More about rotons in superfluid helium 4

In a recent paper(1) it was argued that rotons in superfluid helium 4 are the soft modes announcing a charge density wave that leads to the crystal: rotons are a normal state property. A small superfluid condensate acts to hybridize quasiparticles and soft density fluctuations - hence a level repulsion that lowers the energy: superfluidity is energetically favourable. A shallow roton implies a very small condensate density, as found in He4: what we need is a saturation mechanism. The clue is depletion due to quantum fluctuations. In (1) we assumed that such a depletion was drawn from the condensate itself: superfluidity then disappears in the liquid if the roton gap is too small. Here we explore an alternate possibility: quantum fluctuations are drawn from the normal fluid. We reach the opposite conclusion: superfluidity persists down to the spinodal limit where the roton gap vanishes, with an unusual power law dependence. We briefly mention the possible extension of that argument to a frozen charge density wave: in a toy 1d model it might shed light on the features that favour supersolids.