Wednesday, July 20, 2011

Being critical about quantum criticality

Recently I read a nice paper by Mike Norman and Andrey Chubukov, High Frequency Behaviour of the infrared conductivity of the cuprates. The paper addresses issue raised by a 2003 Nature paper Quantum critical behaviour in a high-Tc superconductor. The Nature paper gave a detailed analysis of the frequency dependence of the real and imaginary parts of the frequency dependent conductivity of an optimally doped cuprate superconductor. In particular, the data is compared to theoretical work of Anderson and Sachdev. A key figure is the one below. The upper panel shows that the magnitude of the conductivity vs. frequency exhibits a power law dependence over approximately 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
  • 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 rate which 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 a large energy scale would imply that the source of the boson spectrum is collective electronic excitations rather 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. 
Aside: To me, this all underscores the importance of the method of multiple alternative hypotheses in theory development.

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