# Adiabatic continuity and discontinuity

Of course it would nice to be able to solve for the electronic properties of a real material exactly, but much of our understanding of real materials is based on the hope that the physics is "continuous" between the (complicated) real material and a (simple) abstract model. Metals provide a striking example of the success of this philosophy. Real metals are strongly interacting and very far in quantitative terms from the free Fermi gas. Yet in describing most metals and insulators, one starts from a picture of noninteracting electrons in e.g. calculating the band structure and other properties. Since the Coulomb interaction energy is actually very large, one might wonder why it is appropriate to assume that noninteracting electrons (a free Fermi gas) make a sensible starting point.

The underlying idea, possibly first phrased in these terms by Landau, is that electrons in a real metal form a "Fermi liquid", which bears the same relation to the "Fermi gas" of free electrons that a normal liquid bears to a normal gas: the interactions are much stronger, but there is no change in symmetry or in the fundamental nature of the state. In particular, the "elementary excitations" of the ground state (those that are found to carry current, heat, and other properties) bear the same quantum numbers as ordinary electrons. We can imagine looking at the full energy spectrum of a many-particle system and trying to identify mobile low-energy excitations: Landau's theory, which we will justify later in this course, explains how these excitations can wind up as electrons

"dressed" by particle-hole pairs, which renormalize the mass (by up to a factor $10^3$ in so-called heavy fermion compounds) and some other properties but not the charge $e$ and fermionic statistics.

It turns out that electrons in a typical metal are stable to strong *repulsive* interactions, but can be unstable to even weak *attractive* interactions. The resulting superconducting state is an example of how adiabatic continuity can be violated: the lowest-energy charged excitations in a traditional superconductor are "Cooper pairs" of charge $2e$.

Example I of discontinuity: The natural energy scale of noninteracting electrons in a solid is the Fermi energy, which can be tens of thousands of kelvins. The natural Coulomb interaction energy scale $e^2 n^{-1/3}$ is comparable to the Fermi energy. Both these energies are very large in comparison to the superconducting transition temperature $T_c$, which for an old-fashioned BCS superconductor is of order 10 K. It turns out that this new small energy scale is a signal of adiabatic discontinuity or "nonperturbative" behavior.

The superconducting gap in BCS theory scales as

\begin{equation}
T_c \sim D e^{-1 / \lambda N(0)},
\end{equation}

where $D$ is a bandwidth or Fermi energy, $\lambda$ is the energy of the attractive electron-electron interaction, and $N(0)$ is the electron DOS at the Fermi level. Looking at this formula, suppose we try to expand it as a power series in $\lambda$ around $\lambda=0$, when the system should be a noninteracting Fermi gas. You will find that all the derivatives at $\lambda=0$ are 0, so the Taylor series looks like

\begin{equation}
T_c \sim 0 + \lambda 0 + {\lambda^2 \over 2!} 0 + \ldots.
\end{equation}

This is often stated as "$T_c$ is zero to all orders in perturbation theory". Its practical meaning is that we need to find a new starting point for the description of the superconductor, rather than just starting from the free Fermi gas and trying to incorporate interactions perturbatively. A large part of this course will be devoted to the new starting points or organizational principles that emerge from the simple rules of nonrelativistic QM and the Coulomb interaction.

Example II of continuity: A superfluid is like a pure (noninteracting) BEC, even though the strong interactions in the superfluid make its quantitative properties very different. For instance, in a noninteracting bosonic gas, at temperature $T=0$ all of the particles are in the lowest eigenstate; for an atomic BEC, about 99 percent or more are in the lowest eigenstate, as the interactions are weak; for superfluid helium-4, only about 10 percent are in the lowest eigenstate. However, helium-4 still shows amazing properties such as an absence of viscosity for low-velocity flows, because in some sense the interactions do not change the basic nature of the state.

Example II of discontinuity: In the simplest "s-wave" superconductors, in which the two electrons in a Cooper pair have an s-orbital structure, there is no phase transition between the molecular or BEC limit where the electrons are closely bound compared to the mean separation between pairs, and the BCS limit where the size of a Cooper pair is much larger than the interparticle separation. (There are certainly strong quantitative differences.) However, in certain other superconductors there is a difference between the strong-pairing and weak-pairing limits.