Wednesday, December 21, 2011

A signature of a "two fluid" picture for a strongly correlated electron system

This post connects two seemingly disparate research topics of interest to me: optical properties of organic molecules and low energy excitations of strongly correlated electron materials. The connecting concept is that of an isosbestic point.

The figure below shows the absorption spectrum of an organic dye  for a range of pH values. Note that there is a wavelength (around 500 nm) at which all the spectra appear to cross. This is called the isosbestic point. Varying the pH varies the relative concentration of two forms of the dye molecule. Each form has a characteristic absorption peak (centred at lambda_1 and lambda_2 in the figure). The isosbestic point occurs at the wavelength at which the absorption due to each of the molecular forms has the same intensity. Hence, at this wavelength varying the relative concentration does not change the absorption intensity.
So what does this have to do with strongly correlated electron materials?
In 1997 Dieter Vollhardt published a PRL, Characteristic Crossing Points in Specific Heat Curves of Correlated Systems which pointed out that in liquid 3He, some heavy fermion materials, and the Hubbard model (solved by dynamical mean-field theory (DMFT)) there was a temperature at which the specific heat curves for different pressures (or Hubbard U or magnetic field) all crossed. Below is shown the data for liquid 3He. The crossing temperature is approximately that at which Fermi liquid theory breaks down.
Below are the curves for the Hubbard model.
This is not an artefact of DMFT since later Chandra, Kollar, and Vollhardt showed it is also present for the exact solution of the one-dimensional model.

I find this quite amazing!

Vollhardt made no mention of isosbestic points which I think would have been well known to many chemists. But, in 2007 there is Isosbestic Points in the Spectral Function of Correlated Electrons by Martin Eckstein, Marcus Kollar and Dieter Vollhardt.
It contains a nice discussion of the essential physics, including the connection to the spectra of dye molecules.
The paper contains the figure below of the spectral density for a Hubbard model (calculated with DMFT). The horizontal scale is frequency (omega) and the dashed curve is the non-interacting density of states.

So what does this all mean?

The key idea is one of "two fluids" i.e., that as one varies the relevant parameter (pressure, magnetic field or Hubbard U) all one is doing is varying the relative concentration of the two fluids. A sum rule requires one fluid must be converted into another. In the case of the Hubbard model the total number of electrons (area under A(omega)) is conserved. As one increases U one decreases the spectral weight in the Fermi liquid component (centred at omega=0) and increases the weight of the Hubbard bands (centred at omega=+-U/2).


  1. With respect to high Tc superconductivity, this should give you some insight on how the enormous (optical) pairing energies are manifested (or alternatively - hidden) from the low energy excitations of the system as it undergoes a BCS to BEC transition (or alternatively a BEC to BCS transition). The specific mechanism of spectral weight transfer is the interesting question, one that Philips ascribes to 'Mottness', but there may be others.

    My pet theory is field induced superconductor to insulator transitions at half filling, something that most would claim doesn't exist.

  2. Sure there is spectral weight transfer, but why should there be an exact isobetic point? That seems to be what is implied in this work. If the crossing is not exact (or is becoming exact in the limit of zero temperature), then it is a bit of a red herring and of limited utility. Better to just talk of spectral weight transfer...

  3. While it is true that isosbestic points and accompanying van't Hoff scaling can be a marker of two state behavior, a much more general interpretation admits that any continuous equilibrium distribution might display this behavior (a result that I have found very interesting).

    This is written about from the perspective of a physical chemist here:

  4. The problem with this dissertation is that something is fluctuating. That implies two phases or states at the very least. In this case, it is the bath coupled to the solute. A microscopic examination of the solute bath composite would reveal the details of the coupling, but certainly it must be there.

    In the case of the high temperature superconductors, at the very least there is a microscopic phase separation, which implies two microscopic competing domains, and percolative behavior. This has recently been demonstrated even for the doped bismuth oxides.