# How do we imagine the transition between phonons and rotons?

In the theory of superfluidity, we have both phonon and roton excitations. Phonons are long-range density fluctuations (sound waves) and rotons are short-range (atomic scale) circulations of the atoms of the fluid.

However, these are merely points along a continuous spectrum and so increasing the momentum of a phonon in a superfluid ought to convert it into a roton. How does this occur? Can it be pictured?

Both the phonon and the roton are features that appear in the spectral function $$\rho(\omega,k)=Im(G_R(\omega,k))$$ of the density-density correlation function $$G_R(\omega,k)=\int d^3xdt\, e^{-i(\omega t-kx)}\langle [\rho(x,t),\rho(0,0)]\rangle \Theta(t).$$ Any excitation in this channel is some form of "sound", and the spectral function will evolve smoothly from small $$k$$ ("phonons") to large $$k$$ rotons.

The phonon part is generic, and occurs in most fluids, superfluids, solids, etc. The phonon dispersion relation is approximatrely linear, $$\omega \simeq c_s k$$ where $$c_s$$ is the speed of sound, and the width of the phonon vanishes as $$k\to 0$$.

What is unusual about helium (and a few other materials) is the existence of a local minimum at relatively large $$k$$ $$\omega \simeq \Delta + \frac{(k-k^*)^2}{2m^*}$$ where the width is again small. Since the dispersion relation is smooth there is a local maximum between the phonon and the roton, known as the "maxon". The maxon is harder to see experimentally, because the width is not necessarily small, and it is not as important for thermodynamics and transport as phonons and rotons.

In the early days of low temperature physics there was a lot of discussion on the nature of the roton. Modern quantum many body physics reproduces the maxon-roton feature, but the physical origin is not always transparent. As a rough explanation, we still have Feynman's variational wave function, which gives $$\omega \simeq \frac{k^2}{2mS(k)}$$ where $$S(k)$$ is the static structure factor (the density-density correlation function at zero frequency). The roton minimum is then related to a peak in the structure factor, and this peak reflects density correlation in a liquid on the verge of solidifying. Modern theory does not really support notions of whirlpools or vortex rings, initially suggested by Landau (the name "roton") and Feynman.

• thanks very much for the considered answer. Could I ask what specifically contradicts the image of a whirlpool that Feynman and Landau suggest? As I understand it there is an excited wavefunction which (when obtained by a variational principle) involves a stationary region carrying probability current. To this, Feynman argued, there must be a correction in which the current is circulated from front to back in a loop.
– Kris
Dec 10 '18 at 17:19
• There is a basic fact which is true for any excitation: If the bare excitation receives an admixture from other states with momentum $+k$, there must be a backflow with momentum $-k$. This is already the case for a dressed phonon, or an ordinary quasi-particle. Dec 11 '18 at 2:37
• With regards to the vortex-ring picture, the basic problem is that the inverse wave vector $k^{-1}$ at the roton minimum corresponds to a microscopic distance (basically, the inter-atomic distance), so that a macroscopic picture (like smoke rings) does not make much sense. Dec 11 '18 at 2:38