The Sum of it all

A characteristic of strongly correlated metals and of phase transitions (e.g. superconducting and the Mott metal-insulator transition) is that they lead to a redistribution of spectral weight (e.g. due to the opening of an energy gap in the excitation spectrum). The total spectral weight is often constrained by sum rules. However, it turns out that when comparing to experimental data one needs to be careful about the high energy cutoffs one uses in these sum rules.

An important sum rule is the Ferrel-Glover-Tinkham sum rule which related the total spectral weight in the optical conductivity in the normal state (NS) to that in the superconducting state (SC) and the superfluid density n_s

Note the integrals extend to infinite frequency.

The schematic figure below shows the frequency dependence of the real part of the optical conductivity in NS (upper curve) and the SC state (lower curve). The green shaded area is the area which collapses into a delta function peak at (energy) omega=0 with weight n_s, characteristic of the infinite conductivity in the superconducting state.  

In a lattice the sum rules become modified because of band effects and the mass has to be replaced by a band mass. Many-body effects can reduce the sum for the NS, by as much as a factor of ten. Roughly this sum scales with the average electronic kinetic energy. Furthermore, one has to worry about finite (energy) frequency cut-offs in the integrals.

Today I read a nice PRB by Maiti and Chubukov which performs a systematic study of how for different model self energies (in both SC and NS) the sums behave, particularly as a function of the cutoff used.
This paper is motivated by some analysis of experimental data for the cuprates which suggested "sum rule violation". This might be expected if there are no quasi-particle poles in the normal state or if superconductivity is extracting spectral weight from energy above 2Delta. [This violation is interpreted as that the kinetic energy decreases in SC, opposite to what happens in BCS, due to particle-hole mixing].

Maiti and Chubukov find that for various model cases the results are quite sensitive to the cutoff. Furthermore, a marginal Fermi liquid model study by Norman and Pepin which did produce a kinetic energy decrease in SC [and so attracted significant attention] turns out to be parameter dependent.

The Figure above is taken from a Nature Physics viewpoint by Basov and Chubukov. They use it to partially justify Homes law which relates the superfluid density to the product of Tc and the dc conductivity at T=Tc.