Lecture 3BEC at finite temperature

Thermal and quantum fluctuations in condensate fraction.

Phase coherence and incoherence in the many particle wave function.

Basic assumption and a priori justification

Consequences

Connection between BEC and two fluid behaviour

Connection between condensate and superfluid fraction

Why BEC implies sharp excitations.

Why sf flows without viscosity while nf does not.

How BEC is connected to anomalous thermal expansion as sf is cooled.

Hoe BEC is connecged to anomalous reduction in pair correlations as sf is cooled.

jj

j FTTf )()(

Thermal Fluctuations

2

),(1

)0( rdsrsdV

nF jjj

Boltzman factorexp(-Ej / T)/Zj

22 )( j

jj fFTf

At temperature T

Δf ~1/√NBasic assumption;

(√f is amplitude of order parameter)

Fj = f ± ~ 1/ √ N

dEEEgE jj

j )()(

All occupied states give samecondensate fraction

g(E)

η(E)

E

ΔE, Δf ~1/ √ N-1/2

As T changes band moves to different energy

“Typical” state gives different f

Can take one “typical” occupied stateas representative of density matrix

Drop subscript j to simplify notation

All occupied states gives same f to ~1/√N

Quantum Fluctuations

2

),(1V

dV

F rsr sss dfP )()(

sss dfFPF 22 )()(

V

dV

f rrs s )(1

)( V

dP rsrs2

),()()(

),()(

s

srrS

P

F = f ±~1/√N f(s) ~ f ± 1/√N

ΔF ~1/√N f(s) = F±~1/√N

width~ħ/L

Weightf

to ~1/√Nfor any

state and any s

Delocalised function of r(non-zero within volume > f V)J. Mayers Phys. Rev. Lett. 84 314 (2000),Phys. Rev.B 64 224521, (2001)

)(rS

2

).exp()()( rrprp S din

Phase correlations in r over distances ~L)(rS0~)( rrS dotherwise

BEC n(p)

Phase coherent

Condensate

Phase incoherent

No condensate

rC

~1/rC

Temperature dependence

At T = 0 , Ψ0(r,s) must be delocalised over volume ~ f0V and phase coherent.

For T > TB occupied states Ψj(r,s) must be either localised or phase incoherent.

What is the nature of the wave functions of occupied states for 0 < T < TB?

BASIC ASSUMPTION

Ψ(r,s) = b(s)Ψ0(r,s) + ΨR(r,s)

• Ψ0(r,s) is phase coherent ground state• ΨR(r,s) is phase incoherent in r • b(s) 0 as T TB for typical occupied state

• ΨR(r,s) 0 as T 0

1. Gives correct behaviour in limits T TB, T 0 2. True for IBG wave functions.3 Bijl-Feynman wave functions have this property4. Implications agree with wide range of experiments

),(),(),( 0 srsrsr

k

nN

nn

k

riksr

1

).exp(),(

Bijl-Feynman wave functionsJ. Mayers, Phys. Rev.B 74 014516, (2006)

),()(),( srsbsr R

b(s) is sum of all terms not containing r = r1

Phase coherent in r.

Fraction of terms in b(s) is (1-M/N) as N M N Θ(r,s) is phase incoherent (T TB)M 0 Θ(r,s) is phase coherent (T 0)

• nk = number of phonon-roton excitations with wave vector k.•M = total number of excitations• sum of NM terms.

k

knM

ΘR(r,s) is sum of terms containing r Phase incoherent in rrC ~1/Δk ~ 5 Å in He4 at 2.17K

2

0

22

),()(1

),(1

rdsrsdsbV

rdsrsdV

f

Consequences

Ψ(r,s) = b(s)Ψ0(r,s) + ΨR(r,s)

• If Δf ~1/N1/2

Nwsb C /1~)(2

rdsrrdsrwrdsr RC

22

0

2),(),(),(

MacroscopicSystem

X)(sX

dr N/1~

),(0 sr

),( srR

Microscopic basis of two fluid behaviour

NsXrdsrrdsrwrdsr RC /1~)(),(),(),(22

0

2

Momentum distribution and liquid flow split into two independent components of weights wC(T), wR(T).

rdsrrdsrwrdsr RC

22

0

2),(),(),(

2

).exp(),(1

)( rdrpisrsdV

pn

)()(0 pnwpnw RRC

Parseval’stheorem

wR = 1- wC.

RRC EwEwE 0

00 VC TES

Thermodynamic properties split into two independentcomponents of weights wc(T), wR(T)

Bijl-Feynman wR determined by number of “excitations”

wc(T) = ρS(T) wR(T) = ρN(T)

• True to within term ~N-1/2

• Only if fluctuations in f, ρS and ρN are negligible.• Not in limits T 0 T TB

)0()()( fTTf S

)0(

)(

f

Tf

o o T. R. Sosnick,W.M.Snow and P.E. Sokol Phys. Europhys Lett 9 707 (1989).X X H. R. Glyde, R.T. Azuah and W.G. Stirling Phys. Rev. B 62 14337 (2000).

NNST

PPV

EP

0

Superfluid has extra “Quantum pressure”

PN = PB

)0(

)(

T

)()(0 qSqS RNS

sdrdrqiN

srqS ni

1

2

2

1 ).exp(1

),()(

α < 1 → S less ordered than SR

SR-1

S-1

q

1)(

1)(

qS

qS

R

1 NS

]1)[(1)( 0 TT S

SR(q) S0(q) → Ψ0(r,s) and ΨR(r,s) 0 for different s

α(T)α0

V.F. Sears and E.C. Svensson, Phys. Rev. Lett. 43 2009 (1979).

]1)[(1)( 0 TT S

For s where Ψ0(r,s) 0 ~7% free volume

Why is superfluid more disordered?

Assume for s where ΨR(r,s) 0 negligible free volume

Ground state more disordered

J. Mayers Phys. Rev. Lett. 84 314 (2000)

Quantitative agreement with measurement at atomic size and N/V in liquid 4He

Phase coherent component Ψ0(r,s)

s such that Ψ0(r,s) is connected

nrdr 2).(

(Macro loops)

Quantised vortices, macroscopic quantum effects

s such that ΨR(r,s) is not connectedLocalised phase incoherent regions.Localised quantum behaviour over length scales rC ~ 5 ÅNo MQE or quantised vortices

Phase incoherent component ΨR(r,s)

Phase incoherent Regions of size ~rC

Normal fluid - momentum of excitations is uncertain to ~ ħ/rC

Superfluid - momentum can be defined to within ~ ħ/L

Excitations

)()(1

),(2

ff

f EEAN

S qq

srrqsrsrq ddiNA ff ).exp(),(),()( *

Momentum transfer = ħqEnergy transfer = ħω

|Aif(q)|2 has minimum widthΔq ~ 1/rC

Anderson and StirlingJ. Phys Cond Matt (1994)

q (Å-1)

ε ( deg K)

0 < T < TB

h/rC

Landau Theory

Basic assumption is that excitations with well definedenergy and momentum exist.

Landau criterion vC = (ω/q)min

Normal fluid vC = 0

ω

q

Only true in presence of BEC

Summary

BASIC ASSUMPTION Ψ(r,s) = b(s)Ψ0(r,s) + ΨR(r,s)

Phase coherentground state

Phase incoherent

• Has necessary properties in limits T0, T TB

• IBG, Bijl-Feynman wave functions have this form

•Simple explanations of•Why BEC is necessary for non-viscous flow

Why Landau theory needs BEC.

Summary

Theory given here explains quantitatively all these features

Existing microscopic theory does not provide even qualitativeexplanations of the main features of neutron scattering data

Why the condensate fraction is accurately proportional to the superfluid fraction

Why spatial correlations decrease as superfluid helium is cooled

Why superfluid helium is the only liquid which contains sharp excitations

Why superfluid helium expands when it is cooled

This is the only experimental evidence of the microscopic nature of Bose condensed helium.