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The instability of a Fermi surface against Ising nematic order destroys thequasiparticle character of the low-energy degrees of freedom. Therefore,observables exhibit deviations from Fermi liquid behavior which gives rise tothe term Ising nematic quantum critical metal. To obtain a theoreticaldescription we use a finite-temperature version of Eliashberg theory whichallows to treat the strong coupling between quantum and thermal fluctuations inthe absence of well-defined quasiparticles. Here, we use this self-consistent,diagrammatic approach to compute, in particular, the nematic susceptibility andthe non-Fermi liquid correlations. Upon decreasing the temperature, thesusceptibility crosses over from a $(T \log T)^{-1}$ to $T^{-2/3}$ behavior,which is induced by the absence of quasiparticles and restores the Isingnematic critical scaling. Correspondingly, the fermions obey a simple$\omega/T$ scaling law at low enough temperatures. However, this regime ischaracterized by strong IR-UV mixing since the proportionality factors exhibita dependence on the spectral width of the non-quasiparticle excitations and onthe underlying lattice. Tuning the parameters of the model, therefore, givesrise to several scenarios for the breakdown of the scaling theory. We discussthem within the Eliashberg approach and estimate the related crossover scales.We also show that the leading order vertex corrections do not change thescaling with temperature or coupling constants.

$\omega/T$ scaling and IR/UV-mixing in Ising-nematic quantum critical metals