## Wednesday, November 9, 2011

### Deconstructing the chemical potential of the cuprate superconductors

I have been reading through the nice review Finite temperature properties of doped antiferromagnets by Jaklic and Prelovsek from 2000. They summarise their studies of the t-J model by the Finite temperature Lanczos method.
At first sight the graph below of the temperature and doping dependence of the chemical potential does not look particularly interesting [at least to me]. However, they highlight its significance.
Here are a few points.
• In a simple Fermi liquid the chemical potential has a positive, quadratic and weak temperature dependence. This is only seen for doping c_h=x=0.3
• For a wide doping range [0.05 < c_h < 0.3] the temperature dependence is approximately linear. The slope changes sign for approximately optimal doping (c_h ~ 0.15).
• The weak temperature dependence for c_h ~ 0.15 means that optimal doping corresponds to maximum entropy!  [This can be deduced via the Maxwell relation below. Don't you love thermodynamics!]
• This relation is also related [approximately] to the thermopower via a relationship [equation 8.6], which is essentially a restatement of the Kelvin formula [discussed  by Peterson and Shastry].
• The latter means the thermopower should change sign around optimal doping, as is indeed observed [more on that later].
• The large entropy near optimal doping emerges from the interplay of the localised spins [from the remnants of the Mott insulator] and frustration of the antiferromagnetic spin interactions via doping.
I would be interested to see a similar calculation for the Hubbard model on the anisotropic triangular lattice at half filling to see how the chemical potential varies as a function of U/t as the Mott insulator is approached from within the metallic phase.

1. So s, T, ch and mu are entropy, temperature, hole concentration and hole chemical potential?

2. yes. sorry i should have defined them.

3. This comment is regarding this post and this post
http://condensedconcepts.blogspot.com.au/2011/12/optimal-doping-corresponds-to-maximum.html

You mention the paper by Jaklic-Prelovsek and Eq. 8.6 in particular--I looked at the arxiv version so hopefully they are the same. Eq. 8.6 is not exactly S_Kelvin but is instead just the so-called Mott-Heikes formula. It originates from the Kubo formula where they argue that the transport terms, the current-current and current-heat-current correlation functions are closely related to one another and are nearly \omega independent. They find that this term just looks like \mu(T=0) and they get the Mott-Heikes formula.

Jaklic-Prelovsek can connect entropy to chemical potential but cannot quite connect entropy to thermopower since their formula for thermopower is must \mu/T. S_Kelvin allows a precise connection from the entropy to the thermopower.

Anyway, I agree that the Jaklic-Prelovsek paper is filled with interesting things and FTLM appears very powerful.

4. Hi Michael,
Thanks for the clarifying comment.
You are correct. Indeed, Jaklic-Prelovsek do not use Kelvin [contrary to my suggestion] but rather the Mott-Heikes formula for thermopower.
Cheers

5. Dear Prof. McKenzie,

This comment is regarding the post "Deconstructing the chemical potential of the cuprate superconductors".

You point out that the T^2 dependence of the chemical potential has a positive coefficient in a Fermi liquid. I'm afraid I do not see how this is true in general: My understanding is that the value of the T^2 coefficient (and consequently the sign as well, I imagine) changes with the dependence of the effective quasiparticle mass on the system density; perhaps even dimensionality plays a role.

Any clarifications will be greatly welcome.

Thanks a lot.