Frank Marsiglio "Issues Concerning the Optical Sum Rule Anomaly below Tc in the Cuprates"

TOC
1. Conventional Theory
2. Experiment
3. Phenomenological Explanation
4. Issues

Conventional Theory

Kubo sum rule
Integral of the real part of conductivity over frequency is
constant -- temperature independent.


Single band sum rule (theoretical construction)
Integral of the real part of conductivity coming from a single band
over frequency (denoted by W(T)) measures the average second
derivative of energy over momentum.

In conventional cases it is proportional to the minus average energy.


In Fermi Gas as temperature increases the distribution function smears and
particles get transferred to higher energy, so W(T) goes down.

If one now decreases the temperature the Superconducting transition occurs,
the distribution function gets smeared, kinetic energy increases and W(T)
goes down.

So W(T) has a maximum at T_c.


Experiment

Experiment shows the decrease W(T) as one decreases temperature through T_c
in overdoped materials, but in optimally doped and underdoped materials
it goes up.

It means that in optimally and underdoped materials the kinetic energy
decreases in superconducting state. That gives us
"kinetic energy driven superconductors".


Phenomenological Explanation

Norman & Pepin (2002) showed that interactions decrease W(T).
Microwave experiments show that there is a collapse of the scattering
rate (scattering rate is due to interactions) below T_c

Taking together those two statements mean that above T_c W(T) is suppressed
by the interactions while below T_c interactions are suppressed and W(T)
goes over to the one of the noniteracting case -- increases.


Issues

There is no issue of the low energy cutoff as Kuzmenko et al explained that
although they cannot measure conductivity at low frequencies accurately
enough, they can measure the contribution to the sum rule.


The main issue is the upper cutoff.
In order to measure the sum rule for the single band one has to introduce
an upper frequency cutoff which is below the frequency of the interband transitions.

Imagine that we have a simple Drude behaviour of the conductivity.
The Drude peak sharpens up as one lowers temperature. If one then checks the
sum rule up to some upper cutoff in frequency one finds that it is more
weight below this frequency. So although the total weight is conserved
the total weight below a frequency cutoff is temperature dependent.

So the normal state ~T^2 behavior can be explained by a mundane upper cutoff
effect. We are currently investigating whether the anomalous rise of W(T)
below T_c can be attributed to a mundane cutoff effect as well.



Questions

Assa: What should the high frequency cutoff be in order to recover full temperature independent sum rule?

F.M. Large, depends what "full" means.


Chubukov: comment, the increase or decrease of W(T) due to cutoff
depends on the valueof \Delta\tau


Pepin: Has anyone investigated the influence of the van Hove singularity
on W(T)?

F.M. Theoretically, last year in a PRB paper we showed that the change below
T_c can be anomalous, using just a BCS approach. As far as I know no one
has measured this same quantity in High T_c samples that are doped beyond the van Hove singularity.