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May 13-14 (2011) GriffinFest, Toronto, Canada Formation of Magnetic Impurities, π-Junctions, and a Spontaneous Current State in a Superfluid Fermi Gas Yoji Ohashi1,3, Takashi Kashimura1 , and Shunji Tsuchiya2,3 1:Department of Physics, Keio University, Japan 2:Department of Physics, Tokyo University of Science, Japan 3:CREST (JST), Japan introduction idea to introduce magnetic impurities to a superfluid Fermi gas physical properties around magnetic impurity local density of states localized excited (bound) states superfluid/ferromagnet/superfluid-junction, and spontaneous current state 1991.4~1994.3 (PhD course : Tokyo Institute of Technology) “Antiferromagnetic spin fluctuations in high-Tc cuprates” 1991.4~1994.3 (PhD course : Tokyo Institute of Technology) “Antiferromagnetic spin fluctuations in high-Tc cuprates” 1991~1994 (PhD course : Tokyo Institute of Technology) 1994.4~1995.3 (PD: Osaka University) “Boundary effects on d-wave superconductors” 1991~1994 (PhD course : Tokyo Institute of Technology) 1994.4~1995.3 (PD: Osaka University) “Boundary effects in d-wave superconductors” 1991~1994 (PhD course : Tokyo Institute of Technology) 1994.4~1995.3 (PD: Osaka University) 1995.4~2006.3 (Tsukuba University) “Carlson-Goldman mode and Josephson plasma in high-Tc cuprates” 1991~1994 (PhD course: Tokyo Institute of Technology) 1994.4~1995.3 (PD: Osaka University) 1995.4~2006.3 (Tsukuba University) “Carlson-Goldman mode and Josephson plasma in high-Tc cuprates” 1991~1994 (PhD course : Tokyo Institute of Technology) 1994.4~1995.3 (PD: Osaka University) 1995.4~2006.3 (Tsukuba University) 2001.8~2002.6 (visiting researcher, Toronto University) “BCS-BEC crossover in a gas of Fermi atoms with a Feshbach resonance” 2004.9~2004.11 (visiting researcher, Toronto University) 2006.4~ (Keio University) 2011.5.13 and 14 (Toronto!) May 13-14 (2011) GriffinFest, Toronto, Canada Formation of Magnetic Impurities, π-Junction, and Spontaneous Current State in a Superfluid Fermi Gas Yoji Ohashi1,3, Takashi Kashimura1 , and Shunji Tsuchiya2,3 1:Department of Physics, Keio University, Japan 2:Department of Physics, Tokyo University of Science, Japan 3:CREST (JST), Japan Introduction Idea to introduce magnetic impurities to a superfluid Fermi gas Properties around pseudo-magnetic impurity density of states localized excited (bound) states superfluid/ferromagnet/superfluid-junction, π-junction, and spontaneous current state 40K and 66Li Fermi gases Superfluid 40 “quantum simulator” to study various physical properties of superconductivity BCS-BEC crossover by tunable interaction 40K strong-coupling effects beyond mean-field theory pseudogap (preformed pairs) Fermi SF Bose SF optical lattice C. A. Regal, et al. PRL 92 (2004) 040403. | 9 / 2, −7 / 2 > | 9/ 2, −9/ 2 > band effect low-dimensional effect Hubbard model and strong correlation population imbalance superconductivity under magnetic field FFLO “Magnetic” effect superfluid Fermi gas metallic superconductivity La3-xGdxIn 40K Tc Tc0 C. A. Regal, et al. PRL 92 (2004) 040403. | 9 / 2, −7 / 2 > | 9/ 2, −9/ 2 > “pseudospin” Crow et al., Phys. Lett. 21, 378 (1966). magnetic impurity concentration ? “magnetic” impurity “spin” Magnetic effects on superconductivity pairing mechanism n spi tions ua t c flu depairing effect pa r e p Coo ir magnetic impurity High-Tc cuprates: antiferromagnetic spin fluctuations Superfluid liquid 3He: ferromagnetic spin fluctuations pair-breaking effect by magnetic impurities Satori, Shiba (1992) Shiba(1968) Abrikosov, Gor’kov (1961) bound state below the gap suppression of SC state by magnetic scattering competition between SC and Kondo effect gapless SC Nonmagnetic impurities do not affect s-wave SC. (Anderson’s theorem) impurity band bound state energy /Δ ρimp × S ( S + 1)ui2mp N (0) / Tc0 superconducting DOS spin doublet Tc Tc0 Kondo singlet Kondo singlet TK = 0.3 ∆ TK / ∆ π-junction Insulator SC SC ferromagnet SC SC ∆ ( x) ∆( x) x 0 “0”-junction x 0 “π”-junction Nb ferromagnet ∆ ( x) SC Nb Al/Al2O3/PdNi T. Kontos et al., PRL 89, 137007 (2002) π-junction Insulator SC SC ferromagnet SC SC ∆ ( x) ∆( x) x 0 x 0 “π”-junction “0”-junction Nb ∆ ( x) F SC SC Nb Al/Al2O3/PdNi T. Kontos et al., PRL 89, 137007 (2002) We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas. Phase sepration in a superfuid Fermi gas with population imbalance pseudo-spin ↑ pseudo-spin ↓ ↑−↓ > 0 ↑ ↑ N SF N ↑↓ G. Partridge et al., PRL 97, 190407 (2006) We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas. Phase sepration in a superfuid Fermi gas with population imbalance pseudo-spin ↑ π-junction ∆( x) pseudo-spin ↓ x ↑−↓ > 0 N SF N ↑↓ G. Partridge et al., PRL 97, 190407 (2006) Super fluid N Ferro magnet ↑ Super fluid ↑ ↑ ↑↓ SF ↑↓ SF We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas. Phase sepration in a superfuid Fermi gas with population imbalance pseudo-spin ↑ pseudo-spin ↓ ↑−↓ > 0 ↑ polarized magnetic impurities ↑ N SF N ↑↓ ↑ ↑ ↑ ↑ G. Partridge et al., PRL 97, 190407 (2006) small nonmagnetic impurity potentials We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas. Phase sepration in a superfuid Fermi gas with population imbalance Ferro magnet ↑ ↑ ↑ ↑ Super fluid N Super fluid ↑ ↑↓ SF ↑↓ SF To examine these ideas in a simple manner, we consider a Hubbard model at T=0. We self-consistently determine the order parameter, particle density, and polarization, around impurity potential and barrier within the mean-field level. We examine the possibilities of magnetization of impurities, SFS-, and π-junction. Formulation two-component 2D Fermi gas population imbalance −t σ =↑, ↓ N↑ > N↓ −U H = −t ∑ ⎡⎣ci†σ c jσ + h.c ⎤⎦ − U ∑ ni↑ ni↓ + ∑ Vi niσ − µ↑ N ↑ − µ↓ N ↓ i, j i i ,σ “nonmagnetic” potential Vi [t ] impurity junction trap Formulation H = −t ∑ ⎡⎣ci†σ c jσ + h.c ⎤⎦ − ∑ ∆ i ⎡⎣ci†↑ ci†↓ + h.c.⎤⎦ − U ∑ niσ ni −σ i, j i i ,σ + ∑ Vi niσ − µ↑ N ↑ − µ↓ N ↓ + EG i ,σ We solve the BdG equations to self-consistently determine ∆ i , niσ , µσ for given Nσ . ni↑ + ni↓ ≅ 0.36 N↑ − N↓ P= N↑ + N↓ phase diagram of 2D uniform Hubbard model Formation of pseudo-magnetic impurity [t ] Γ2 Vi = V0 (R − R imp ) 2 + Γ 2 V0 / t = 0.25 Γ /t =1 U /t = 6 N ↑ = 300, N ↓ = 300 per 41× 41sites ∆ is not destroyed by impurity potential (Anderson’s theorem). N ↑ = 301, N ↓ = 300 per 41× 41sites ∆ is remarkably damaged by impurity potential. Formation of pseudo-magnetic impurity S z = ni↑ − ni↓ Local magnetization ∆N = N ↑ − N ↓ = 0 ∆N = 1 ∆N = 5 Nonmagnetic potential is magnetized by excess atoms! Magnetization of nonmagnetic potential “pseudo”-magnetic impurity To minimize the condensation energy loss by the depairing effect, excess atoms are localized around the region where ∆ is small from the begining. magnetic impurity in metal Strong electron correlation excludes double occupancy of ↑ and↓spins. Coulomb repulsion ∆( x) Vimp H = − J Sz σ z (classical spin) (no exchange term) x H = − JS ⋅ σ (quantum spin) Bound states by off-diagonal pair-potential well V ( x) ∆N = 5 pair-potential well ∆(R) bound states potential well V(R) We can expect that this well structure of off-diagonal pair potential ∆(Rxx,Ryy) induces bound states around the impurity potential, as in the case of ordinary (diagonal) potential well V(Rxx,Ryy) . local density of states ρ↑ ↑(ω,R) BCS gap uniform Fermi gas N↑ = N↓ N ↑ = 301, N ↓ = 300 Ry × impurity ρ↑(ω,R) × Rx In-gap states appear around “pseudo-magnetic” impurity. (γ / t = 0.05) ω /t bound state wavefunctions below the energy gap | Ψ particle ( Rx , Ry ) |2 L=1(p) L=2 (d) L=3 (f) 2nd lowest states 3rd lowest states 4th lowest states L=0 (s) Ry lowest bound state Rx The pair-potential well ∆(Rxx,Ryy) works as a cylindrical potential well, so that bound states can be classified by angular momentum L. Superfluid/ferromagnet/superfluid (SFS) junction V (R ) Ry Rx S z (R ) = ni↑ − ni↓ Population imbalance N↑ − N↓ P= = 0.11 N↑ + N↓ Ry Rx S F S U /t = 7 N = 99 ∆N = N ↑ − N ↓ = 11 π-phase in a superfluid Fermi gas S z (R ) N↑ − N↓ P= = 0.11 N↑ + N↓ Ry S ∆( Rx , Ry ) π-junction S F ∆( Rx , Ry ) _ + 0-junction Ry Rx E / t = −875.00 + + Ry Rx E / t = −874.78 stability of π-junction −1 [(tLy ) ] Eπ − junction − E0− junction π-junction 0-junction S z / Ly The π-junction becomes stable when magnetization is large to some extent. Effects of trap potential 2 2D cigar trap S z ( Rx , Ry ) N↑ = 32 N↓ = 28 U = 4t Vx = 0.1t ∆ ( Rx , Ry ) + Vy = 0.001t − VB = 1.0t σ =6 spontaneous current state in a ring trap Because of the single-valueness of ∆, the π-junction twists the phase θ of the order parameter by π along the ring. N↑ > N↓ Spontaneous current +∆ πi −∆ = ∆e J ~ ∇θ > 0 nonmagnetic potential barrier + localized excess ↑-spin atoms || π-junction ∆( x) + _ Rx “corner-junction” spontaneous current state in a ring trap J ~ ∇θ > 0 1D-ring trap ( N ↑ , N ↓ ) = (41, 40) +∆ −∆ = ∆eπ i nonmagnetic potential barrier + localized excess ↑-spin atoms || π-junction ∆( x) + _ Rx θ(x)/π N↑ > N↓ | ∆( x) | eiθ ( x ) δθ ~ π spontaneous flow Rx Summary We have discussed an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas. Using the phase separation of a polarized Fermi superfluid, we have shown that nonmagnetic potential is magnetized in the sense that some of excess atoms are localized around the potential. pseudo-magnetic impurities polarized classical spin suppression of ∆ around impurity in-gap states “quantum dot” gapless Fermi superfluid S superfluid/ferromagnet/superfluid junction ∆( Rx , Ry ) π-junction FFLO _ + Ry Rx F S ∆ FFLO ( x) S π spontaneous supercurrent Summary T. Kashimura, S. Tsuchiya, Y. Ohashi, Phys. Rev. A, 82, 033617 (2010) Y. Ohashi, Phys. Rev. A (2011), in press, cond-mat/1103.1942. T. Kashimura, S. Tsuchiya, and Y. Ohashi, in preparation pseudo-magnetic impurities polarized classical spin suppression of ∆ around impurity in-gap states “quantum dot” gapless Fermi superfluid S superfluid/ferromagnet/superfluid junction ∆( Rx , Ry ) π-junction FFLO _ + Ry Rx F S ∆ FFLO ( x) S π spontaneous supercurrent