SUPERSOLIDS? minnesota, july 2007 acknowledgments to: Moses Chan &Tony Clark David Huse & Bill Brinkman Phuan Ong& Yayu Wang Hari Kojima& many other exptlists

1Why not a supersolid? Some generalities 2Model for a supersolid: incompressible flow x-y model  vortex liquid 3Theory of vortex liquid: incompressible vorticity  NLRS 4Experiments: a) vortex liquid above Tc; b) supersolid? 1--why not a supersolid? Kohn’s observation: a solid (insulator) has local gauge symmetry: energy independent of phase. for Fermions, solids are easy: either band solid--energy gap in spectrum + Pauli principle--or Mott-Hubbard Solid: fill every site with one spin. topics

(filled band has well-known gauge symmetry) Bosons are harder. Hartree-Fock theory works fine at first sight: unlike Fermions, particle self-energy does not drop out and a big gap appears between localized hole states and Bloch waves for particles (PWA book, Ch IV) So one writes ground state for H-F theory here  * creates boson at r,  is the localised state at I. But  ’s are not orthogonal,so can interfere--state depends on phase? number on site i not well-defined? it seems hard to see how a ground state which must be positive everywhere can have energy not dependent on phase.

topic 2: whatever--if energy depends on phase, will be like So for He to rotate, we must introduce vortices which uniquely determine the flow: the superflow is a vortex liquid.

theory of vortex liquid above Tc: almost all experiments are consistent with vortex liquid. superflows at any time described by a fluctuating tangle of vortices; with curl and div J=0, uniquely so. In the following I use the Kosterlitz-Thouless 2D model as exemplar but the key point generalizes to 3D The energy is (aside from cores) all flow energy:

theory of vortex liquid cont. evaluating this using a lower cutoff a for size of vortex cores, and an upper cutoff at the sample size, gives simply This is not the expression used by K-T. That omitted the first term: assuming no net vorticity (or ignoring vortex self-energy). It depends on sample size.

The derivative is divergent as n V  0; this means the system is incompressible for vorticity. But you, with almost everyone who has seen this result,will say, “but the background density of vortex pairs must screen this energy out!”

Below Tc, the thermal vortices are bound in pairs, and give the tiny K-T correction; (in 3D, are loops--v Williams). Above Tc, the logarithmic entropy dominates the logarithmic energy so latter has no effect on their distribution, just their net density. (similar in 3D, but linear not log) in 2D; I don’t know how 3D works but in the pseudogap vortex liquid there seems to be a definable phase with average U  T caused by vortex loops. Conclusion: The Bose Vortex liquid has a vorticity-incompressible phase above Tc, and a nonlinear response falling off as ln .

experiments: some comments 1: Is it vortices? evidence 1) order of magnitude of crit velocity. 2)log dependence on velocity(see next slide) 3) sensitivity to He3: must be quantum effect 2: What happens at high T? vortex energy = kT c (M)x large log<<kT, nonlinearity goes away--see Kubota data! 3: Dissipation peak is where  (osc) =relaxation rate of vortices. (old slide) T dependence sign OK (Kojima) 4: Where is Tc? Must be below where any of this is. (remember, you heard this here first) conjecture: it is below Moses’ 60 mK line, which is just where true VL NLRS sets in. 5: Has Kojima seen it? gigantic hysteresis around 30 mK.

Strong and ‘universal’ velocity dependence in all samples v C ~ 10µm/s =3.16µm/s for n=1 ω R

Huse Proposal log Rate of Vortex motion  T Tc? He3 concentration dissi pation peak

conclusions, conjectures, confusions It is a quantum effect, probably vortices I believe most of what has been seen is NLRS Much of data confusion is strong nonlinearity-- you’re going too fast, guys Real Tc may have appeared I have no idea how He3 operates Or what relation of 2-level centers to vortices is Or whether PURE He4 is supersolid thanks for listening!