Electron bubbles and Weyl fermions in chiral superfluid 3 He-A - PowerPoint PPT Presentation

Electron bubbles and Weyl fermions in chiral superfluid 3 He-A Oleksii Shevtsov James A. Sauls Northwestern University, Evanston, IL 60208 USA October 12, 2016 (1) Introduction to superfluid 3 He and e -bubbles (2) Transport and AHE of e -bubbles in 3 He-A: Experiments (3) Structure and transport of e -bubbles in 3 He-A: Theory [O. Shevtsov and J. A. Sauls, PRB 94, 064511 (2016)]

Symmetry group of normal-state 3 He : G = SO ( 3 ) S × SO ( 3 ) L × U ( 1 ) N × P × T BW state (B-phase): J = 0 , J z = 0 34 A 30 T AB 24 B p/ bar 18 p PCP 12 A µ j = ∆ δ µ j T c 6 ABM state (A-phase): 0 L z = 1 , S z = 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 T/ mK J. J. Wiman and J. A. Sauls PRB 92 , 144515 (2015) Spin-triplet p -wave order parameter: ∆ αβ ( k ) = � d ( k ) · ( i � σ σ y ) αβ , d µ ( k ) = A µ j k j A µ j = ∆ˆ d µ ( ˆ m + i ˆ n ) j

Magnetic field B : suppresses the | ↑↓� + | ↓↑� spin component in the order parameter makes the A-phase ( | ↑↑� + | ↓↓� ) more favorable at critical temperature T c Ikegami et al. J. Phys. Soc. Jpn. 84 , 044602 (2015)

Magnetic field B : suppresses the | ↑↓� + | ↓↑� spin component in the order parameter makes the A-phase ( | ↑↑� + | ↓↓� ) more favorable at critical temperature T c Ikegami et al. J. Phys. Soc. Jpn. 84 , 044602 (2015) Edge states spectrum:

Electron bubbles in liquid 3 He Bubble with R ≃ 1 . 5 nm, QPs mean free path l ≫ R , λ f ≪ R ≪ ξ 0 Knudsen limit Effective mass M ≃ 100 m 3 Normal-state mobility is const ( m 3 – atomic mass of 3 He) below T = 50 mK

Electron bubbles in chiral superfluid 3 He-A k ) = ∆ k x + ik y ∆ A (ˆ k f v AH v E � �� � ���� µ AH E × ˆ Electric current: v = µ ⊥ E + l Salmelin et al. PRL 63 , 868 (1989) Hall ratio: tan α = v AH / v E = | µ AH /µ ⊥ |

Mobility of e -bubbles in 3 He-A (Ikegami et al., RIKEN) Science 341 , 59 (2013); JPSJ 82 , 124607 (2013); JPSJ 84 , 044602 (2015) v AH v E � �� � ���� µ AH E × ˆ Electric current: v = µ ⊥ E + Hall ratio: tan α = v AH / v E = | µ AH /µ ⊥ | l ;

Mobility of e -bubbles in 3 He-A (Ikegami et al., RIKEN) Science 341 , 59 (2013); JPSJ 82 , 124607 (2013); JPSJ 84 , 044602 (2015) v AH v E � �� � ���� µ AH E × ˆ Electric current: v = µ ⊥ E + Hall ratio: tan α = v AH / v E = | µ AH /µ ⊥ | l ;

Mobility of e -bubbles in 3 He-A (Ikegami et al., RIKEN) Science 341 , 59 (2013); JPSJ 82 , 124607 (2013); JPSJ 84 , 044602 (2015) v AH v E � �� � ���� µ AH E × ˆ Electric current: v = µ ⊥ E + Hall ratio: tan α = v AH / v E = | µ AH /µ ⊥ | l ;

Mobility of e -bubbles in 3 He-A (Ikegami et al., RIKEN) Science 341 , 59 (2013); JPSJ 82 , 124607 (2013); JPSJ 84 , 044602 (2015) v AH v E � �� � ���� µ AH E × ˆ Electric current: v = µ ⊥ E + Hall ratio: tan α = v AH / v E = | µ AH /µ ⊥ | l ; tan α

Equation of motion for e -bubbles in 3 He-A: (i) M d v dt = e E + F QP , F QP – force due to quasiparticle scattering

Equation of motion for e -bubbles in 3 He-A: (i) M d v dt = e E + F QP , F QP – force due to quasiparticle scattering (ii) F QP = − ↔ ↔ η · v , η – generalized Stokes tensor

Equation of motion for e -bubbles in 3 He-A: (i) M d v dt = e E + F QP , F QP – force due to quasiparticle scattering (ii) F QP = − ↔ ↔ η · v , η – generalized Stokes tensor   η ⊥ η AH 0 (iii) ↔ in superfluid 3 He-A with ˆ η = − η AH η ⊥ 0 l � e z   0 0 η �

Equation of motion for e -bubbles in 3 He-A: (i) M d v dt = e E + F QP , F QP – force due to quasiparticle scattering (ii) F QP = − ↔ ↔ η · v , η – generalized Stokes tensor   η ⊥ η AH 0 (iii) ↔ in superfluid 3 He-A with ˆ η = − η AH η ⊥ 0 l � e z   0 0 η � (iv) M d v dt = e E − η ⊥ v + e for E ⊥ ˆ c v × B eff , l

Equation of motion for e -bubbles in 3 He-A: (i) M d v dt = e E + F QP , F QP – force due to quasiparticle scattering (ii) F QP = − ↔ ↔ η · v , η – generalized Stokes tensor   η ⊥ η AH 0 (iii) ↔ in superfluid 3 He-A with ˆ η = − η AH η ⊥ 0 l � e z   0 0 η � (iv) M d v dt = e E − η ⊥ v + e for E ⊥ ˆ c v × B eff , l (v) B eff = − c B eff ≃ 10 3 − 10 4 T !!! e η AH ˆ l ,

Equation of motion for e -bubbles in 3 He-A: (i) M d v dt = e E + F QP , F QP – force due to quasiparticle scattering (ii) F QP = − ↔ ↔ η · v , η – generalized Stokes tensor   η ⊥ η AH 0 (iii) ↔ in superfluid 3 He-A with ˆ η = − η AH η ⊥ 0 l � e z   0 0 η � (iv) M d v dt = e E − η ⊥ v + e for E ⊥ ˆ c v × B eff , l (v) B eff = − c B eff ≃ 10 3 − 10 4 T !!! e η AH ˆ l , (vi) d v − 1 v = ↔ ↔ µ = e ↔ dt = 0 µ E , where η � µ � = e η ⊥ η AH , µ ⊥ = e , µ AH = − e η � η 2 ⊥ + η 2 η 2 ⊥ + η 2 AH AH

Scattering of Bogoliubov QPs off the negative ion

Scattering of Bogoliubov QPs off the negative ion (i) Lippmann-Schwinger equation for the retarded T -matrix ( ε = E + i η , η → 0 + ): � d 3 k ′′ � � ˆ S ( k ′ , k , E )= ˆ ( 2 π ) 3 ˆ ˆ S ( k ′′ , E ) − ˆ ˆ T R T R N ( k ′ , k ) + T R N ( k ′ , k ′′ ) G R G R N ( k ′′ , E ) T R S ( k ′′ , k , E )

Scattering of Bogoliubov QPs off the negative ion (i) Lippmann-Schwinger equation for the retarded T -matrix ( ε = E + i η , η → 0 + ): � d 3 k ′′ � � ˆ S ( k ′ , k , E )= ˆ ( 2 π ) 3 ˆ ˆ S ( k ′′ , E ) − ˆ ˆ T R T R N ( k ′ , k ) + T R N ( k ′ , k ′′ ) G R G R N ( k ′′ , E ) T R S ( k ′′ , k , E ) � � − ∆(ˆ � k ) | 2 , ξ k = � 2 k 2 1 ε + ξ k k ) ˆ k + | ∆(ˆ G R ξ 2 S ( k , E ) = , E k = 2 m ∗ − µ ε 2 − E 2 − ∆ † (ˆ k ) ε − ξ k k

Scattering of Bogoliubov QPs off the negative ion (i) Lippmann-Schwinger equation for the retarded T -matrix ( ε = E + i η , η → 0 + ): � d 3 k ′′ � � ˆ S ( k ′ , k , E )= ˆ ( 2 π ) 3 ˆ ˆ S ( k ′′ , E ) − ˆ ˆ T R T R N ( k ′ , k ) + T R N ( k ′ , k ′′ ) G R G R N ( k ′′ , E ) T R S ( k ′′ , k , E ) � � − ∆(ˆ � k ) | 2 , ξ k = � 2 k 2 1 ε + ξ k k ) ˆ k + | ∆(ˆ G R ξ 2 S ( k , E ) = , E k = 2 m ∗ − µ ε 2 − E 2 − ∆ † (ˆ k ) ε − ξ k k (ii) Normal-state T -matrix: � � N (ˆ k ′ , ˆ t R k ) 0 ˆ N (ˆ k ′ , ˆ T R k ) = in p-h (Nambu) space N ( − ˆ k ′ , − ˆ − [ t R k )] † 0

Scattering of Bogoliubov QPs off the negative ion (i) Lippmann-Schwinger equation for the retarded T -matrix ( ε = E + i η , η → 0 + ): � d 3 k ′′ � � ˆ S ( k ′ , k , E )= ˆ ( 2 π ) 3 ˆ ˆ S ( k ′′ , E ) − ˆ ˆ T R T R N ( k ′ , k ) + T R N ( k ′ , k ′′ ) G R G R N ( k ′′ , E ) T R S ( k ′′ , k , E ) � � − ∆(ˆ � k ) | 2 , ξ k = � 2 k 2 1 ε + ξ k k ) ˆ k + | ∆(ˆ G R ξ 2 S ( k , E ) = , E k = 2 m ∗ − µ ε 2 − E 2 − ∆ † (ˆ k ) ε − ξ k k (ii) Normal-state T -matrix: � � N (ˆ k ′ , ˆ t R k ) 0 ˆ N (ˆ k ′ , ˆ T R k ) = in p-h (Nambu) space, where N ( − ˆ k ′ , − ˆ − [ t R k )] † 0 ∞ k ) = − 1 � k ′ · ˆ N (ˆ k ′ , ˆ ( 2 l + 1 ) e i δ l sin δ l P l (ˆ t R k ) , P l – Legendre polynomial π N f l = 0

Scattering of Bogoliubov QPs off the negative ion (i) Lippmann-Schwinger equation for the retarded T -matrix ( ε = E + i η , η → 0 + ): � d 3 k ′′ � � ˆ S ( k ′ , k , E )= ˆ ( 2 π ) 3 ˆ ˆ S ( k ′′ , E ) − ˆ ˆ T R T R N ( k ′ , k ) + T R N ( k ′ , k ′′ ) G R G R N ( k ′′ , E ) T R S ( k ′′ , k , E ) � � − ∆(ˆ � k ) | 2 , ξ k = � 2 k 2 1 ε + ξ k k ) ˆ k + | ∆(ˆ G R ξ 2 S ( k , E ) = , E k = 2 m ∗ − µ ε 2 − E 2 − ∆ † (ˆ k ) ε − ξ k k (ii) Normal-state T -matrix: � � N (ˆ k ′ , ˆ t R k ) 0 ˆ N (ˆ k ′ , ˆ T R k ) = in p-h (Nambu) space, where N ( − ˆ k ′ , − ˆ − [ t R k )] † 0 ∞ k ) = − 1 � k ′ · ˆ N (ˆ k ′ , ˆ ( 2 l + 1 ) e i δ l sin δ l P l (ˆ t R k ) , P l – Legendre polynomial π N f l = 0 Hard-sphere model � tan δ l = j l ( k f R ) / n l ( k f R ) , j l , n l – spherical Bessel fn-s k f R – the only adjustable parameter (to be determined)!

From T-matrix to drag force (i) QP scattering rate – Fermi’s golden rule: Γ( k ′ , k ) = 2 π � W (ˆ k ′ , ˆ k ) δ ( E k ′ − E k )

From T-matrix to drag force (i) QP scattering rate – Fermi’s golden rule: Γ( k ′ , k ) = 2 π k ) = 1 � � � W (ˆ k ′ , ˆ k ) δ ( E k ′ − E k ) , W (ˆ k ′ , ˆ |� out k ′ ,σ ′ | ˆ T S | in k ,σ �| 2 2 σ,σ ′ = ↑ , ↓ in , out

From T-matrix to drag force (i) QP scattering rate – Fermi’s golden rule: Γ( k ′ , k ) = 2 π k ) = 1 � � � W (ˆ k ′ , ˆ k ) δ ( E k ′ − E k ) , W (ˆ k ′ , ˆ |� out k ′ ,σ ′ | ˆ T S | in k ,σ �| 2 2 σ,σ ′ = ↑ , ↓ in , out (ii) Drag force from QP-ion collisions (linear in v ): Baym et al. PRL 22 , 20 (1969) � � � � �� − ∂ f k ′ − ∂ f k � � ( k ′ − k ) � k ′ v f k Γ( k ′ , k ) F QP = − − � kv ( 1 − f k ′ ) ∂ E ∂ E k , k ′