Thesis defense Excitations in topological superfluids and - PowerPoint PPT Presentation

Thesis defense Excitations in topological superfluids and superconductors Hao Wu Supervisor: James A. Sauls Northwestern University December 3, 2016 DMR-1106315

Introduction – superconductivity and superfluidity Heike Onnes: superconductor & liquid helium ”Mercury has passed into a new state, which on account of its extraordinary electrical properties may be called the superconductive state” – Onnes 1911 ”his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium” – Nobel prize committee 1913 Superconductivity – zero conductivity, Meissener effect Ginzburg-Landau Complex order parameter Ψ Gauge symmetry breaking U ( 1 ) Bardeen-Cooper-Schrieffer (BCS) theory Electron-phonon interaction Cooper pair – Bound state of electrons | p , "i , | � p , #i I Fermi statistics I attractive interaction Bogoliubov quasiparticles ( u ( p ) , v ( p )) T

Introduction – superconductivity and superfluidity Superfluid 4 He – frictionless flow, quantized vortices Bose-Einstein condensation: macroscopically occupied ground state Gauge symmetry breaking Superfluid 3 He Anisotropic Cooper pair of 3 He atoms I Repulsive interaction non-swave I Spin susceptibility spin-triplet pairing Attractive interaction – spin exchange Unconventional superfluid

Spin-Triplet, P-wave Pairing - Superfluid Phases of 3 He Symmetry Group of Normal 3 He : G = SO ( 3 ) S ⇥ SO ( 3 ) L ⇥ U ( 1 ) N ⇥ P ⇥ T Chiral ABM State ~ Phase Diagram of Bulk 3 He l = ˆ m ⇥ ˆ n 30 A 3 He 25 Superfluid PCP 20 P [bar] 15 A µ i = ∆ ˆ d µ ( ˆ m + i ˆ n ) i B L z = 1 , S z = 0 10 “Isotropic” BW State 5 0 0.0 0.5 1.0 1.5 2.0 2.5 T [mK] Spin-Triplet, P-wave Order Parameter : A µ i = ∆ δ µ i ∆ αβ ( p ) = ~ d ( p ) · ( i ~ σσ y ) αβ d µ ( p ) = A µ i p i J = 0 , J z = 0

Symmetry and topology of superfluid phases ✓ ◆ ξ ( p ) c ( p x ± ip y ) m ( p ) · b Nambu-Bogoliubov Hamiltonian for 3 He-A : b H = = ~ ~ τ � ξ ( p ) c ( p x ⌥ ip y ) I Symmetry: C ⇥ U ( 1 ) N + L I Topological Invariant G.E. Volovik, JETP 67(9), 1804-1811 (1988) : ✓ ∂ ˆ ◆ Z d 2 p m ⇥ ∂ ˆ m N 2D = π m ( p ) · = ± 1 ( 2 π ) 2 ˆ ∂ p x ∂ p y Nambu-Bogoliubov Hamiltonian for 3 He-B : b H = ξ ( p ) b σ b τ 3 + c p · ˆ τ 1 I Symmetry: C ⇥ T ⇥ SO ( 3 ) L + S I Topological Invariant G.E. Volovik, JETP Lett. 90: 398 (2009) : Z n o d 3 p Γ ( b H � 1 ∂ p i b H ) ⇥ ( b H � 1 ∂ p j b H )( b H � 1 ∂ p k b N 3D = 24 π 2 ε i jk Tr H ) = 2 , Γ = CT Bulk-edge correspondence I Nontrivial bulk topology gapless boundary states I Bogoliubov quasiparticle, particle-hole mixing Majorana Fermion

Superfluid in confined geometry Porous medium – aerogels Phase diagram Polar phase V. Dmitriev et al., PRL 115, 165304 (2015) J. Pollanen et al., Nature Physics 8, 317-320 (2012) Slab geometry L. Levitin et al., Science 340, 841-844 (2013) A. Vorontsov and J. A. Sauls, PRL 98, 045301 (2007)

Confinement induced inhomogeneous superfluid phase Missing Stripe phase Exist in slab geometry A. Vorontsov and J. A. Sauls, PRL 98, 045301 (2007) Exist with diffusive surface K. Aoyama, arXiv:1605.07302 (2016) Generic phenomena, in cylinders L. K. Aoyama, PRB 89, 140502 (2014) Levitin et al., Science 340, 841-844 (2013) J. Wiman and J. A. Sauls, to be published (2016) Strong-coupling effect J. Wiman and J. A. Sauls, JLTP 184, 10541070 (2016) Chiral stripe phase H. Wu and J. A. Sauls, to be published (2016) I Periodic chiral domain walls (PDW) I Competing phases – Polar, PDW, ABM

Chiral superconductivity Chiral superconductors Complex order parameter: ∆ ( p ) = | ∆ ( p ) | e i φ ( p ) I Phase winding: d φ ( p ) = n ⇥ 2 π C p Time reversal symmetry breaking m ( p ) · b ∆ ( p ) = ∆ ( p x + ip y ) Nontrivial bulk topology – b H = ~ ~ τ Majorana Fermions ϕ = π /4 Chiral Weyl Fermion ϕ =0 ϕ = π /2 Spontaneous charge current ϕ = π Half quantum vortex ϕ =3 π /4 Phase vortex + d -vector rotation D. Ivanov, PRL 86, 268 (2001) Non-abelian statistics quantum computing J. Jang et al., Science 331, 186-188 (2011)

Is Sr 2 RuO 4 a chiral superconductor? Sr 2 RuO 4 T c = 1 . 15 K Evidence for chiral superconductivity I Spin triplet pairing S = 1 NMR I Broken time reversal symmetry I Two dimensional E 1 u representation, p x ± ip y Evidence against chiral superconductivity I Missing edge current I No second transition R. Cava, Chem. Commun. (2005) Kerr rotation Uniaxial strain J. Xia et al. - PRL 97, 167002 (2006) J. Xia et al., PRL 97, 167002 (2006) cooled in -43 Oe ! warmup in ZF C. Hicks et al., Science 344, 283-285 (2014)

Outline Goal : Search for Majorana Fermions Novel superfluid phases under confinement Indentification of chiral superconductor Content : Introduction of quasiclassical theory (Ch 2) 1 Majorana states and currents on superfluid 3 He-B (Ch 3-4) 2 H. Wu and J. Sauls, PRB 88, 184506 (2013) Edge spectrum, currents and phase transitions in 3 He-A film (Ch 5-6) 3 H. Wu and J. Sauls, to be published (2016) Collective modes and power absoptions in chiral superconductor (Ch 7-8) 4 H. Wu and J. Sauls, to be published (2016) Conclusion (Ch 9) 5

Elements of quasi-classical theory ! ˆ g ˆ f Quasiclassical Green’s function: b G ( R , p ; ε ) = , ∆ ⌧ E f , ¯ h / p f ⌧ ξ 0 ... ˜ ˆ ˜ f g ˆ Z ϖ Z d r e i p · r ⌦ ↵ Tr Ψ ( R + r / 2 ) Ψ † ( R � r / 2 ) g αβ ( R , p ) = � � ϖ d ξ ( p ) p = p f ˆ p Z ϖ Z d r e i p · r h Tr Ψ ( R + r / 2 ) Ψ ( R � r / 2 ) i f αβ ( R , p ) = � � ϖ d ξ ( p ) Equilibrium Eilenberger’s transport equation h i τ 3 � b ∆ ( R , p ) , b h v ( p ) · ∇ R b ε ˆ G ( R , p ; ε ) + i ¯ G ( R , p ; ε ) = 0 Self-consistent equation Z Z ϖ ˆ d p 0 V ( p , p 0 ) � ϖ d ε ˆ f ( R , p 0 ; ε ) ∆ ( R , p ) = Linear response theory G ( R , p ; ε , t ) = b b G ( R , p ; ε ) eq + δ b ∆ ( R , p ; t ) = b b ∆ ( R , p ) eq + δ b G ( R , p ; ε , t ) , ∆ ( R , p ; t )

Superfluid 3 He-B in a film with width D z ∆ y 3 He-B x D Bulk 3 He-B Surface 3 He-B q I Fully gapped E ( p ) = ξ 2 ( p )+ ∆ 2 I Gapless Fermions E ( p ) = ± cp k I Ising spin state I Helicity eigenstate

Quasi-classical theory for confined 3 He-B B Phase Order parameter ˆ ∆ = ~ d · ( i ~ σσ y ) d x = ∆ k ( z ) p x , d y = ∆ k ( z ) p y , d z = ∆ ? ( z ) p z , ! ~ g + ~ g · ~ σ f · ( i ~ σσ y ) Nambu Green’s function b G ( p , z ; ε ) = ˜ ~ g + ˜ σ tr f · ( i σ y ~ σ ) ˜ ~ g · ~ Eilenberger’s equation h i H A , b h v p · ∇ b b + i ¯ G = 0 G Andreev Hamiltonian b τ 3 � b H A = ε b ∆ Integrate along trajectories p , p

Green’s function for specular 3 He-B surface – Quasiparticle propagator Quasiparticle Green’s function (Scalar) " # ? cos 2 θ p ∆ 2 πε k sin 2 θ � ε 2 e � 2 ∆ 2 � ε 2 z / v z p g = � 1 + ∆ 2 � ε 2 ∆ 2 Local density of states N ( p , z ; ε ) = � 1 π Img ( p , z ; ε ) Quasiparticle Green’s function (Spin vector) g = � π ∆ ? ∆ k sin θ cos θ p ∆ 2 � ε 2 z / v z ~ e � 2 ~ e 2 k sin 2 θ � ε 2 ∆ 2 Local spin density of states S ( p , z ; ε ) = � 1 ~ g ( p , z ; ε ) π Im ~

Green’s function for specular 3 He-B surface – Cooper pair propagator Anomalous Green’s function " # p p ∆ 2 � ε 2 p f z = � π ∆ ? cos θ ∆ 2 � ε 2 z / v z + i ε ∆ 2 � ε 2 z / v z 1 � e � 2 k sin 2 θ � ε 2 e � 2 p ∆ 2 � ε 2 ∆ 2 " # ∆ ? cos 2 θ p π ∆ k sin θ ∆ 2 � ε 2 z / v z k sin 2 θ � ε 2 e � 2 f k = p 1 + ∆ 2 � ε 2 ∆ 2 1 . 2 1 . 0 I ξ ∆ = v f / 2 ∆ 0 . 8 I Supression of p z orbital 0 . 6 I Enhancement of p k orbital 0 . 4 I Change in length scale ξ ∆ ∆ k / ∆ 0 . 2 ∆ ? / ∆ 0 . 0 0 1 2 3 4 5 6 z/ ξ ∆

Local density of states Surface bound states Bound state energy N ( p || , ε ) 6 ε b = ± cp k , c = ( ∆ k / p f ) 4 Spectral weight 2 π ∆ ? cos θ e � 2 ∆ ? z / v f 0 2 1 . 0 Continuous spectrum 0 . 5 p k | ε | > ∆ 0 . 0 / p f z ! ∞ N c ( p , z ; ε ) = N bulk 2 lim � 0 . 5 1 0 � 1 ε / ∆ � 1 . 0 � 2

Spontaneous spin current Spin spectral function: S = π ∆ ? cos θ [ δ ( ε + c p k ) � δ ( ε � c p k )] e � 2 ∆ ? z / v f ~ ~ e 2 2 Confining length: ξ ∆ = v f / 2 ∆ ? Spin polarized along ~ e 2 = z ⇥ p Spectral Spin current: ⇥ ⇤ ~ S α ( p , z ; ε ) � S α ( p 0 , z ; ε ) J α ( p , z ; ε ) = 2 N f v p Time reversal p ! p 0 : S α ( p , z ; ε ) = � S α ( p 0 , z ; ε ) , ~ J α ( p 0 , z ; ε ) ~ J α ( p , z ; ε ) = 0 1 0 � 1 0 @ A spin current tensor J ( p , z ; ε ) ⇥ + 1 0 0 0 0 0

Surface spin current Spin current structure on the surface 1 . 0 Spin polarization Flow direction 0 . 5 y 0 . 0 − 0 . 5 − 1 . 0 − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 x

Spectral spin current density p k = 0 . 5 p f 2 Fermi Distribution 1 0 − 1 − 2 5 4 3 − 2 − 1 2 0 ε 1 / ∆ 1 0 2