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Bose-Einstein Condensation Bose-Einstein Condensation and Superfluidity of and Superfluidity of Strongly Correlated Bose Strongly Correlated Bose Fluid in a Random Potential Fluid in a Random Potential

Michikazu Kobayashi and Makoto Tsubota Osaka-City-University, Japan

Experiment Experiment

• K. Yamamoto, H. Nakashima, Y. Shibayama and K. Shirahama,

cond-mat 0310375

Liquid 4He in porous Gelsil glass porous Gelsil glass

Pore size : ~ 25Å Filling rate : ~ 30% Pore area : ~ 130m²/cm³ Porous glass have Wormhole-like structure

Measurement of Superfluidity b y Torsional Oscillator

10 20 30 40 50 0.5 1 1.5 2 P [MPa] T [K] Superfluid Nonsuperfluid Vycor Tc Vycor Freezing Pc Bulk T Bulk Freezing 10 20 30 40 0.5 1 1.5 2 n [ mol/m2] T [K] Nonsuperfluid Superfluid nc

Vanishing of superfluidity at Vanishing of superfluidity at high pressures high pressures P > ~35 MPa wi wi thout freezing thout freezing

The effect of particle The effect of particle correlation and randomness correlation and randomness The effect of particle The effect of particle correlation and randomness correlation and randomness

This phenomenon can be described analytically? This phenomenon can be described analytically?

Model Model

3-dimensional Bose Fluid in a Random Potential

Perturbation of KI Perturbation of KI

Calculation of Green Function and Self-energy Calculation of Green Function and Self-energy Avoidance of divergence due to one-bubble Avoidance of divergence due to one-bubble

Self-energy : bubble approximation

1 2 3 0.5 1 1.5 Calculation of self-energy (na3)=0.1 (na3)=0.5 Free Boson 2ma2/2[k0+ (0,k)- ] ka 1 2 3 0.5 1 1.5 Calculation of self-energy (na3)=0.1 (na3)=0.5 Free Boson 2ma2/2[k0+ (0,k)- ] ka

0.4 0.42 0.44 0.46 0.48 0.5 0.5 1 1.5  n1/3a 0.4 0.42 0.44 0.46 0.48 0.5 0.5 1 1.5  n1/3a

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Calculation of critical temperature Tc Calculation Bulk liquid 4He Tc/Tc n1/3a 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Calculation of critical temperature Tc Calculation Bulk liquid 4He Tc/Tc n1/3a

For small a : increase of Tc →increase of the excitation For large a : decrease of Tc →increase of the effective mass Difference between the calculation and liquid Difference between the calculation and liquid 4

4He

He → →T This may be caused by the long-range attraction of his may be caused by the long-range attraction of liquid liquid 4

4He.

He.

Perturbation of KR Perturbation of KR

Perturbation : Second-order

Random potential : taking ensemble average

|V(k)|2/V k R0 kp |V(k)|2/V k R0 kp

We assume that th We assume that th e quenched random e quenched random potential potential U Uk

k decays

decays above above k kp

p

Determination of Strength

• f the Random Potential R0

Determination of Strength

• f the Random Potential R0

Quantitative comparison of the critical adsorbed coverage with an experiment of dilute 4He in porous glass

• M. Kobayashi and M. Tsubota, Phys. Rev. B66 174516 (2002)

0.0 0.50 1.0 1.5 2.0 2.5 3.0 3.5 17.0 18.0 19.0 20.0 Calculation of superfluid density

R0=1.0×10-75 [J2m3] R0=5.0×10-75 [J2m3] R0=2.5×10-74 [J2m3]

ns [×10-3nbulk] Coverage [mg] 0.0 0.50 1.0 1.5 2.0 2.5 3.0 3.5 17.0 18.0 19.0 20.0 Calculation of superfluid density

R0=1.0×10-75 [J2m3] R0=5.0×10-75 [J2m3] R0=2.5×10-74 [J2m3]

ns [×10-3nbulk] Coverage [mg] 0.0 2.0 4.0 6.0 8.0 10 17.0 18.0 19.0 20.0 Experimental measurement

• f superfluidity

∆P [nsec] Coverage of 4He [mg] 0.0 2.0 4.0 6.0 8.0 10 17.0 18.0 19.0 20.0 Experimental measurement

• f superfluidity

∆P [nsec] Coverage of 4He [mg]

By the comparison, we can obtain R0=5.0×10-75

Other quantitative parameters Other quantitative parameters

m = 6.6×10-27 kg : mass of 4He a = 5 Å : s-wave scattering length of 4He nbulk = 2.1×1026 m-3 : density of bulk liquid 4He V = 1 cm3 : volume of porous glass kp = 25 Å : pore size of porous glass

Calculation of BEC and Superfluidity Calculation of BEC and Superfluidity

Dyson- equation : one- bubble

Near Tc

Superfluidity : Linear response theory Superfluidity : Linear response theory

Only the normal fluid density which have viscosity responds to dragging this pipe. n=ns+nn ns : superfluid density nn : normal fluid density

Results Results

0 100 2 10-3 4 10-3 6 10-3 8 10-3 0.96 0.97 0.98 0.99 1

n/nbulk=0.09 n/nbulk=0.18 n/nbulk=0.27

ns / n T/Tc 0 100 2 10-3 4 10-3 6 10-3 8 10-3 0.96 0.97 0.98 0.99 1

n/nbulk=0.09 n/nbulk=0.18 n/nbulk=0.27

ns / n T/Tc 0.05 0.1 0.15 0.2 0.96 0.97 0.98 0.99 1

n/nbulk=0.09 n/nbulk=0.18 n/nbulk=0.27

n0 / n T/Tc 0.05 0.1 0.15 0.2 0.96 0.97 0.98 0.99 1

n/nbulk=0.09 n/nbulk=0.18 n/nbulk=0.27

n0 / n T/Tc 1 2 0.1 0.2 0.3

Dependence of the critical temperature on the density BEC Superfluidity

Tc [K] n/nbulk 1 2 0.1 0.2 0.3

Dependence of the critical temperature on the density BEC Superfluidity

Tc [K] n/nbulk

Disappearance of Disappearance of BEC and BEC and superfluidity at high superfluidity at high densities densities : Qualitative : Qualitative agreement with the agreement with the experimental result experimental result

Summary Summary

1. We compare the model of 3-dimensional Bose fluid in a random potential with the recent experiment by Yamamoto et. al. 2. By using the perturbation of repulsive interaction and the random potential, we can

• btain the BEC and superfluid critical

temperatures. 3. BEC and superfluidity disappear at high

• densities. This is qualitatively consistent with

the experimental result.