absence of a spatial scale and the freeing of the scale of frequency of the critical fluctuations from the requirement that they tend to lower values as T → 0, removes the two deleterious effect of the Gaussian critical fluctuations on T$_{c}$ : the prefactor in the expression for T$_{c}$ remains at ω$_{c}$ and there is essentially no extra pair-breaking due to inelastic scattering for ω$_{c}$ >> T$_{c}$ . The locality of the spatial scale, which in practice means that the spatial scale is similar to k − 1 F has the additional consequence that the ℓ = 0 and ℓ ̸ = 0 couplings are similar in magnitude. In fact, given a total spectral weight χ$_{0}$ and the ℓ in which pairing is favored, it is not possible within Eliashberg theory to have a better spectrum for high T$_{c}$ than such a spectrum. 

## D. Calculations of T$_{c}$ for the Hubbard Model 

Following the suggestion by Anderson [33] that the essential physics of the Cuprates is described by the Hubbard model, there have been innumerable attempts by a variety of methods to calculate the properties of the Hubbard model, including T$_{c}$ . The approximate methods (RPA [34], varieties of dynamical mean-field theories [35, 36], variational Monte- carlo [37, 38]) in comparison with the best numerically precise method, Monte-Carlo without 

