Is the Roton in Superfluid ⁴He the Ghost of a Bragg Spot?

Abstract

It has always been assumed that the roton in ⁴He had to do with local vorticity—hence the name! We present here an alternate view: the roton is viewed as a “soft mode”, precursor of a crystallization instability. In such a picture the liquid is “nearly solid”, and the long observed similarities of heat propagation in liquid and solid phases are naturally explained. In this qualitative paper we consider three models successively. A lattice gas with one atom per site displays a Mott localization transition, as shown by the Bangalore group. The important result is the vanishing of the superfluid order parameter (condensate fraction N _o) at the transition. There is no breakdown of translational symmetry and consequently no soft mode. Another lattice model with half filling, with an added nearest neighbour repulsion, was studied by Matsubara and Matsuda in the early 1950s it displays a first order transition between a superfluid and a localized charge density wave state. The excitation spectrum has a soft mode at zone edge near the transition, signalling the proximity of the CDW instability. Finally, we consider the realistic situation of a continuous system with no preexisting lattice. We approach the problem from the limit N _o = 0 instead of the ideal gas N _o = N. When N _o = 0 the quasiparticle spectrum and the charge density spectrum are decoupled. The latter should have a soft mode ω=ω_m if crystallization is close. That soft mode is a normal state property that has nothing to do with superfluidity. A small N _o acts to hybridize quasiparticles and density fluctuations: the resulting anticrossing lowers ω_m ² as well as the ground state energy. We show that N _o is bounded for two reasons: (i) if ω _m ² turns negative, the liquid is unstable towards freezing (ii) depletion due to quantum fluctuation exceeds N _o if the latter is too large. The resulting upper bound noindent for N/N _{o is} ≪ 1, a consequence of the deep roton minimum. The whole paper is qualitative, based on outrageous simplifications in order to make algebra tractable.