Low Temperature Expansion of Chemical Potential

[QuasiParticle]

I'm trying to derive a low temperature series expansion for the chemical potential of a weakly interacting Fermi gas. The starting point is, of course, the Fermi-Dirac distribution function (p is the particle momentum):

$$f(p) = \frac{1}{e^{\beta(\epsilon(p) - \mu)}+1} ,$$

where, in the Hartree-Fock approximation, we have

$$\epsilon(p) = \frac{p^2}{2m} + n V(0) - \frac{1}{(2\pi \hbar)^3} \int d^3p' V(\textbf{p} - \textbf{p}' ) f(p').$$

Here, $$m$$ is the effective mass, $$n$$ is the particle density, $$V(0)$$ is the interaction potential $$V(q)$$ at zero momentum transfer. The potential may be assumed to depend only on the momentum transfer $$V(\textbf{p} - \textbf{p}' ) = V(| \textbf{p} - \textbf{p}' | ) = V(q)$$. The F-D distribution $$f(p')$$ in the exchange term may be approximated with the non-interacting one. The chemical potential is determined by the condition (spin-1/2):

$$n = \frac{2}{(2\pi \hbar)^3} \int d^3p f(p) = \frac{1}{\pi^2 \hbar^3} \int_0^\infty p^2 f(p) dp$$

Now, the right-hand side should somehow be expanded as a series in $$( k_B T/ \mu)^2$$, which can then be inverted to give $$\mu$$ as a function of $$T$$. It seems that the Sommerfeld method used for a non-interacting system is not easy to use in this case. I know the result should be the following:

$$\mu (T) = \mu_F (T) + n V(0) - \frac{1}{2} n \left[ F + G \frac{\pi^2}{12} \left( \frac{T}{T_F} \right)^2 \right] ,$$

where
$$F = \frac{3}{2 k_F^3} \int_0^{2 k_F} k^2 \left( 1 - \frac{k}{2 k_F} \right) V(k) dk .$$
and
$$G = 3 \left( V(2 k_F) - \frac{1}{4} \int_0^{2 k_F} \frac{k^3}{k_F^4} V(k) dk \right) .$$

The potential is now written as a function of the Fermi wave vector ($$p = \hbar k$$). $$\mu_F (T)$$ is the chemical potential of a non-interacting Fermi gas. The zero temperature limit, i.e. $$F$$, is rather simple to derive.

Has anyone come across this problem or know any good references? I would really appreciate any assistance.

 

[QuasiParticle]

I guess I could update this thread a little bit. I was able to derive the requested expansion somewhat after posting the above message. My approach was, however, slightly different. The result also had an extra term and reads:

$$G = 3 \left( V(2 k_F) + \frac{1}{4} \int_0^{2 k_F} \left( \frac{k}{k_F^2} - \frac{3}{2} \frac{k^3}{k_F^4} \right) V(k) dk \right) .$$

The reason for this small discrepancy is unclear. By comparing the two results to numerical calculations using the exact equations, I find that my $$G$$ is a better approximation.

 

[charng]

Thank you for sharing your work on deriving a low temperature series expansion for the chemical potential of a weakly interacting Fermi gas. This is a complex and important problem in the field of condensed matter physics. I am not an expert in this specific area, but I can provide some general insights and suggestions.

Firstly, it is important to note that the Fermi-Dirac distribution function and the Hartree-Fock approximation are commonly used in many-body systems, and your approach seems reasonable. The use of an exchange term with a non-interacting distribution is also a common approximation in these types of calculations.

Regarding your question about expanding the integral as a series in (k_B T/\mu)^2, this is known as a Sommerfeld expansion and is a common technique in the study of Fermi gases. However, as you have mentioned, this method may not be straightforward for interacting systems. Some possible approaches you can consider are perturbation theory or using a diagrammatic technique such as the Feynman diagrams.

I am not aware of any specific references for this particular problem, but some relevant texts that may be helpful are "Quantum Theory of Many-Particle Systems" by Fetter and Walecka and "Methods of Quantum Field Theory in Statistical Physics" by Abrikosov, Gorkov, and Dzyaloshinski. These texts cover many-body systems and perturbation theory, respectively.

Additionally, you may want to consult with colleagues or experts in the field for guidance and potential collaborations. It is always beneficial to have discussions with others who are familiar with the problem and can offer different perspectives and insights.

Overall, your work on deriving the low temperature expansion for the chemical potential of a weakly interacting Fermi gas is a valuable contribution to the field. I wish you the best of luck in your future research.