Quantum Properties of Two-dimensional Helium Systems

Transcription

1 Quantum Properties of Two-dimensional Helium Systems Hiroshi Fukuyama Department of Physics, Univ. of Tokyo 1. Quantum Gases and Liquids 2. Bose-Einstein Condensation 3. Superfluidity of Liquid 4 He 4. Elementary Excitations in Superfluid 4 He 5. Critical Phenomena of Lambda Transition 6. Kosterlitz-Thouless Transition in 4 He Thin Films 7. Gas-Liquid Transition of He Monolayers 8. Quantum Solids (magnetism and defectons) 9. Two Dimensional Solid 3 He (quantum spin liquid) Intensive Nov. 2017

2 1 Quantum Gases and Liquids Lennard-Jones type potential: Principle of corresponding state: Phase diagrams of classical rare gases scale with the reduced P*, V *and T*.* strong short-range repulsion (hardcore) weak long-range attraction (van der Waals force) Phase diagram of classical materials U (r) s e r Pressure 0 Solid triple point (T t ) Liquid Temperature critical point (T c ) Gas

3 Quantum parameter Quantum parameter: K: kinetic energy U : potential energy experimental reduced critical temperature: T c * T c * 1.5 Xe, Kr Ar Ne H 2 3D 2D 4 He 3 He T c * ! quantum parameter: h

4 Phase diagram of 4 He composite boson second most common element in universe most pure and simple condensed matter large quantum effects (zero-point energy) superfluidity (T < T l ) Bose-Einstein condensation

5 2 Bose-Einstein Condensation Bose-Einstein condensation (BEC) When the thermal de Broglie wavelength (l T ) becomes comparable to mean interparticle distance (a), a phase transition to an unusual phase with the Bose-Einstein condensate (BEC) takes place at T = T c in ideal Bose gases. 2 f (e) C V / R l T a gives: T / T c µ e 0 e Fermi degeneracy emerges in Fermion systems at nearly the same temperature as T BEC.

6 Historical papers on BEC Satyendra N. Bose ( ) S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift fu r Physik 26, 178 (1924) thermal de Broglie wavelength ( mean inter-particle distance (a) ) macroscopic wave-fn. Albert Einstein ( ) A. Einstein, Akademie der Wissenschaften, Berlin, Sitzungsberichte, 3 (1925) ; (idid. 261 (1924)) However, it is evident that at low temperatures l becomes of the order of magnitude of s for the gases Hydrogen and Helium, and it indeed seems that the coefficient of friction will suffer the influence which we have to expect according to the theory. Therefore, close to that temperature an accelerated decrease of the viscosity with decreasing temperature will set in rather abruptly. An estimate of that temperature on the basis of the relation l = s yields 56 for H2, 40 for He. lt one particle wave-fn.

7 Lambda transition of liq. 4 He below 2.17 K l-transition in liq. 4 He should be a realization of Bose-Einstein condensation (BEC) in strongly interacting Bose systems. F. London, Nature 141, 643 (1938); Phys. Rev. 54, 947 (1938) Nobel prizes in Physics phenomenological theory (1962) experimental finding of superfluidity (1978) Fritz London ( ) M.J. Buckingham and W.M. Fairbank (1961) T l = 2.17 K Lev D. Landau ( ) phonon-roton dispersion Pyotr Kapitza ( ) h (T < T l ) 10 3 h (T > T l ) T BEC (ideal gas ) = 3.1 K L.D. Landau, J. Phys. USSE 11, 91 (1947) P. Kapitza, Nature 141, 74 (1938)

8 BEC in dilute alkali-atom vapors (quantum gas) Nobel prize in Physics 2001 velocity distribution Laser cooling and trapping Eric A. Cornell (1961 ) Wolfgang Ketterle (1957 ) Carl E. Wieman (1951 ) M.H. Anderson et al., Science 269, 198 (1995) J.R. Ensher et al, PRL 77, 4984 (1996) Weakly interacting quantum gas Nearly ideal BEC Condensate fraction 100%, as T 0. Finite numbers of atom trapped in a harmonic potential T / T 0 (N) 87 Rb T c = 170 nk n = cm -3 (N 4,000 at T c ) finite size liq. 4 He N 0 /N = 1 - (T/T c ) 3

9 Interactions change specific heat anomalies at BEC transition BEC in dilute Bose gas theory (finite number ideal Bose gas) Lambda transition in liq. 4 He exp. M.J. Buckingham and W.M. Fairbank (1961) T l = 2.17 K J. R. Ensher et al., PRL 77, 4984 (1996) T BEC (ideal gas ) = 3.1 K Dilute Bose gas: n cm -3 T BEC << 10-6 K Liq. 4 He: n cm -3 T l = 2 K (high-t superfluidity!)

10 Fabrication of 3D dilute 4 He gas d = 4-8 nm adsorption of 4 He thin film J.D. Reppy et. ai., PRL 84,2060 (2000) porous glass (Vycor) 3D connected 4 He gas d < l T at low-t Measurement of superfluid density with torsional oscillator torsion rod sample µ superfluid density ideal Bose gas r s (1 - T/T c ) 2/3 : strongly interacting 3D Bose system torsional oscillator high density

11 PIMC cal. Interaction effects on BEC T 0 = T BEC (ideal Bose gas) experimental (dilute 4 He gas) RG cal. exp. (liq. 4 He) n : density a : hardcore diam. Path integral Monte Carlo (PIMC) calculation P. Gruter, D. CeperIey and F. LaIoe, PRL 79, 3549 (1997) g = 0.34 ± 0.06 At low n, repulsive (hardcore) interactions suppress spatial density fluctuations. At high n, atomic hindrance suppresses phase coherence. 1 H ( 10) 4 He Renormalization group theory g = 4.66 only applicable to low n region H.T.C. Stoof, PRA 45, 8398 (1992) 23 Na 87 Rb Experimental J.D. Reppy et. ai., PRL 84,2060 (2000) g = 5.1 ± 0.9 m*/m = 1.34 ± 0.2