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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 50 Vycor y Torsional Oscillator Freezing Bulk 40 Freezing P c Nonsuperfluid 30 P [MPa] Vanishing of superfluidity at Vanishing of superfluidity at high pressures P > ~35 MPa wi wi high pressures 20 Bulk T  Vycor T c thout freezing thout freezing Superfluid 10 0 0 0.5 1 1.5 2 40 T [K] Superfluid The effect of particle The effect of particle The effect of particle The effect of particle 30 correlation and randomness correlation and randomness correlation and randomness correlation and randomness n [  mol/m 2 ] n c 20 Nonsuperfluid This phenomenon can be This phenomenon can be 10 described analytically? described analytically? 0 0 0.5 1 1.5 2 T [K]

Model Model 3-dimensional Bose Fluid in a Random Potential

Perturbation of K I Perturbation of K I 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 Calculation of self-energy 0.5 Calculation of self-energy 0.5 3 3 0.48 2 ma 2 /  2 [  k 0 +  (0, k )-  ] (na 3 )=0.1 0.48 2 ma 2 /  2 [  k 0 +  (0, k )-  ] (na 3 )=0.1 (na 3 )=0.5 (na 3 )=0.5 2 0.46 2 0.46 Free Boson Free Boson   0.44 0.44 1 1 0.42 0.42 0 0.4 0 0.4 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 n 1/3 a n 1/3 a ka ka

Calculation of critical temperature T c Calculation of critical temperature T c 1.2 1.2 1 1 0.8 0.8 For small a : increase of T c 0 T c / T c 0 0.6 T c / T c 0.6 → increase of the excitation 0.4 0.4 Calculation Calculation For large a : decrease of T c 0.2 0.2 Bulk liquid 4 He Bulk liquid 4 He 0 0 → increase of the effective 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 mass n 1/3 a n 1/3 a Difference between the calculation and liquid 4 He Difference between the calculation and liquid 4 He This may be caused by the long-range attraction of his may be caused by the long-range attraction of → T → liquid 4 He. liquid 4 He.

Perturbation of K R Perturbation of K R Random potential : taking ensemble average We assume that th We assume that th R 0 R 0 e quenched random e quenched random potential U decays potential k decays U k Perturbation above k above k p : Second-order | V ( k )| 2 / V p | V ( k )| 2 / V k p k p k k

Determination of Strength Determination of Strength of the Random Potential R 0 of the Random Potential R 0 Quantitative comparison of the critical adsorbed coverage with an experiment of dilute 4 He in porous glass M. Kobayashi and M. Tsubota, Phys. Rev. B66 174516 (2002) Calculation of superfluid Experimental measurement Calculation of superfluid Experimental measurement density of superfluidity density of superfluidity 3.5 10 3.5 10 R 0 =1.0 × 10 -75 [J 2 m 3 ] R 0 =1.0 × 10 -75 [J 2 m 3 ] 3.0 By the comparison, 3.0 8.0 R 0 =5.0 × 10 -75 [J 2 m 3 ] 8.0 R 0 =5.0 × 10 -75 [J 2 m 3 ] 2.5 n s [ × 10 -3 n bulk ] 2.5 we can obtain n s [ × 10 -3 n bulk ] R 0 =2.5 × 10 -74 [J 2 m 3 ] R 0 =2.5 × 10 -74 [J 2 m 3 ] 6.0 ∆ P [nsec] 2.0 6.0 ∆ P [nsec] 2.0 R 0 =5.0×10 -75 1.5 4.0 1.5 4.0 1.0 1.0 2.0 2.0 0.50 0.50 0.0 0.0 0.0 0.0 17.0 18.0 19.0 20.0 17.0 18.0 19.0 20.0 17.0 18.0 19.0 20.0 17.0 18.0 19.0 20.0 Coverage of 4 He [mg] Coverage [mg] Coverage of 4 He [mg] Coverage [mg]

Other quantitative parameters Other quantitative parameters m = 6.6×10 -27 kg : mass of 4 He a = 5 Å : s-wave scattering length of 4 He n bulk = 2.1×10 26 m -3 : density of bulk liquid 4 He V = 1 cm 3 : volume of porous glass k p = 25 Å : pore size of porous glass

Calculation of BEC and Superfluidity Calculation of BEC and Superfluidity Near T c Dyson- equation : one- bubble

Superfluidity : Linear response theory Superfluidity : Linear response theory Only the normal fluid density which have viscosity responds to dragging this pipe. n = n s + n n n s : superfluid density n n : normal fluid density

Dependence of the critical Results Dependence of the critical Results temperature on the density Disappearance of temperature on the density Disappearance of 2 2 BEC and BEC and superfluidity at high superfluidity at high densities T c [K] densities T c [K] 1 1 : Qualitative : Qualitative BEC agreement with the agreement with the BEC Superfluidity Superfluidity experimental result experimental result 0 0 0 0.1 0.2 0.3 0 0.1 0.2 0.3 n / n bulk n / n bulk 0.2 8 10 -3 0.2 8 10 -3 n/n bulk =0.09 n/n bulk =0.09 n/n bulk =0.09 n/n bulk =0.09 n/n bulk =0.18 n/n bulk =0.18 0.15 6 10 -3 n/n bulk =0.18 0.15 6 10 -3 n/n bulk =0.27 n/n bulk =0.18 n/n bulk =0.27 n/n bulk =0.27 n/n bulk =0.27 n 0 / n n s / n n 0 / n 4 10 -3 0.1 n s / n 4 10 -3 0.1 2 10 -3 0.05 2 10 -3 0.05 0 10 0 0 0 10 0 0 0.96 0.97 0.98 0.99 1 0.96 0.97 0.98 0.99 1 0.96 0.97 0.98 0.99 1 0.96 0.97 0.98 0.99 1 T / T c T / T c T / T c T / T c

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 obtain the BEC and superfluid critical temperatures. 3. BEC and superfluidity disappear at high densities. This is qualitatively consistent with the experimental result.