1. Lecture 3 BEC 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.

2. Thermal Fluctuations Boltzman factor exp(-E j / T)/Z j At temperature T Δf ~1/√N Basic assumption; (√ f is amplitude of order parameter) F j = f ± ~ 1/ √ N

3. All occupied states give same condensate fraction g(E) η(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 state as representative of density matrix Drop subscript j to simplify notation All occupied states gives same f to ~1/√N

4. Quantum Fluctuations F = f ±~1/√N f(s) ~ f ± 1/√N ΔF ~1/√N f(s) = F±~1/√N

5. width~ħ/L Weight f to ~1/√N for 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) Phase correlations in r over distances ~L otherwise BEC n(p)

6. Phase coherent Condensate Phase incoherent No condensate rCrC ~1/r C

7. Temperature dependence At T = 0, Ψ 0 (r,s) must be delocalised over volume ~ f 0 V and phase coherent. For T > T B occupied states Ψ j (r,s) must be either localised or phase incoherent. What is the nature of the wave functions of occupied states for 0 < T < T B ?

8. BASIC ASSUMPTION Ψ(r,s) = b(s)Ψ 0 (r,s) + Ψ R (r,s) Ψ 0 (r,s) is phase coherent ground state Ψ R (r,s) is phase incoherent in r b(s)  0 as T  T B for typical occupied state Ψ R (r,s)  0 as T  0 1.Gives correct behaviour in limits T  T B, T  0 2.True for IBG wave functions. 3Bijl-Feynman wave functions have this property 4. Implications agree with wide range of experiments

9. Bijl-Feynman wave functions J. Mayers, Phys. Rev.B 74 014516, (2006) b(s) is sum of all terms not containing r = r 1 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  T B ) M  0 Θ(r,s) is phase coherent (T  0) n k = number of phonon-roton excitations with wave vector k. M = total number of excitations sum of N M terms. Θ R (r,s) is sum of terms containing r Phase incoherent in r r C ~1/Δk ~ 5 Å in He4 at 2.17K

10. Consequences Ψ(r,s) = b(s)Ψ 0 (r,s) + Ψ R (r,s) If Δf ~1/N 1/2

11. Macroscopic System X Microscopic basis of two fluid behaviour

12. Momentum distribution and liquid flow split into two independent components of weights w C (T), w R (T). Parseval’s theorem w R = 1- w C.

13. Thermodynamic properties split into two independent components of weights w c (T), w R (T) Bijl-Feynman w R determined by number of “excitations” w c (T) = ρ S (T) w R (T) = ρ N (T) True to within term ~N -1/2 Only if fluctuations in f, ρ S and ρ N are negligible. Not in limits T  0 T  T B

14. 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).

15. Superfluid has extra “Quantum pressure” P N = P B

16. α < 1 → S less ordered than S R S R -1 S-1 q

17. S R (q)  S 0 (q) → Ψ 0 (r,s) and Ψ R (r,s)  0 for different s α(T) α0α0 V.F. Sears and E.C. Svensson, Phys. Rev. Lett. 43 2009 (1979).

18. For s where Ψ 0 (r,s)  0 ~7% free volume Why is superfluid more disordered? Assume for s where Ψ R (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 4 He

19. Phase coherent component Ψ 0 (r,s) s such that Ψ 0 (r,s) is connected (Macro loops) Quantised vortices, macroscopic quantum effects

20. s such that Ψ R (r,s) is not connected Localised phase incoherent regions. Localised quantum behaviour over length scales r C ~ 5 Å No MQE or quantised vortices Phase incoherent component Ψ R (r,s)

21. Phase incoherent Regions of size ~r C Normal fluid - momentum of excitations is uncertain to ~ ħ/r C Superfluid - momentum can be defined to within ~ ħ/L Excitations Momentum transfer = ħq Energy transfer = ħω |A if (q)| 2 has minimum width Δq ~ 1/r C

22. Anderson and Stirling J. Phys Cond Matt (1994) q (Å -1) ε ( deg K) 0 < T < T B h/r C

23. Landau Theory Basic assumption is that excitations with well defined energy and momentum exist. Landau criterion v C = (ω/q) min Normal fluid v C = 0 ω q Only true in presence of BEC

24. Summary BASIC ASSUMPTION Ψ(r,s) = b(s)Ψ 0 (r,s) + Ψ R (r,s) Phase coherent ground state Phase incoherent Has necessary properties in limits T  0, T  T B IBG, Bijl-Feynman wave functions have this form Simple explanations of Why BEC is necessary for non-viscous flow Why Landau theory needs BEC.

25. Summary Theory given here explains quantitatively all these features Existing microscopic theory does not provide even qualitative explanations 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.