Quantum Fluids and Solids. Christian Enss (4He) & Henri Godfrin (3He)

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1 Quantum Fluids and Solids Christian Enss (4He) & Henri Godfrin (3He)

2 What Do We Consider as Quantum Matter? φ Lennard Jones Potential σ ε r De Boer quantum parameter: De Broglie wavelength:

3 Introduction Phase Diagrams He-II Experimental Observations Superfluidity Viscosity Second sound Two Fluid Model Hydrodynamics Thermodynamical effect Sound propagation Bose-Einstein Condensate Condensate fraction Macroscopic wave function Quantized vortices Normal Fluid Component Excitation spectrum Critical Velocity Quantum Crystals Overview Fermions: 3 He Phase diagram Melting curve Normal Fluid 3 He Fermi liquids Heat capacity Magnetic susceptibility Superfluid 3 He Cooper pairs A and B phases Vortices Dirac multi-spin exchange U2D2 phase CNAF phase Susceptibility and heat capacity 2D Systems Heisenberg ferromagnet Spin liquid Conclusions

4 Discovery of Helium Pierre Jules César Janssen Sir William Ramsay

5 Phase Diagrams of 3 He and 4 He

6 Solid Phases bcc phase much larger for 3 He less dense because of higher zero point energy zero point energy unimportant at high temperatures

7 Specific Heat at the Superfluid Transition

8 Superfluidity: Flow Through Thin Capillaries Hagen-Poiseuille law Flow velocity Experimental result: flow velocity is independent of pressure and increases with decreasing diameter

9 Temperature Dependence of Viscosity flow through capillary rotary viscosimeter oscillating disc Three different results and all experiments are correct! What is wrong?

10 Beaker Experiments

11 Beaker Experiments

12 Fountain Effect

13 Thermomechanical Effect direction of mass transport is opposite to heat transport very thin capillary (superleak)

14 Second Sound: Temperature waves Thermometer

15 Two Fluid Model basic idea T = T λ T = 0

16 Two Fluid Hydrodynamics mass flow mass conservation Euler equation entropy conservation equation of motion superfluid compontent

17 Viscosity Experiments Flow Through Capillaries: Superfluid component is flowing at critical velocity no viscosity below Lambda point Rotary Viscosimeter: Torque: Normal fluid component is observed below Lambda point viscocity depends on mean free path of excitations Oscillating Disc Viscosimeter: Torque: Product of normal fluid density and normal fluid viscocity is observed

18 Beaker flow: Helium films Film thickness at saturated vapor: For Film flows at the critical velocity

19 Thermo-mechanical Effect In equilibrium: (H.) London equation:

20 Sound Propagation From two-fluid hydrodynamics: Can be transformed into: two weakly coupled modes with and

21 First Sound Second Sound

22 Impossible d afficher l image. Bose-Einstein Condensation

23 Specific Heat Specific Heat ideal Bose-Gas liquid Helium Temperature Kelvin

24 Macroscopic Quantum State Macroscopic wave function Quantized circulation

25 Quantization of Circulation

26 Vortices with Quantized Circulation T = 0,1 K d = 2 mm l = 25 mm (a) (l) 3 8 Rev./min

27 Determination of ρ n Andronikashvili Aluminium discs Thickness 13 µm Diameter 3,5 cm Distance 210 µm Andronikashvili Second Sound

28 Excitation Spectrum of superfluid 4He Well-defined collective excitations maxons Phonons and Rotons No single particle excitations rotons phonons Theory: Landau

29 Concept of a Critical Velocity one excitation with: energy and momentum energy conservation momentum conservation Critical velocity

30 Solid 4 He

31 Crystallization waves in helium-4. Photograph taken by S. Balibar, C. Guthmann and E. Rolley in Paris. Growing helium-3 crystal at 2.2 mk Eleven different types of facets observed on the surface of helium-3 crystals. from Wagner et al., Phys Rev. Lett. 76, 263 (1996).

32 Nuclear magnetism of 3 He 3 He atom : nuclear spin ½ Fermion! Nuclear magnetic moment : µ = µ n µ/k B = Kelvin/Tesla γ/2π = MHz/Tesla Spin ½, large magnetic moment, good nucleus for NMR!!! In the solid phases the atoms are quasi-localized Zero point energy is comparable to the potential well depth, about 10 K (structural energies!). Large tunneling of atoms (frequency of order MHz) Quantum exchange interactions : J ~ 1 mk. Small dipole-dipole interactions E D ~ µ 2 /a 3 ~ µk

33 Phase Diagram of 3 He Erkki Thuneberg

34 Melting curve T > 1 K : classical phase diagram T < 1 K : the entropy of the solid is larger than that of the liquid Disordered solid (spin entropy Liquid is ordered in k-space (FermiLiquid ) Minimum of the melting curve (Clausius-ClapeyronEquation ) Solid nuclear order at T ~ 1 mk

35 Landau theory of Fermi liquids C = γt (T<<T F ) χ = c/t F ** η α T -2 Effective mass m* κ α T -1 C/C id = m*/m 2,8 to 5,8 increases with pressure Reinforced magnetism 9,2 to 23,7 increases with pressure «Fermi Gas» with renomalised parameters Theory is subtle! See Pines and Nozières books C C v id v χ 0 χ * m = m 1 =1+ 3 F ( 0) m* 1 = a ( 0) m 1+ F 0 s 1

36 Heat Capacity D.S. Greywall (1983)

37 Magnetic susceptibility χ 1 P = 2,90 MPa P = 1,85 MPa P = 1,10 MPa P=0,29 MPa χ 2 Cette thèse P=2,9MPa (2001) Thomson et al. P=3,0MPa (1962) Ramm et al. P=2,7MPa (1970) T (mk) T (mk) 1 + F 0 a 0,32 0,3 0,28 0,26 0,24 0,22 0,2 0,18 Presque localisé p=1,08 Paramagnons p=0,9 Cette thèse Renforcement magnétique Renforcement magnétique Cette thèse Ramm et al. 0,16 0,15 0,2 0,25 0,3 0,35 m / m* m* / m (PhD thesis S. Triqueneaux, 2001; V. Goudon, 2006)

The Nobel Prize in Physics 1996 "for their discovery of superfluidity in helium-3" David M. Lee Douglas D.

Richardson The figure shows the pressure inside a sample containing a mixture of liquid helium-3 and solid

The sample is first subjected to increasing external pressure for about 40 minutes, whereafter the external

38 The Nobel Prize in Physics 1996 "for their discovery of superfluidity in helium-3" David M. Lee Douglas D. Osheroff Robert C. Richardson The figure shows the pressure inside a sample containing a mixture of liquid helium-3 and solid helium-3 ice. The sample is first subjected to increasing external pressure for about 40 minutes, whereafter the external pressure is reduced. Note the changes in the slope of the curve at A and B and the temperatures at which these occur. The graph resembles that published by D.D. Osheroff, R.C. Richardson and D.M. Lee in Physical Review Letters 28, 885 (1972) in which the new helium-3 phase transitions were first reported..

39 Low temperature phase diagram

42 A-phase: iϕ µ i = Δe dµ ( m in ) A + i i

44 B-phase: A iϕ µ i n θ µ i µ = Δ e R(ˆ, ), i = 1,2,3

45 Rotation and vortices

46 Vortex in A phase Vortex in B phase

48 Superfluid 3 He bolometry Copper box Sintered silver 60 µm hole Vibrating Wires (5 µm and 13 µm)

50 Topological defects creation and fast transition: E Superconductors, 4 He, A e iφ

51 Phase Diagram of 3 He Erkki Thuneberg

52 Multi-spin exchange: of quasi-localized Fermions - Identical particles - Hamiltonian without explicit spin-dependent interactions Pauli principle: the spin state is coupled to the parity of the wave function Permutation of spins & particles: Dirac (1947) : Effective Hamiltonian on spin variables H ex = -ΣP (-1) p Jp P Two-particle permutations: P 2 = (1 + σi.σj) (Heisenberg Hamiltonian) Multi-spin exchange in solid 3He (Thouless, 1965) Three-particle exchange is also Heisenberg P3 = (1 + σi.σj+ σj.σk+ σk.σi) Four-spin exchange introduces new physics: P4 = (1 + Σ σµ.σν + Σ ((σi.σj).(σk.σl) + (σi.σl).(σj.σk) - (σi.σk).(σj.σl))) All exchange coefficients J are positive

53 Nuclear order: U2D2 phase

54 Nuclear order: CNAF phase

55 Heat capacity

56 Multi-spin exchange in 2D

57 2D Heisenberg Ferromagnet

58 Spin-liquid phase

59 Conclusions Quantum fluids constitute interesting, simple, quantum many-body systems Elementary excitations Superfluidity Crystallisation Magnetism Supersolidity?

PHYS 393 Low Temperature Physics Set 2: Liquid Helium-4

PHYS 393 Low Temperature Physics Set 2: Liquid Helium-4 Christos Touramanis Oliver Lodge Lab, Office 319 c.touramanis@liverpool.ac.uk He 4 atom Two protons, two neutrons in nucleus: I=0 Two electrons in