Spontaneous Symmetry Breaking & Topology of the Superfluid - PowerPoint PPT Presentation

International Conference on Ultra-Low Temperature Physics, Heidelberg, Germany Spontaneous Symmetry Breaking & Topology of the Superfluid Phases of 3 He J. A. Sauls Department of Physics & Astronomy Northwestern University, Evanston, Illinois, USA August 18, 2017 ◮ Symmetry & Broken Symmetry of 3 He ◮ Topology of the Bulk Phases ◮ Dynamical Consequences: ◮ Signatures: Bosonic Spectrum Weyl & Majorana Fermions • Research supported by US National Science Foundation Grant DMR-1508730

Spin-Fluctuation Mediated Pairing � Odd-Parity, Spin-Triplet Pairing for 3 He A. Layzer and D. Fay, Int. J. Magn. 1, 135 (1971) ◮ 3 . 0 − g/ 4 V sf = 2 . 5 p ′ ↑ 1 − g χ ( q ) − p ′ ↑ g 2 . 0 = V exch ( q ) = 1 − g χ ( q ) 1 . 5 p ↑ − p ↑ � d Ω ˆ � d Ω ˆ 1 . 0 p ′ p 4 π V sf ( p , p ′ ) P p ′ ) − g l = ( 2 l + 1 ) l ( ˆ p · ˆ 0 . 5 q ≈ ¯ h/ξ sf q/p f 4 π 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 − g l is a function of g ≈ 0 . 75 ◮ & ξ sf ≈ 5 ¯ h / p f ◮ l = 1 (p-wave) is dominant pairing channel p e + i φ ˆ p ◮ p x + i ˆ ˆ p y ∼ sin θ ˆ � l z = + 1 ◮ p z ∼ cos θ ˆ ˆ � l z = 0 p p e − i φ ˆ ◮ p x − i ˆ p y ∼ sin θ ˆ p � l z = − 1 ˆ ◮ S = 1 , S z = 0 , ± 1 pairing fluctuations

Spin-Fluctuation Mediated Pairing � Odd-Parity, Spin-Triplet Pairing for 3 He A. Layzer and D. Fay, Int. J. Magn. 1, 135 (1971) ◮ 3 . 0 − g/ 4 V sf = 2 . 5 p ′ ↑ 1 − g χ ( q ) − p ′ ↑ g 2 . 0 = V exch ( q ) = 1 − g χ ( q ) 1 . 5 p ↑ − p ↑ � d Ω ˆ � d Ω ˆ 1 . 0 p ′ p 4 π V sf ( p , p ′ ) P p ′ ) − g l = ( 2 l + 1 ) l ( ˆ p · ˆ 0 . 5 q ≈ ¯ h/ξ sf q/p f 4 π 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 − g l is a function of g ≈ 0 . 75 ◮ ◮ Feedback on V s f � & ξ sf ≈ 5 ¯ h / p f Multiple Stable Superfluid Phases ◮ l = 1 (p-wave) is dominant ∆ ⋆ − p ′ ↑ pairing channel p ′ ↑ p ↑ − p ↑ p e + i φ ˆ p ◮ p x + i ˆ ˆ p y ∼ sin θ ˆ � l z = + 1 ∆ δ χ pair ◮ p z ∼ cos θ ˆ ˆ � l z = 0 p p e − i φ ˆ ◮ p x − i ˆ p y ∼ sin θ ˆ p � l z = − 1 ˆ χ A ≈ χ N > χ B � 1 3 χ N � Superfluid A-phase ◮ S = 1 , S z = 0 , ± 1 pairing W. Brinkman, J. Serene, & P. Anderson, PRA 10, 2386 (1974) ◮ fluctuations

Spin-Fluctuation Mediated Pairing � Odd-Parity, Spin-Triplet Pairing for 3 He A. Layzer and D. Fay, Int. J. Magn. 1, 135 (1971) ◮ 3 . 0 − g/ 4 V sf = 2 . 5 p ′ ↑ 1 − g χ ( q ) − p ′ ↑ g 2 . 0 = V exch ( q ) = 1 − g χ ( q ) 1 . 5 p ↑ − p ↑ � d Ω ˆ � d Ω ˆ 1 . 0 p ′ p 4 π V sf ( p , p ′ ) P p ′ ) − g l = ( 2 l + 1 ) l ( ˆ p · ˆ 0 . 5 q ≈ ¯ h/ξ sf q/p f 4 π 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 − g l is a function of g ≈ 0 . 75 ◮ ◮ Feedback on V s f � & ξ sf ≈ 5 ¯ h / p f Multiple Stable Superfluid Phases ◮ l = 1 (p-wave) is dominant ∆ ⋆ − p ′ ↑ pairing channel p ′ ↑ p ↑ − p ↑ p e + i φ ˆ p ◮ p x + i ˆ ˆ p y ∼ sin θ ˆ � l z = + 1 ∆ δ χ pair ◮ p z ∼ cos θ ˆ ˆ � l z = 0 p p e − i φ ˆ ◮ p x − i ˆ p y ∼ sin θ ˆ p � l z = − 1 ˆ χ A ≈ χ N > χ B � 1 3 χ N � Superfluid A-phase ◮ S = 1 , S z = 0 , ± 1 pairing W. Brinkman, J. Serene, & P. Anderson, PRA 10, 2386 (1974) ◮ fluctuations Liquid 3 He is near a Mott transition & Solid is AFM Ordered ◮ Not the Whole Story: ◮ Normal 3 He: an almost localized Fermi liquid, D. Vollhardt, RMP 56, 99 (1984)

Spin-Fluctuation Mediated Pairing � Odd-Parity, Spin-Triplet Pairing for 3 He A. Layzer and D. Fay, Int. J. Magn. 1, 135 (1971) ◮ 3 . 0 − g/ 4 V sf = 2 . 5 p ′ ↑ 1 − g χ ( q ) − p ′ ↑ g 2 . 0 = V exch ( q ) = 1 − g χ ( q ) 1 . 5 p ↑ − p ↑ � d Ω ˆ � d Ω ˆ 1 . 0 p ′ p 4 π V sf ( p , p ′ ) P p ′ ) − g l = ( 2 l + 1 ) l ( ˆ p · ˆ 0 . 5 q ≈ ¯ h/ξ sf q/p f 4 π 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 − g l is a function of g ≈ 0 . 75 ◮ ◮ Feedback on V s f � & ξ sf ≈ 5 ¯ h / p f Multiple Stable Superfluid Phases ◮ l = 1 (p-wave) is dominant ∆ ⋆ − p ′ ↑ pairing channel p ′ ↑ p ↑ − p ↑ p e + i φ ˆ p ◮ p x + i ˆ ˆ p y ∼ sin θ ˆ � l z = + 1 ∆ δ χ pair ◮ p z ∼ cos θ ˆ ˆ � l z = 0 p p e − i φ ˆ ◮ p x − i ˆ p y ∼ sin θ ˆ p � l z = − 1 ˆ χ A ≈ χ N > χ B � 1 3 χ N � Superfluid A-phase ◮ S = 1 , S z = 0 , ± 1 pairing W. Brinkman, J. Serene, & P. Anderson, PRA 10, 2386 (1974) ◮ fluctuations Liquid 3 He is near a Mott transition & Solid is AFM Ordered ◮ Not the Whole Story: ◮ Normal 3 He: an almost localized Fermi liquid, D. Vollhardt, RMP 56, 99 (1984) Poster Fri-038, Joshua Wiman

Superfluid Phases of 3 He Maximal Symmetry G = SO ( 3 ) S × SO ( 3 ) L × U ( 1 ) N × P × T × C → “Isotropic” BW Chiral AM State � l = ˆ z J. Wiman & J. A. Sauls, PRB 92, 144515 (2015) State 34 A 30 T AB 24 B p/ bar 18 p PCP L z = 1 , S z ′ = 0 12 J = 0 , J z = 0 T c 6 H = U ( 1 ) S × U ( 1 ) L z -N × Z 2 H = SO ( 3 ) J × T 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 T/ mK � Ψ ↑↑ � Ψ ↑↓ � ← Ψ αβ ( p ) = � ψ α ( p ) ψ β ( − p ) � Spin-Triplet Condensate Amplitudes : Ψ = Ψ ↑↓ Ψ ↓↓ � � p x − ip y ∼ e − i φ p z � Ψ BW = ∆ p x + ip y ∼ e + i φ p z � � p x + ip y ∼ e + i φ 0 � Ψ AM = ∆ p x + ip y ∼ e + i φ 0 Ψ AM = | ∆ | 2 sin 2 θ Fully Gapped: � Ψ † BW � Ψ BW = | ∆ | 2 Nodal Points: � Ψ † AM �

Dynamical Consequences of Spontaneous Symmetry Breaking New Bosonic Excitations

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 � � 2 c 2 | ∇∆ | 2 � 0 . 3 α | ∆ | 2 + β | ∆ | 4 + 1 0 . 2 U [ ∆ ] = dV α > 0 0 . 1 0 . 0 − 0 . 1 − 0 . 2 � − 0 . 3 | α | / 2 β ◮ Broken Symmetry State: ∆ = − 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 ( − ) = 0 t D (+) − c 2 ∇ 2 D (+) + 4 ∆ 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 � � α | ∆ | 2 + β | ∆ | 4 + κ | ∇∆ | 2 � 0 . 3 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 ∆

Ginzburg-Landau Functional for Superfluid 3 He ◮ Maximal Symmetry of 3 He: G = SO ( 3 ) L × SO ( 3 ) S × U ( 1 ) N × P × T × C ◮ 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

Ginzburg-Landau Functional for Superfluid 3 He ◮ Maximal Symmetry of 3 He: G = SO ( 3 ) L × SO ( 3 ) S × U ( 1 ) N × P × T × C ◮ 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 α i � 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 α j ∂ i A ∗ α j + κ 2 ∂ i A α i ∂ j A ∗ α j + κ 3 ∂ i A α j ∂ j A ∗ + α i ◮ Mermin, Ambegaokar, Brinkman, Anderson, circa 1974