We predicted that vF is essentially independent of doping from VMC calculations  of projected wavefunctions. (This required the use of higher moment sum rules of the spectral function [10,11] not discussed here). Our prediction was confirmed by later ARPES experiments  (see Fig. 2.7). Optical measurements  found a mass renormalization of 2 consistent with our results. The Gutzwiller approximation vF, also shown in Fig. 2.7, is not in quantitative agreement with the VMC and experimental results, in that it has considerable x-dependence. However, it does capture an important qualitative aspect of the VMC results in that vF goes to a constant as x ! 0.
Fig. 2.7 The nodal Fermi velocity from Gutzwiller approximation, variational
Monte Carlo  and ARPES experiments 
We thus find the remarkable result that although the weight Z ! 0 as x ! 0, the nodal vF, or effective mass m* = kF/vF, remains finite. This is very unusual and different from well-studied examples such as electron-phonon interactions in metals and conventional superconductors, or electron-electron interactions in three-dimensional transition metal oxides or the heavy fermions. In all of these cases, the electron self-energy has a weak k-dependence and thus a vanishing Z goes hand in hand with a divergent m*.
Along the nodal direction, where the gap vanishes, one can relate the nodal quasiparticle weight Z and the nodal Fermi velocity vF to the real part of self-energy E' using
Z"1 = 1 - 9E'(k, !)/@! and vF = Z (vF C 9E'(k, !)/3k), (2.27)
where all derivatives are evaluated at the node (k = kF,! = 0) and vF is the bare (noninteracting) Fermi velocity. The vanishing of Z ~ x seen from (2.23) implies a singularity in 9E(k,!)/ ~ 1/ x. The only way in which vF can remain nonzero in this limit is for 9E(k,!)/ 9k to have a compensating ~1/x singularity! The vanishing of Z upon approaching a Mott insulator implies loss of coherence due to strong correlations. The finite vF as x ! 0 arises from the fact that the Fock shift contribution /1 to the renormalized dispersion in (2.18) remains finite (of order J) as x ! 0; (all other k-dependent contributions to £k vanish in this limit).
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