strongly correlated limit of the electron gas

Question

In chapter 7 of "introduction to many-body physics" by Coleman, the author calculated the Hartree-Fock contribution to the energy of the electron gas, following which it is claimed that the most strongly correlated limit of the electron gas is the dilute limit. This statement appears quite counter-intuitive to me and the author didn't explain why. Any help is greatly appreciated!

Answer 1

Yes, it is indeed counterintuitive, because it is a purely quantum result with no classical counterpart. The reason is simple and stems from the Heisenberg uncertainty relation.

The Hamiltonian takes the form $$H = K + U := \sum_{i=1}^N \frac{p_i^2}{2m} + \sum_{i,j \neq i}^N \frac{q^2}{|{\bf r}_i - {\bf r}_j|}.$$ Specifying the density is equivalent to specifying the mean interelectron spacing $a$ (not the Bohr radius!). By the uncertainty principle, the typical momentum $p$ satisfies $p a \sim \hbar$, so the kinetic energy scales like $a^{-2}$, while the potential energy scales like $a^{-1}$. So at low densities where $a$ is large, the potential energy dominates.

The physical intuition is that packing in electrons very tightly is kind of like confining each one in a small box. And the HUP tells you that if they're confined to a small box, they will have high momentum, so they'll be moving so fast that their interactions will be negligible - they'll just fly right past each other without having enough time close together to be deflected very much. Even though the potential energy becomes stronger for denser systems, the kinetic energy becomes stronger even faster, and it's the relative energy that matters.

The crossover value of $a$ for which the two energies are comparable is around the Bohr radius $a_0$. Unfortunately, in real metals the interelectron spacing is often of this order of magnitude, so neither the kinetic nor the potential energy dominates and we need to consider both together. This is why some metals are strongly correlated and others aren't.