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 poin**t. 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).