Michikazu Kobayashi. Kyoto University

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1 Topological Excitations and Dynamical Behavior in Bose-Einstein Condensates and Other Systems Michikazu Kobayashi Kyoto University Oct. 24th, 2013 in Okinawa International Workshop for Young Researchers on Topological Quantum Phenomena in Condensed Matter with Broken Symmetries 2013

2 Contents 1. Bose-Einstein condensates with internal degrees of freedom 2. Spin-2 spinor BEC 3. Vortices in spinor BEC 4. Dynamics of vortices in spinor BEC 5. Summary

3 Contents 1. Bose-Einstein condensates with internal degrees of freedom 2. Spin-2 spinor BEC 3. Vortices in spinor BEC 4. Dynamics of vortices in spinor BEC 5. Summary

4 Bose-Einstein Condensate with Internal Degrees of Freedom Scalar BEC without internal degrees of freedom y(x)= y(x) exp[i (x)] : broken U(1) symmetry of global phase shift

5 Bose-Einstein Condensate with Internal Degrees of Freedom BEC with internal degrees of freedom 1. Multi-component BEC (ex. 87 Rb and 41 K BECs or different hyperfine level) 2. Spinor BEC (ex. 87 Rb spin-1 and spin-2 BECs) magnetic trap : spin degrees of freedom is frozen scalar BEC laser trap : spin degrees of freedom is alive spinor BEC Hyperfine spin : F = I + S 87 Rb, 23 Na, 7 Li, 41 K F=1, 2 I : nuclear spin S : electron spin 85 Rb F=2, Cs F=3, 4 52 Cr F=3

6 Symmetry and Topological Excitation in BEC Order parameter manifold (G/H) Topological excitation Scalar BEC U(1)/1 U(1) vortex 2-component BEC (miscible) U 1 U 1 /Z 2 vortex 2-component BEC (inmiscible) U 1 U 1 /(U(1)/Z 2 ) O(2) U(1) Z 2 vortex & domain wall Spin-1 BEC (ferro) (U(1) SO(3))/U(1) SO(3) vortex Spin-1 BEC (polar) (U(1) SO(3))/(U(1) Z 2 ) (U(1) S 2 )/Z 2 Spin-2 BEC (ferro) (U(1) SO(3))/(U(1) Z 2 ) SO(3)/Z 2 vortex & monopole vortex Spin-2 BEC (uniaxial nematic) U(1) SO(3)/(U(1) Z 2 ) U(1) RP 2 vortex & monopole Spin-2 BEC (biaxial nematic) (U 1 SO 3 )/D 4 vortex (non-abelian) Spin-2 BEC (cyclic) (U 1 SO 3 )/T vortex (non-abelian)

7 Symmetry and Topological Excitation in BEC Order parameter manifold (G/H) Topological excitation Scalar BEC U(1)/1 U(1) vortex 2-component BEC (miscible) U 1 U 1 /Z 2 vortex 2-component BEC (inmiscible) U 1 U 1 /(U(1)/Z 2 ) O(2) vortex & domain wall Spin-1 BEC (ferro) (U(1) SO(3))/U(1) SO(3) vortex Spin-1 BEC (polar) (U(1) SO(3))/(U(1) Z 2 ) (U(1) S 2 )/Z 2 Spin-2 BEC (ferro) (U(1) SO(3))/(U(1) Z 2 ) SO(3)/Z 2 vortex & monopole vortex Spin-2 BEC (uniaxial nematic) U(1) SO(3)/(U(1) Z 2 ) U(1) RP 2 vortex & monopole Spin-2 BEC (biaxial nematic) (U 1 SO 3 )/D 4 vortex (non-abelian) Spin-2 BEC (cyclic) (U 1 SO 3 )/T vortex (non-abelian)

8 Contents 1. Bose-Einstein condensates with internal degrees of freedom 2. Spin-2 spinor BEC 3. Vortices in spinor BEC 4. Dynamics of vortices in spinor BEC 5. Summary

9 Theory of Spinor BEC H = dx 1 ħ 2 Hamiltonian of Bose system with spin 2M Ψ m (x 1 ) Ψ m (x 1 ) dx 2 Ψ m1 x 1 Ψ m2 x 2 V m1 m 2 m 1 m 2 (x 1 x 2 )Ψ m2 (x 2)Ψ m1 (x 1) Low energy contact interaction (l = 0) V m1 m 1 m 1 m 2 x 1 x 2 = δ x 1 x 2 g F FM FM O m1 m 2 O m1 m 2 F=even m 1 m 2 m 1 m 2 M Coupling constant depends on total spin of two colliding particles

10 Theory of Spinor BEC For spin-2 case H = dx ħ2 2M Ψ m Ψ m + c 0 2 : n2 : + c 1 2 : F2 : + c 2 2 A A 20 c 0 = 4g 2 + 3g 4 7, c 0 = g 4 g 2 7, c 0 = 7g 0 10g 2 + 3g 4 35 n = Ψ m Ψ m : number density operator F = Ψ m F mn Ψ n : spin density operator A 20 = ( 1) m Ψ m Ψ m : time reversal operator (singlet-pair amplitude)

11 Theory of Spinor BEC Mean-field theory at T = 0 : ψ = ψ m a m,k=0 N 0 : all particles condense into a single-particle ground state H = dx ħ2 2M ψ m ψ m + c 0 2 n2 + c 1 2 F2 + c 2 2 A A 20 n = ψ m ψ m : number density F = ψ m F mn ψ: spin density A 20 = ( 1) m ψ m ψ m : singlet-pair amplitude

12 Phase Diagram for Ground State H = dx ħ2 2M ψ m ψ m + c 0 2 n2 + c 1 2 F2 + c 2 2 A A 20 Uniaxial Nematic: c 1 Cyclic: ψ m U = T degenerate Biaxial Nematic: ψ B m = T 87 Rb ψ C m = 1 T 2 i i Ferromagnetic: ψ F m = T c 2 c 2 = 4c 1 A. Widera et al. NJP 8, 152 (2006)

13 Spherical Harmonic Representation ψ θ, φ = 2 m= 2 ψ m Y 2,m (θ, φ)

14 Phase Diagram for Ground State H = dx ħ2 2M ψ m ψ m + c 0 2 n2 + c 1 2 F2 + c 2 2 A A 20 Uniaxial Nematic: ψ m U = T D : cylindrical symmetry Biaxial Nematic: ψ m B = T D 4 : square symmetry c 2 = 4c 1 87 Rb Cyclic: T : tetrahedral symmetry Ferromagnetic: c 1 ψ m C = 1 2 i i T ψ m F = T c 2 U 1 Z 2 : oriented toroidal symmetry

15 Symmetry of cyclic state Spin rotates by p 0-2p/3 2p/3 Phase shift by 2p/3 and spin rotates by 2p/3 -p p

16 Symmetry of cyclic state Spin rotations keeping cyclic state invariant form a non Abelian tetrahedral symmetry

17 Phase Diagram for Ground State H = dx ħ2 2M ψ m ψ m + c 0 2 n2 + c 1 2 F2 + c 2 2 A A 20 Uniaxial Nematic: ψ m U = T D : cylindrical symmetry Biaxial Nematic: ψ m B = T D 4 : square symmetry c 2 = 4c 1 Cyclic: T : tetrahedral symmetry Ferromagnetic: c 1 ψ m C = 1 2 i i T ψ m F = T c 2 U 1 Z 2 : oriented toroidal symmetry

18 Phase Diagram for Ground State H = dx ħ2 2M ψ m ψ m + c 0 2 n2 + c 1 2 F2 + c 2 2 A A 20 Uniaxial Nematic: c 1 Non-Abelian vortices appear due to non-abelian discrete symmetry Cyclic: T : tetrahedral symmetry ψ m C = 1 2 i i T Biaxial Nematic: ψ m B = T D 4 : square symmetry Ferromagnetic: c 2 c 2 = 4c 1

19 Contents 1. Bose-Einstein condensates with internal degrees of freedom 2. Spin-2 spinor BEC 3. Vortices in spinor BEC 4. Dynamics of vortices in spinor BEC 5. Summary

20 Quantized Vortices in BEC Quantized vortex for m = +1 y For scalar BEC : y= y e im Topological charge can be characterized by widing number m (additive group of integers) -p p Arg(y)

21 Non-Abelian Vortex Topological charge of vortices Scalar BEC Integer (winding of phase by 2p multiple) Cyclic phase in spin-2 spinor BEC Component of tetrahedral group

22 Vortices in cyclic state 1/2 spin vortex 1/3 vortex p 2p/3 ψ = 1 2 ieiφ ie iφ T ψ = 1 3 eiφ T

23 Vortices in biaxial nematic state 1/2 spin vortex 1/4 vortex 1/2 vortex 12 eiφ e iφ T 12 eiφ T 12 0 eiφ T

24 Contents 1. Bose-Einstein condensates with internal degrees of freedom 2. Spin-2 spinor BEC 3. Vortices in spinor BEC 4. Dynamics of vortices in spinor BEC 5. Summary

25 Gross-Pitaevskii Equation Coherent dynamics of mean-field : Gross-Pitaevskii equation iħ ψ ±1 t iħ ψ ±2 t = ħ2 2M 2 ψ ±2 + c 0 nψ ±2 + c 1 F ψ 1 ± 2F z ψ ±2 + c 2 5 A 00ψ 2 = ħ2 6 2M 2 ψ ±1 + c 0 nψ ±1 + c 1 2 F 0ψ 0 + F ± ψ ±2 ± F z ψ ±1 c 2 5 A 00ψ 1 iħ ψ 0 t = ħ2 2M 2 ψ 0 + c 0 nψ c 1 F ψ 1 + F + + c 2 5 A 00ψ 0

26 Collision Dynamics For Abelian vortex Reconnect : All Abelian vortices such as those in scalar BEC Pass through : Rarely seen for quantized vortices, and sometimes seen for disclination in liquid crystals or cosmic strings

27 Collision Dynamics for Cyclic Phase There are 12 kinds of vortices in cyclic phase For same kinds of vortices : +2p/3 & +2p/3 reconnection For different and commutative vortices : +2p/3 & -2p/3 pass through What happens for non-commutative vortices for different spin rotations?

28 Collision Dynamics for Cyclic Phase New rung vortex appears bridging two colliding vortices

29 Collision of Vortex B A AB BA B A A ABA -1 BA -1 AB=BA B A=B B -1 AB A ABA -1 A B A A

30 Monopole Confined in Vortex Junction Vortex core has usually internal structure different from cyclic state depending on the charge of vortex (ex. ferromagnetic state with F 0) Monopole (divf 0) appears at the Y-shape junction point Magnetization and its divergence is confined to only vortex lines confined monopole (charge is classified by the tetrahedral symmetry)

31 Monopole Confined in Vortex Junction Each arrow shows the direction of magnetization Confined monopoles appear at the junctions points of vortices as a form of monopole-antimonopole pair

32 Collision Dynamics in Biaxial Nematic Phase 1/4 vortex 1/2 vortex Rung vortex burst and disappears Non-Abelian property cannot be seen

33 Degeneration between Uniaxial and Biaxial Nematic Phases H = dx ħ2 2M ψ m ψ m + c 0 2 n2 + c 1 2 F2 + c 2 2 A A 20 Uniaxial Nematic: c 1 degeneration with another continuous degree of freedom 12 cosη 0 2sinη 0 cosη T ψ m U = T D : cylindrical symmetry Biaxial Nematic: ψ m B = T c 2 D 4 : square symmetry c 2 = 4c 1

34 Degeneration between Uniaxial and Biaxial Nematic Phases Uniaxial Nematic: Biaxial Nematic: 12 cosη 0 2sinη 0 cosη T Very large order-parameter manifold : (U 1 S 4 )/Z 2 Several vortices in both phases are topologically unstable due to η

35 Quasi-Nambu-Goldstone Mode 12 cosη 0 2sinη 0 cosη T η is not the symmetry of the Hamiltonian (accidental symmetry) Gapless excitation mode due to η is not the true Nambu-Goldstone mode (Quasi-Nambu-Goldstone mode)

36 Thermal Phase Diagram Quasi-Nambu-Goldstone mode easily becomes gapful through (quantum or) thermal fluctuation Uniaxial Nematic: c 1 Cyclic: ψ m U = T phase boundary at finite T Biaxial Nematic: ψ B m = T 87 Rb ψ C m = 1 T 2 i i Ferromagnetic: ψ F m = T c 2 c 2 = 4c 1

37 BEC at Finite Temperature Stochastic Gross-Pitaevskii equation (complex Langevin equation) iħ ψ m t = 1 γ δh δψ m + ξ ξ x, t ξ(x, t ) = k B Tδ x x δ(t t ) Condensate fraction and its fluctuation

38 Collision Dynamics at Finite Temperature T = 0.1T c T = 0.4T c Non-Abelian property is restored by massive quasi- Nambu-Goldstone mode due to thermal fluctuation

39 Summary 1. Spin-2 spinor BEC can have exotic non-abelian vortex due to non-abelian discrete symmetry. 2. Collision of non-abelian vortex in the cyclic phase show the formation of rung vortex bridging colliding vortices. 3. After the collision, there appears a monopole confined in the Y-shaped junction. 4. Non-Abelian property disappears in the biaxial nematic phase due to the quasi-nambu-goldstone mode, and is restored by thermal fluctuations at finite temperatures.

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