TMS Intensive-Interactive Meeting, Keio University, November 17-18, 2016 Spontaneous Symmetry Breaking & Topological Order in Superfluid 3 He J. A. Sauls Northwestern University Supported by National Science Foundation Grant DMR-1508730 • Oleksii Shevtsov • Hao Wu • Joshua Wiman • Takeshi Mizushima (Osaka University) ◮ Spontaneous Symmetry Breaking in 3 He ◮ Topological Order in Chiral Superfluids ◮ Nambu-Goldstone & Higgs Modes ◮ Chiral Fermions & Edge Currents ◮ Anomalous Hall Effect in 3 He-A ◮ Nambu’s Fermion-Boson Mass Relation
Ferromagnetic Spin Fluctuations � Odd-Parity, Spin-Triplet Pairing for 3 He A. Layzer and D. Fay, Int. J. Magn. 1, 135 (1971) ◮ − p ′ ↑ p ′ ↑ − g / 4 V sf ( p , p ′ ) = = 1 − g χ ( p − p ′ ) p ↑ − p ↑ � d Ω ˆ � d Ω ˆ p p ′ V sf ( p , p ′ ) P l (ˆ p ′ ) − g l = (2 l + 1) p · ˆ 4 π 4 π − g l is a function of g ≈ 0 . 75 and ξ sf ≈ 5 � /p f ◮ 3 . 0 ◮ l = 1 (p-wave) is dominant pairing channel − g/ 4 V sf = 2 . 5 1 − g χ ( q ) ◮ p-wave basis functions: 2 . 0 p z ∼ cos θ ˆ ˆ p 1 . 5 p e + iφ ˆ p x + i ˆ ˆ p y ∼ sin θ ˆ p 1 . 0 p e − iφ ˆ p x − i ˆ ˆ p y ∼ sin θ ˆ p 0 . 5 q ≈ ¯ h/ξ sf q/p f ◮ S = 1 pairing fluctuations in V sf � 0 . 0 Multiple P-wave Superfluid Phases 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 W. Brinkman, J. Serene, and P. Anderson, PRA 10, 2386 (1974)
The 3 He Paradigm: Maximal Symmetry G = SO ( 3 ) S × SO ( 3 ) L × U ( 1 ) N × P × T Ψ αβ ( p ) = � ψ α ( p ) ψ β ( − p ) � BCS Condensate Amplitude : “Isotropic” BW State J. Wiman & J. A. Sauls, PRB 92, 144515 (2015) 34 A 30 T AB 24 B p/ bar 18 p PCP 12 J = 0 , J z = 0 T c 6 H = SO ( 3 ) J × T 0 Chiral AM State � l = ˆ z 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 T/ mK � � � Ψ ↑↑ � p x − ip y ∼ e − iφ Ψ ↑↓ p z = Ψ ↑↓ Ψ ↓↓ p x + ip y ∼ e + iφ p z BW � � � Ψ ↑↑ � p x + ip y ∼ e + iφ Ψ ↑↓ 0 L z = 1 , S z = 0 = Ψ ↑↓ Ψ ↓↓ p x + ip y ∼ e + iφ 0 AM H = U ( 1 ) S × U ( 1 ) L z -N × Z 2
Ginzburg-Landau Functional for Superfluid 3 He ◮ Maximal Symmetry of 3 He: G = SO ( 3 ) L × SO ( 3 ) S × U ( 1 ) N × P × T ◮ Order Parameter for P-wave ( L = 1 ), Spin-Triplet ( S = 1 ) Pairing Orbital Basis � �� � Spin Basis A xx A xy A xz p x ˆ � �� � � � � × Ψ(ˆ p ) = S x S y S z × A yx A yy A yz p y ˆ A zx A zy A zz p z ˆ ◮ GL Functional: A αi � vector under both SO ( 3 ) S [ α ] and SO ( 3 ) L [ i ] � � � AA † � � � AA † �� 2 + β 1 | Tr { AA tr }| 2 + β 2 d 3 r U [ A ] = α ( T ) Tr Tr � ( AA † ) 2 � � AA † ( AA † ) ∗ � β 3 Tr { AA tr ( AA tr ) ∗ } + β 4 Tr + + β 5 Tr � κ 1 ∂ i A αj ∂ i A ∗ αj + κ 2 ∂ i A αi ∂ j A ∗ αj + κ 3 ∂ i A αj ∂ j A ∗ + αi
Dynamical Consequences of Spontaneous Symmetry Breaking New Bosonic Excitations
Dynamical Consequences of Spontaneous Symmetry Breaking Higgs Boson with mass M = 125 GeV -1 -1 CMS s = 7 TeV, L = 5.1 fb s = 8 TeV, L = 5.3 fb S/(S+B) Weighted Events / 1.5 GeV Events / 1.5 GeV Unweighted 1500 1500 1000 1000 120 130 m (GeV) γ γ Data 500 S+B Fit B Fit Component 1 ± σ 2 ± σ 0 110 120 130 140 150 m (GeV) γ γ
Dynamical Consequences of Spontaneous Symmetry Breaking Scalar Higgs Boson (spin J = 0 ) [P. Higgs, PRL 13, 508 1964] Energy Functional for the Higgs Field F [Ψ] 0 . 4 � 0 . 3 � 2 c 2 | ∇ ∆ | 2 � α | ∆ | 2 + β | ∆ | 4 + 0 . 2 1 U [∆] = dV α > 0 0 . 1 0 . 0 − 0 . 1 − 0 . 2 � − 0 . 3 ◮ Broken Symmetry State: ∆ = | α | / 2 β − 0 . 4 α < 0 0 . 5 Im Ψ 0 . 0 − 0 . 5 0 . 0 − 0 . 5 Re Ψ 0 . 5 − 1 . 0 1 . 0 Space-Time Fluctuations about the Broken Symmetry Vacuum State ∆( r , t ) = ∆ + D ( r , t ) ◮ Eigenmodes: D ( ± ) = D ± D ∗ (Conjugation Parity) � 1 � � D ( − ) ) 2 ] − 2∆ 2 ( D (+) ) 2 − 1 D (+) ) 2 + ( ˙ 2[ c 2 ( ∇ D (+) ) 2 + c 2 ( ∇ D ( − ) ) 2 ] d 3 r 2[( ˙ L = t D (+) − c 2 ∇ 2 D (+) + 4∆ 2 D (+) = 0 t D ( − ) − c 2 ∇ 2 D ( − ) = 0 ◮ ∂ 2 ◮ ∂ 2 Massless Nambu-Goldstone Mode Massive Higgs Mode: M = 2∆
Dynamical Consequences of Spontaneous Symmetry Breaking BCS Condensation of Spin-Singlet ( S = 0 ), S-wave ( L = 0 ) “Scalar” Cooper Pairs Ginzburg-Landau Functional F [Ψ] 0 . 4 � 0 . 3 � α | ∆ | 2 + β | ∆ | 4 + κ | ∇ ∆ | 2 � 0 . 2 F [∆] = dV α > 0 0 . 1 0 . 0 − 0 . 1 − 0 . 2 � − 0 . 3 ◮ Order Parameter: ∆ = | α | / 2 β − 0 . 4 α < 0 0 . 5 Im Ψ 0 . 0 − 0 . 5 0 . 0 − 0 . 5 Re Ψ 0 . 5 − 1 . 0 1 . 0 Space-Time Fluctuations of the Condensate Order Parameter ∆( r , t ) = ∆ + D ( r , t ) ◮ Eigenmodes: D ( ± ) = D ± D ∗ (Fermion “Charge” Parity) � 1 � � D ( − ) ) 2 ] − 2∆ 2 ( D (+) ) 2 − 1 D (+) ) 2 + ( ˙ 2[ v 2 ( ∇ D (+) ) 2 + v 2 ( ∇ D ( − ) ) 2 ] d 3 r 2[( ˙ L = t D ( − ) − v 2 ∇ 2 D ( − ) = 0 t D (+) − v 2 ∇ 2 D (+) + 4∆ 2 D (+) = 0 ◮ ∂ 2 ∂ 2 ◮ Anderson-Bogoliubov Mode Amplitude Higgs Mode: M = 2∆
Dynamical Consequences of Spontaneous Symmetry Breaking First Reported Observations of Higgs Bosons in BCS Condensates
Dynamical Consequences of Spontaneous Symmetry Breaking Higgs Mode with mass: M = 3 meV and spin J = 0 in NbSe 2 Raman Absorption in NbSe 2 M. Me´ asson et al. PRB B 89, 060503(R) (2014) R. Sooyakumar & M. Klein, PRL 45, 660 (1980) ◮ � ω γ 1 = � ω γ 2 + 2∆ ◮ Amplitude Higgs - CDW Phonon Coupling ◮ Theory: P. Littlewood & C. Varma, PRL 47, 811 (1981)
Lagrangian Field Theory for Bosonic Excitations of Superfluid 3 He-B 1 3 He-B: B αi = √ 3∆ δ αi L = 1 , S = 1 � J = 0 ◮ Symmetry of 3 He-B: H = SO ( 3 ) J × T � D J,m ( r , t ) t ( J,m ) ◮ Fluctuations: D αi ( r , t ) = A αi ( r , t ) − B αi = αi J,m ◮ Lagrangian: � � � � D † � � DD † � 5 3 � � d 3 r D ˙ ˙ L = τ Tr − α Tr − β p u p ( D ) − K l v l ( ∂ D ) p =1 l =1 J,m = 1 J,m ( q ) 2 D ( C ) t D ( C ) J,m + E ( C ) τ η ( C ) ∂ 2 J,m with J = { 0 , 1 , 2 } , m = − J . . . + J , C = ± 1 ◮ Time-Dependent Ginzburg-Landau Theory for Superfluid 3 He-B : JAS & T. Mizushima, arXiv:1611.07273 (2016)
Spectrum of Bosonic Modes of Superfluid 3 He-B : Condensate is J C = 0 + ◮ 4 Nambu-Goldstone Modes & 14 Higgs modes � � � 2 E ( C ) c ( C ) M 2 J,m ( q ) = J, C + J, | m | | q | Mode Symmetry Mass Name D (+) J = 0 , C = +1 2∆ Amplitude Higgs 0 ,m D ( − ) J = 0 , C = − 1 0 NG Phase Mode 0 ,m D (+) J = 1 , C = +1 0 NG Spin-Orbit Modes 1 ,m D ( − ) J = 1 , C = − 1 2∆ AH Spin-Orbit Modes 1 ,m � 2 + AH Modes D (+) 8 J = 2 , C = +1 5 ∆ 2 ,m � 2 − AH Modes D ( − ) 12 J = 2 , C = − 1 5 ∆ 2 ,m ◮ Vdovin, Maki, W¨ olfle, Serene, Nagai, Volovik, Schopohl, McKenzie, JAS ...
Collective Mode Spectrum for 3 He-B M
Dynamical Consequences of Spontaneous Symmetry Breaking Higgs Mode with mass: M = 500 neV and spin J = 2 at LASSP-Cornell ◮ R. Giannetta et al., PRL 45, 262 (1980)
Dynamical Consequences of Spontaneous Symmetry Breaking Higgs Mode with mass: M = 500 neV and spin J C = 2 + at ULT-Northwestern Group Velocity T/T c ◮ D. Mast et al. Phys. Rev. Lett. 45, 266 (1980).
Dynamical Consequences of Spontaneous Symmetry Breaking Superfluid 3 He Higgs Detector at ULT-Northwestern 3 He- 4 He Dilution + Adiabatic Demagnetization Stages � T min ≈ 200 µ K
J = 2 − , m = ± 1 Higgs Modes Transport Mass and Spin ◮ “Transverse Waves in Superfluid 3 He-B”, G. Moores and JAS, JLTP 91, 13 (1993) �� 1 � � � ω 2 2 F s ρ n ( ω ) + 2 1 C t ( ω ) = 15 v f 5 ρ s ( ω ) ( ω + i Γ) 2 − 12 5 ∆ 2 − 2 5 ( q 2 v 2 f ) � �� � D ( − ) 2 , ± 1 Transverse Zero Sound Propagation in Superfluid 3 He-B: Cavity Oscillations of TZS B − − − − − − − − − − − → ◮ Y. Lee et al. Nature 400 (1999)
Faraday Rotation: Magneto-Acoustic Birefringence of Transverse Currents ◮ “Magneto-Acoustic Rotation of Transverse Waves in 3 He-B”, J. A. Sauls et al., Physica B, 284,267 (2000) 1 2 F s 15 ρ n ( ω ) + 2 F s ω 2 1 1 C RCP LCP ( ω ) = v f 75 ρ s ( ω ) ( ω + i Γ) 2 − Ω ( − ) 2 , ± ( q ) � �� � D ( − ) 2 , ± 1 � Ω ( − ) 12 2 , ± ( q ) = 5 ∆ ± g 2 − γH eff ◮ Circular Birefringence = ⇒ C RCP � = C LCP = ⇒ Faraday Rotation � C RCP − C LCP � � γH eff � ≃ g 2 − C t ω ◮ Faraday Rotation Period ( γH eff ≪ ( ω − Ω ( − ) ) ): 2 4 πC t λ H ≃ g 2 − γH ≃ 500 µm , H = 200 G Discovery of the acoustic Faraday effect in superfluid 3 He-B , Y. Lee, et al. Nature 400, 431 (1999) ◮
Large Faraday Rotations vs. ``Blue Tuning’’ B = 1097 G B = 1097 G 810 o 630 o 270 o 1170 o 990 o (2n + 1) x 90 o (2n + 1) x 90 o C. Collett et al., Phys. Rev. B 87, 024502 (2013)
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