Christian Veit 4th of June, 2013
2 / 29 The discovery of superfluidity Early 1930 s: Peculiar things happen in 4 He below the λ-temperature T λ = 2.17 K 1938: Kapitza, Allen & Misener measure resistance to the flow of liquid helium Superfluidity below T λ 1938: Fritz London - Superfluidity is related to Bose-Einstein condensation Fountain effect. 1 until 1941: Laszlo Tisza and Lev Landau - two fluid model 1972: Superfluidity in 3 He also in fermionic systems 1 Nature 141, 243-244 (1938)
3 / 29 Superfluidity in 4 He Phase diagram of 4 He. 2 Heat capacity of 4 He. 2 2 from: Tony Guénault, Basic Superfluids
4 / 29 Outline The Landau criterion Two fluid model Off-diagonal long-range order Quantization of flow First & second sound Supersolids Superfluidity in 3 He
5 / 29 The Landau criterion Where does the friction come from? Dissipation arises from elementary excitations At what velocity is it possible to create excitations?
6 / 29 The Landau criterion What is the requirement to create an excitation?
The Landau criterion Landau critical velocity: E(p) v c = min p p Ideal Bose gas: v c = 0 ms 1 no superfluidity! Weakly interacting Bose gas: E(p) = ( p2 2m )2 + gn m p2 v c c Ideal and weakly interacting Bose gas. Liquid helium: v c 60 ms 1 Dispersion relation from neutron scattering experiments 3 from: Tony Guénault, Basic Superfluids (altered) Liquid 4 He. 3 7 / 29
8 / 29 Two fluid model Torsional oscillator: oscillation frequency ω = K I Andronikashvili s experiment. 4 Experimental Results. 4 Two fluid modell: Mass density: ρ = nm = ρ N + ρ S Viscosity: η N > 0, η S = 0 4 from: Tony Guénault, Basic Superfluids
9 / 29 Two fluid model Component densities Can we give an explanation for the behavior of the component densities? Uniform fluid at finite, small temperature in a capillary Just thermal excitations Assume: gas of noninteracting excitations (quasiparticles) in thermal equilibrium Mass flow associated with quasiparticles is not superfluid
10 / 29 Two fluid model Component densities At equilibrium: mean velocity v N of excitations equals velocity of capillary Relative velocity of superfluid and capillary: v S v N Excitation energy in capillary frame: E(p) + p(v S v N ) Equilibrium distribution of excitations: f p = exp 1 ( E(p)+p (vs v N ) kt ) 1
11 / 29 Two fluid model Component densities Mass density: ρ = nm = ρ S + ρ N Mass current: mj = ρ S v S + ρ N v N Total momentum carried by the fluid: P = M v S + p i mj = ρv S + i d 3 p ρ N (v N v S ) = (2π ) 3 pf p d 3 p (2π ) 3 pf p For small relative velocities v S v N : ρ N = 1 d 3 p df p(e, v S v N = 0) 3 (2π ) 3 p2 de
12 / 29 Two fluid model Component densities Low temperatures: just phonon part contributes E(p) = cp ρ N = 2π2 (kt ) 4 45 3 c 5 from: Tony Guénault, Basic Superfluids
13 / 29 Off-diagonal long-range order
14 / 29 Off-diagonal long-range order One particle density matrix describes correlation between the particles: ρ 1 (r r ) = ˆψ (r) ˆψ(r ) r r 0 : ρ 1 (r r ) n r r : ρ 1 (r r ) n 0 Off-diagonal long-range order Calculation via quantum Monte Carlo method Liquid helium: n 0 (T = 0) 0.1n n 0 (T > T λ ) 0 from: J. F. Annett, Superconductivity, Superfluids, and Condensates
Off-diagonal long-range order Momentum distribution N k = a k a k = N 0 δ k,0 + Ñ(k) from: J. F. Annett, Superconductivity, Superfluids, and Condensates (altered) At T=0 the superfluid density approaches ρ but just 10% of the helium is condensed into the groundstate! 15 / 29
16 / 29 Off-diagonal long-range order Order parameter Points are statistically independent for r r : ˆψ (r) ˆψ(r ) ˆψ (r) ˆψ(r ) Order parameter of the system: ψ 0 (r) = ˆψ(r) = n 0 e iθ
17 / 29 Thermodynamic effects Consequences of the two fluid model Entropy is carried by normal component: S S = 0 heat transport exclusively due to ρ N very efficient heat transport (frictionless counterflow) Fountain effect: from: Tony Guénault, Basic Superfluids from: J. F. Annett, Superconductivity, Superfluids and Condensates (altered)
18 / 29 Quantization of flow Order parameter: ψ 0 = n 0 e iθ Superfluid velocity: v s = m θ Irrotational flow: v s = 0 Circulation: κ = dr v s = m dr θ = m θ from: J. F. Annett, Superconductivity, Superfluids and Condensates Quantisation: θ = 2πn κ = h m n
Quantization of flow dr v s = h m n Rotate slowly with ω at T > T λ Fluid velocity: v = ωr Moment of inertia: I cl = NmR 2 Cool through T λ ω N = ω ω S = mr 2 n n takes value closest to ω/ω c for n=1: 2πR ω c R = h m ω << ω c : I(T ) < I cl ω c = mr 2 19 / 29
20 / 29 Vortices In cylindrical coordinates: v s = 0 r 0 if 1 r v s = r (rv Φ) = 0 κ 2πr e Φ No singularities in v s ψ 0 vanishes at r = 0 Vortex cores in helium: 1Å Visualization: tracer particles from: J. F. Annett, Superconductivity, Superfluids and Condensates
21 / 29 First & second sound Superfluid: Two degrees of freedom associated with normal and superfluid component two sound-like modes Coupled wave equations: 2 ρ t 2 = 2 p 2 s t 2 = ρ s s 2 2 T Sound velocities: ρ n First sound: pressure/density wave Second sound: temperature/entropy wave c1 2 = p ρ, c2 2 = ρ st s 2 ρ n c
22 / 29 First & second sound First sound: Density variations driven by pressure variations Two components in phase from: Phys. Today 62, 10, 34 (2009) Second sound: Density constant Composition of density varies Entropy variations driven by temperature variations Two components counter-oscillate
from: R. Srinivasan: Second Sound: Waves of Entropy and Temperature. Resonance 3: 16 24 23 / 29 Second sound Experiment by Peshkov (1944) Ac current creates temperature variations Glass tube acts as resonator standing wave Standing wave patterns are traced with a movable thermometer
24 / 29 Supersolids Science 24 September 2004: 305 (5692), 1941-1944 Superfluid behavior in crystal Off-diagonal + diagonal long-range order Non-classical rotational inertia
25 / 29 Supersolids Torsional oscillator experiment Superfluid fraction decouples from oscillation Reduced moment of inertia: I(T ) = I classical ρn(t ) ρ Resonance frequency: ω r = K I from: Science 24 September 2004: 305 (5692), 1941-1944
26 / 29 Supersolids Experimental results Non-Classical Rotational Inertia Fraction. 5 Phase diagram. 5 5 from: Science 24 September 2004: 305 (5692), 1941-1944
27 / 29 Supersolids Models and problems Models: Zero motion creates a gas of zero-point vacancies undergoing BEC superfluidity of vacancies = superflow of particles Exchange processes between neighboring atoms But: Not clear if interpretation of experiments is correct! existence of supersolids in nature not verified
28 / 29 Superfluidity in 3 He 3 He: fermion T c 2.5 mk Phase diagram: without B-field: 6 with B-field: 6 Superfluid state linked with condensation pairing mechanism 6 from: Tony Guénault, Basic Superfluids
29 / 29 Summary Superfluidity below Landau critical velocity v c Two fluid model: ρ = ρ N + ρ S Off-diagonal long-range order Irrotational flow of superfluid vortices, NCRI First & second sound: counter-oscillating ρ N and ρ S Supersolids: superfluid behavior in solids, NCRI Superfluidity in 3 He pairing mechanism