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In Fundamentals of Many-body Physics by W. Nolting, 1e, the author arrives at the following formula for the electron-electron contribution to the Hamiltonian of Jellium:

$$ \hat{\mathcal{H}}_{ee}=\frac{2\pi q^2}{\alpha^2 V}\left(\hat{N}^2-\hat{N}\right) $$

where $q$ is the electronic charge, $\hat{N}$ is the particle number operator, and $\alpha$ is a small parameter needed to make the integrals converge. The term involving $\hat{N}^2$ exactly cancels other divergent contributions to the Hamiltonian. Regarding the term linear in $\hat{N}$, the author says:

The [term linear in $\hat{N}$] leads to an energy per particle which vanishes in the thermodynamic limit.

I understand $2\pi q^2/\alpha^2 V$ (ie. the energy per particle) vanishes if we first send $V\rightarrow\infty$, then send $\alpha\rightarrow 0$. But if we commute the limits, it will diverge. How do we know $V\rightarrow\infty$ then $\alpha\rightarrow 0$ is the correct order in which to take the limits?

Tobias Fünke
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1 Answers1

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To quote the standard reference on the Jellium:$^1$

The physical hamiltonian is recovered by letting $\kappa \to 0$ after going to the thermodynamic limit $L\to \infty$. The procedure is justified a posteriori since the physical quantities obtained by taking the limits in this order are finite and independent of $\kappa$.

But in principle you don't have to introduce the regularization, although it helps in deriving the Fourier transform of the Coulomb potential, see e.g. this PSE post. References 2 and 3 derive the Jellium Hamiltonian (in the thermodynamic limit) without the "issue" of the order of the limits. In reference 4, the author, after proceeding just as Nolting, states that

The reader may feel uneasy about the results we obtained; they rely on the mathematical artifact of introducing an exponential damping term, and on the sequence in which the limits are taken. The reader may rest assured that the results are correct. In fact, the same results are obtained without introducing the exponential term.

and then derives the results in the same spirit as references 2 and 3.

Very roughly, the idea is to work with a finite volume and periodic boundary conditions. This is also done in reference 1, but there the Fourier coefficients (in the series of the Coulomb potential) are replaced by the Fourier transform, making the regularization necessary due to the $k=0$ term. Note that this replacement is fine in the thermodynamic limit, see e.g. this PSE post.

Instead, e.g. reference 3 makes this replacement only for the $k\neq 0$ terms (eq. $(10.1)$ therein), and then shows that the $k=0$ contributions (finite for finite volume) in the total Hamiltonian vanish in the thermodynamic limit.

References:

  1. Quantum Theory of the Electron Liquid. G. Giuliani and G. Vignale. Section 1.3.2, p. 14
  2. Many-Body Problems and Quantum Field Theory. An Introduction. P. Martin and F. Rothen. Section 4.2.1, p.133
  3. Many-Particle Theory. E. Gross and E. Runge. Chapter 10, p.79
  4. Feynman Diagram Techniques in Condensed Matter Physics. R. Jishi. Section 4.1, p. 68
Tobias Fünke
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