Bose-Einstein Condensation and Superfluidity. Lecture 1. T=0 Motivation. Bose Einstein condensation (BEC) Implications of BEC for properties of ground state many particle WF. Feynman model Superfluidity and supersolidity. Lecture 2 T=0
only existing experimental evidence about the
microscopic nature of superfluid helium
A vast amount of neutron data has been collected
from superfluid helium in the past 40 years.
This data contains unique features, not observed in any other fluid.
These features are not explained even qualitatively
by existing microscopic theory
J. S. Brooks and R. J. Donnelly, J Phys. Chem. Ref. Data 6 51 (1977).
Normalised condensate fraction
o o T. R. Sosnick,W.M.Snow and P.E. Sokol Europhys Lett 9 707 (1989).
x xH. R. Glyde, R.T. Azuah and W.G. Stirling Phys. Rev. B 62 14337 (2000).
What is connection between condensate fraction and superfluid fraction?
Accepted consensus is that size of condensate
fraction is unrelated to size of superfluid fraction
more ordered as it is heated
in superfluid helium is
zero as T → 0. Why?
J. Mayers J. Low. Temp. Phys 109 135 (1997)
109 153 (1997)
J. Mayers Phys. Rev. Lett.80, 750 (1998)
84 314 (2000)
92 135302 (2004)
J. Mayers, Phys. Rev.B64 224521, (2001)
74 014516, (2006)
D. S. Durfee and W. Ketterle Optics Express 2, 299-313 (1998).
Phys Rev B 41 11185 (1989)
Kinetic energy of helium atoms.
J. Mayers, F. Albergamo, D. Timms
Physica B 276 (2000) 811
BEC in Liquid He4
f =0.07 ±0.01
N atoms in volume V
Periodic Boundary conditions
Each momentum state occupies volume ħ3/V
n(p)dp = probability of momentum p →p+dp
Number of atoms in single momentum state (p=0) is proportional to N.
Probability f that randomly chosen atom occupies p=0 state is independent of system size.
Number of atoms in any p state is independent of system size
Probability that randomly chosen atom occupies p=0 state is ~1/N
Quantum mechanical expression for n(p) in ground state
What are implications of BEC for
properties of Ψ?
= overall probability of configuration
s = r2, …rN of N-1 particles
ψS(r) is many particle wave function normalised over r
for given s
Condensate fraction for given s
|Ψ(r,s)|2 = P(r,s)= probability of configuration r,s of N particles
|ψS(r)|2isconditional probability that particle is at r, given s
Probability of momentum ħp given s
Implications of BEC for ψS(r)
ψS(r) non-zero function of r over length scales ~ L
ψS(r) is not phase incoherent in r – trivially true in ground state
Phase of ψS(r) is the same for all r in the
ground state of any Bose system.
Phase of Ψ(r,s,) is independent of r and s
Phase of ψS(r) is independent of r
Not true in Fermi systems
ψS(r) = 0 if |r-rn| < a
ψS(r) = cS otherwise
Feynman model for 4He ground state wave function
Ψ(r1,r2, rN) = 0 if |rn-rm| < a a=hard core diameter of He atom
Ψ(r1,r2, rN) = C otherwise
ΩS is total volume within which ψSis non-zero
Calculation of Condensate fraction in Feynman model
Take a=hard core diameter of He atom
N / V = number density of He II as T → 0
Generate random configurations s
(P(s) = constant for non-overlapping
spheres, zero otherwise)
Calculate “free” volume fraction for each randomly generated s with P(s) non-zero
Bin values generated.
J. Mayers PRL 84 314, (2000)
Has same value for all
possible s to within terms
Periodic boundary conditions.
Line is Gaussian with same mean and standard deviation as simulation.
f ~ 8%
O. Penrose and L. Onsager
Phys Rev 104 576 (1956)
What does “possible” mean?
Probability of deviation of 10-9 is
T. R. Sosnick,W.M.Snow
and P.E. Sokol
Europhys Lett 9 707 (1989).
In generalψS(r) is non –zero within volume >fV.
For any given fψS(r) non-zero within vol >fV
Feynman model -ψS(r) is non –zero within volume fV.
Assume ψS(r) is non zero within volume Ω
ψS = constant within Ω→ maximum value of f = Ω/V
Any variation in phase or amplitude within Ω
gives smaller condensate fraction.
eg ideal Bose gas → f=1 for ψS(r) =constant
For any possible sψS(r) must connected over macroscopic length scales
Loops in ψS(r) over macroscopic
ψS(r) must be non-zero within volume >fV.
In any Bose condensed system
ψS(r) must be phase coherent in r in the ground state
Loops in ψS(r) over
macroscopic (cm) length scales
Rotation of the container creates
a macroscopic velocity field v(r)
if BEC is preserved
Quantisation of circulation
Macrocopic ring of He4 at T=0
At low rotation
SupersolidityψS(r) in solid
Can still be connected over macro length
scales if enough vacancies are present
But how can a solid flow?
In frame rotating with ring
Leggett’s argument (PRL 25 1543 1970)
Ω = angular velocity of ring rotation
R = radius of ring
container is slowly rotated
Simplified model for ψS
Mass density conserved
In ring frame if
No mass rotates with ring
ρ2→0 → F=0
100% of mass rotates
with the ring.
Superfluid fraction determined by amplitude in connecting regions.
Can have any value between 0 and 1.
Condensate fraction determined by volume in which ψ is non-zero
ψ1→ 0 →50% supersolid fraction in model
connectivity suggests f~10% in hcp lattice.
O single crystal high purity He4
X polycrystal high purity He4
□ 10ppm He3 polycrystal
J. Mayers, F. Albergamo, D. Timms Physica B 276 (2000) 811
M. A. Adams, R. Down ,O. Kirichek,J Mayers
Phys. Rev. Lett. 98 085301 Feb 2007
Supersolidity not due to BEC
in crystalline solid
for all s
Superfluidity and Supersolidity
BEC in the ground state implies that;
ψS(r) is a delocalised function of r. – non zero over a volume ~V
Mass flow is quantised over macroscopic length scales