Quantum many-body systems are characterised by many different energy scales (e.g. Fermi energy, Debye frequency, superconducting energy gap, Kondo temperature, ....). However, in many systems properties are "universal" in that they are determined by a single energy scale. This means that the frequency (omega) and temperature (T) dependence of a spectral function can be written in a form such as

where here T_ K is the relevant energy scale and I set hbar =1 and k_B = 1.

However, what happens in the limit where the relevant energy scale T_K goes to zero, for example near a quantum critical point? Then the only energy scale present is that defined by the temperature T and we now expect a functional dependence of the form

This is omega/T scaling.

In one dimension the form of the scaling function is specified by conformal field theory and for quantum impurity problems (e.g. Kondo) by boundary conformal field theory.

In 1989 Varma et al. showed that many of the anomalous properties of the metallic phase of the cuprate superconductors at optimal doping could be described in terms of a “marginal Fermi liquid” self energy. They associate this with a spin (and charge) fluctuation spectrum that exhibited omega/T scaling (for all wave vectors). Specifically, the spectral function was linear in frequency at low frequencies, up to a frequency of order T.

Some claims about quantum criticality in cuprates are debatable, as discussed here.

Finding concrete realistic theoretical microscopic fermion models that exhibit such scaling has proven challenging.

In his Quantum phase transitions book Sachdev reviews several spin models (e.g. transverse field Ising model in one dimension) that exhibit omega/T scaling in the quantum critical region, associated with a quantum critical point.

In 1999 Parcollet and Georges considered a particular limit of a random Heisenberg model which had a spin liquid ground state and a local spin susceptibility chi’’(omega) that exhibited a form consistent with that conjectured in the marginal Fermi liquid scenario.

Local quantum criticality has been observed in a few heavy fermion compounds. Specifically, in 2000 Schroder et al. observed inelastic neutron scattering gives the following \omega/T scaling,

In 2008 Kirchner and Si showed that near the quantum critical point in the Ising-anisotropic Bose-Fermi Kondo model (BFKM) with a sub-ohmic bath (i.e. a very specific model!) they obtained omega/T scaling similar to that associated with boundary conformal field theory, even though the model has no obvious conformal invariance.

This is my potted history and understanding. I welcome corrections and clarifications.

The first neutron scattering evidence of omega/T scaling in a heavy fermion was in UCu5-xPdx - Aronson et al, Phys Rev Lett 75, 725 (1995), and that really was local quantum criticality, since it was completely Q-independent, although we missed the opportunity to call it that at the time. Some interpreted it as Kondo disorder, but UCu4Pd can be highly ordered and later muSR work showed that the scaling was robust with annealing.

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