Transcript SUPERSOLIDS? minnesota, july 2007

SUPERSOLIDS? minnesota,
july 2007
acknowledgments to:
Moses Chan &Tony Clark
David Huse & Bill Brinkman
Phuan Ong& Yayu Wang
Hari Kojima& many other
exptlists
topics
1 Why not a supersolid? Some generalities
2 Model for a supersolid: incompressible flow
x-y modelvortex liquid
3 Theory of vortex liquid: incompressible
vorticityNLRS
4 Experiments: 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.
(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
  ( bi *)vac
i
bi * 
3
d
 ri (r) * (r)
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
topic 2: whatever--if energy depends on phase, will be like
E   t ij g cos(i   j ) which coarse- grains to
2
ij
 d r ( ) , and implies a supercurrent
3
2
s
J =  s . [1]  can vary in space- -may be due t o defects?
Current can't accumulate: crucial that
  J  0. I've separated mat rix element int o kinetic
energy t and small overlap2 .g s 

Becauseof [1]
  J  0!!
2
/2M, M  m He
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
ˆ /|rr |
Ji  i  qi
i
i
qi  1 Here i is a vortex,not a site.
The energy is (aside from cores) all flow
energy:
1
U
2

d 2 r( J i ) 2
i
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
U / 2   qi2 ln(R / a)  qi qj ln R / rij
i j
i
 ( qi ) ln R / a   qi qj ln rij / a
2
i

i j
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.

A mismatch in vortex numbers means that the sample is rotating as
a whole (or, in the superconducting case, that it is experiencing an
external B-field). As has been understood since the ‘50’s, the
minimum energy configuration will be a uniform array of vortices
which is the closest mimic of rigid rotation; and the divergent selfenergy for r>the lattice constant of this array may be cancelled
against whatever source of energy is causing the rotation; but there
still remains the energy caused by quantization of the vorticity,
which leads to a nonuniform local velocity. This energy is (if the
density of extra vortices is nV) proportional to
2
nV ln(1/ nV a )
The derivative is divergent as nV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)
npr  (1/ a 2 )exp[Ec /(T  Tc )]
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 = kTc(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
ω
R
vC~ 10µm/s
h
 vs dl  m  n
h
vs 
n
2Rm
=3.16µm/s
for n=1
Huse Proposal
He3 concentration
log Rate
of Vortex
motion
dissi
pation
peak

Tc?
T
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!