**one decade of frequency**with a fractional exponent. Anderson has also invoked this power law dependence as evidence for his Hidden Fermi liquid theory.

The lower panel shows that the phase angle of the conductivity is approximately constant and equal to 60 degrees over a large frequency range. This is what was predicted by Anderson.

However, Norman and Chubukov show that there are two alternative explanations of this data, both involving

**electrons interacting with a broad spectrum of bosons.**They work in terms of a frequency dependent (and momentum independent) self energy and no vertex corrections to the conductivity. This formalism was originally developed by Allen in 1971 to describe the effects of the electron-phonon interaction on the optical spectrum of metals. Norman and Chubukov obtain quantitative fits to the data with a self energy which has either a Marginal Fermi liquid form or from a Lorentzian spectrum of overdamped bosons (e.g., spin fluctuations). What are the implications of this work? Here is my paraphrase of their own conclusions

Aside: To me, this all underscores the importance of the method of multiple alternative hypotheses in theory development.

- the apparent scaling behavior over a wide frequency range may be unrelated to quantum criticality but rather just a consequence of the
flattening of the frequency dependence of the scattering ratewhich is accompanied by a corresponding decrease in the real part of the self energy.- the behavior of the single particle and optical self-energies is very similar for the marginal Fermi-liquid phenomenology, and the Lorentzian model used in microscopic fermion-boson theories.
- the data give strong evidence for an
upper cutoff of the boson spectrum of around 300 meV. A cutoff of this scale was suggested in the original marginal Fermi-liquid phenomenology. Such alarge energy scalewould imply that the source of the boson spectrum iscollective electronic excitationsrather than phonons. Inelastic neutron scattering data show magnetic spectral weight up to this energy scale and, thus, spin fluctuations or possibly other electronic excitations are a natural explanation for the boson spectrum.

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