# Category Archives: Fermi Surface

## Neither Energy Gap Nor Meissner Effect Imply Superflow

I have read several times in lecture notes, textbooks and online forums that the persistent current in a superconductor of annular geometry is a result of either:

1. The opening of a superconducting gap at the Fermi surface
2. The Meissner Effect

This is not correct, actually.

The energy gap at the Fermi surface is neither a sufficient nor necessary condition for the existence of persistent supercurrents in a superconducting ring. It is not sufficient because gaps can occur for all sorts of reasons — semiconductors, Mott insulators, charge density wave systems all exhibit energy gaps separating the occupied states from the unoccupied states. Yet these systems do not exhibit superconductivity.

Superconductivity does not require the existence of a gap either. It is possible to come up with models that exhibit superconductivity yet do not have a gap in the single-particle spectra (see de Gennes Chapter 8 or Rickayzen Chaper 8). Moreover, the cuprate and heavy fermion superconductors possess nodes in their single-particle spectra and still exhibit persistent currents.

Secondly, the Meissner effect is often conflated with superflow in a superconductor, but it is an equilibrium phenomenon, whereas persistent currents are a non-equilibrium phenomenon. Therefore, any conceptual attempts to make a conclusion about persistent currents in a superconducting ring from the Meissner effect is fraught with this inherent obstacle.

So, obviously, I must address the lurking \$64k question: why does the current in a superconducting ring not decay within an observable time-frame?

Getting this answer right is much more difficult than pointing out the flaws in the other arguments! The answer has to do with a certain “topological protection” of the current-carrying state in a superconductor. However one chooses to understand the superconducting state (i.e. through broken gauge symmetry, the existence of a macroscopic wavefunction, off-diagonal long-range order, etc.), it is the existence of a particular type of condensate and the ability to adequately define the superfluid velocity that enables superflow:

$\textbf{v}_s = \frac{\hbar}{2m} \nabla \phi$

where $\phi$ is the phase of the order parameter and the superfluid velocity obeys:

$\oint \textbf{v}_s \cdot d\textbf{l} = n\hbar/2m$

The details behind these ideas are further discussed in this set of lecture notes, though I have to admit that these notes are quite dense. I still have some pretty major difficulties understanding some of the main ideas in them.

I welcome comments concerning these concepts, especially ones challenging the ideas put forth here.

## Kohn Anomalies and Fermi Surfaces

Kohn anomalies are dips in phonon dispersions that arise because of the presence of a Fermi surface. The presence of the Fermi surface renormalizes the bare phonon frequencies and causes an anomly in the phonon dispersion, as seen below for lead (taken from this paper):

Why this happens can be understood using a simplified physical picture. One can imagine that the ions form some sort of ionic plasma in the long-wavelength limit and we can use the classical harmonic oscillator equation of motion:

$m\frac{d^2\textbf{x}}{dt^2} = \frac{-NZ^2e^2\textbf{x}}{\epsilon_0}$

One can take into account the screening effect of the electrons by including an electronic dielectric function:

$m\frac{d^2\textbf{x}}{dt^2} = \frac{-NZ^2e^2\textbf{x}}{\epsilon(\textbf{q},\omega)\epsilon_0}$

The phonon frequencies will therefore be renormalized like so:

$\omega^2 = \frac{\Omega_{bare}^2}{\epsilon(\textbf{q},\omega)}$

and the derivative in the phonon frequency will have the form:

$\frac{d\omega}{d\textbf{q}} \propto -\frac{d\epsilon(\textbf{q},\omega)}{d\textbf{q}}$.

Therefore, any singularities that arise in the derivative of the dielectric function will also show up in the phonon spectra. It is known (using the Lindhard function) that there exists such a weak logarithmic singularity that shows up in 3D metals at $\textbf{q} = 2k_F$. This can be understood by noting that the ability of the electrons to screen the ions changes suddenly due to the change in the number of electron-hole pairs that can be generated below and above $\textbf{q}=2k_F$.

The dip in the phonon dispersion can be thought of as the phonon analogue of the “kinks” that are often seen in the electron dispersion relations using ARPES (e.g. see here). In the case here, the phonon dispersion is affected by the presence of the electrons, whereas in the “kink” case, the electronic dispersion is affected by the presence of the phonons (though kinks can arise for other reasons as well).

What is remarkable about all this is that before the advent of high-resolution ARPES, it was difficult to map out the Fermi surfaces of many metals (and still is for samples that don’t cleave well!). The usual method was to use quantum oscillations measurements. However, the group in this paper from the 60s actually tried to map out the Fermi surface of lead using just the Kohn anomalies! They also did it for aluminum. In both cases, they observed pretty good agreement with quantum oscillation measurements — quite a feat!