Frontiers in Quantum Matter Symmetry, Topology & Strong - PowerPoint PPT Presentation

International Conference on Quantum Fluids and Solids, University of Tokyo, July 25, 2018 Frontiers in Quantum Matter Symmetry, Topology & Strong Correlation Physics J. A. Sauls Northwestern University • Wave Ngampruetikorn • Takeshi Mizushima • Robert Regan • Oleksii Shevtsov • Joshua Wiman ◮ Strong Correlation Physics in 3 He ◮ Chiral Fermions & Anomalous Hall Transport ◮ Low Temperature Physics at 10 8 Kelvin ◮ Quanta of a Superfluid Vacuum ◮ Supported by National Science Foundation Grant DMR-1508730

Chiral Quantum Matter Molecular Chiral Enantiomers Handedness: Broken Mirror Symmetry

Chiral Quantum Matter Chiral Diatomic Molecules Molecular Chiral Enantiomers Ψ( r ) = f ( r ) ( x + iy ) Mirror Broken Mirror Symmetries Π zx Ψ( r ) = f ( r ) ( x − iy ) Handedness: Broken Mirror Symmetry

Chiral Quantum Matter Chiral Diatomic Molecules Molecular Chiral Enantiomers Ψ( r ) = f ( r ) ( x + iy ) Mirror Broken Mirror Symmetries Π zx Ψ( r ) = f ( r ) ( x − iy ) Broken Time-Reversal Symmetry Handedness: Broken Mirror Symmetry T Ψ( r ) = f ( r ) ( x − iy )

Chiral Quantum Matter Chiral Diatomic Molecules Molecular Chiral Enantiomers Ψ( r ) = f ( r ) ( x + iy ) Mirror Broken Mirror Symmetries Π zx Ψ( r ) = f ( r ) ( x − iy ) Broken Time-Reversal Symmetry Handedness: Broken Mirror Symmetry T Ψ( r ) = f ( r ) ( x − iy ) Realized in Superfluid 3 He-A & possibly the ground states in unconventional superconductors

Chiral Quantum Matter Chiral Diatomic Molecules Molecular Chiral Enantiomers Ψ( r ) = f ( r ) ( x + iy ) Mirror Broken Mirror Symmetries Π zx Ψ( r ) = f ( r ) ( x − iy ) Broken Time-Reversal Symmetry Handedness: Broken Mirror Symmetry T Ψ( r ) = f ( r ) ( x − iy ) Realized in Superfluid 3 He-A & possibly the ground states in unconventional superconductors Signatures: Chiral, Edge Fermions � Anomalous Hall Transport

Chiral Superconductors Ground states exhibiting: ◮ Emergent Topology of a Broken-Symmetry Vacuum of Cooper Pairs ◮ Weyl-Majorana excitations of the Vacuum ◮ Ground-State Edge Currents and Angular Momemtum ◮ Broken P and T � Anomalous Hall-Type Transport

Chiral Superconductors Ground states exhibiting: ◮ Emergent Topology of a Broken-Symmetry Vacuum of Cooper Pairs ◮ Weyl-Majorana excitations of the Vacuum ◮ Ground-State Edge Currents and Angular Momemtum ◮ Broken P and T � Anomalous Hall-Type Transport Where are They? ◮ 3 He-A: definitive chiral p-wave condensate; quantitative theory-experimental confirmation ◮ Sr 2 RuO 4 : proposed as the electronic analog of 3 He-A; evidence of chirality ◮ UPt 3 : electronic analog to 3 He: Multiple Superconducting Phases; evidence of chirality ◮ Other candidates: URu 2 Si 2 ; SrPtAs, doped graphene ...

The Pressure-Temperature Phase Diagram for Liquid 3 He Superfluid Phases of 3 He Maximal Symmetry: G = SO ( 3 ) S × SO ( 3 ) L × U ( 1 ) N × P × T → 34 A 30 T AB 24 B p/ bar 18 p PCP 12 T c 6 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 T/ mK ◮ J. Wiman & J. A. Sauls, PRB 92, 144515 (2015)

Realization of Broken Time-Reversal and Mirror Symmetry by the Vacuum State of 3 He Films ◮ Length Scale for Strong Confinement: SO ( 3 ) S × SO ( 3 ) L × U ( 1 ) N × T × P ξ 0 = � v f / 2 πk B T c ≈ 20 − 80 nm ⇓ ◮ L. Levitov et al., Science 340, 6134 (2013) SO ( 2 ) S × U ( 1 ) N-L z × Z 2 ◮ A. Vorontsov & J. A. Sauls, PRL 98, 045301 (2007) Chiral ABM State � 1 l = ˆ z 0.8 B A 0.6 0.4 0.2 Stripe Pha se 0 0 10 20 � � � Ψ ↑↑ � p x + ip y ∼ e + iφ Ψ ↑↓ 0 = Ψ ↑↓ Ψ ↓↓ p x + ip y ∼ e + iφ 0 ABM

Realization of Broken Time-Reversal and Mirror Symmetry by the Vacuum State of 3 He Films ◮ Length Scale for Strong Confinement: SO ( 3 ) S × SO ( 3 ) L × U ( 1 ) N × T × P ξ 0 = � v f / 2 πk B T c ≈ 20 − 80 nm ⇓ ◮ L. Levitov et al., Science 340, 6134 (2013) SO ( 2 ) S × U ( 1 ) N-L z × Z 2 ◮ A. Vorontsov & J. A. Sauls, PRL 98, 045301 (2007) Chiral ABM State � 1 l = ˆ z 0.8 B A 0.6 0.4 0.2 Stripe Pha se L z = 1 , S z = 0 0 0 10 20 Ground-State Angular Momentum � � L z � = N � Ψ ↑↑ � � � 2 � ? p x + ip y ∼ e + iφ Ψ ↑↓ 0 = Ψ ↑↓ Ψ ↓↓ p x + ip y ∼ e + iφ Open Question 0 ABM

Momentum-Space Topology Topology in Momentum Space Winding Number of the Phase: Ψ( p ) = ∆( p x ± ip y ) ∼ e ± iϕ p L z = ± 1 � N 2D = 1 1 d p · | Ψ( p ) | Im [ ∇ p Ψ( p )] = L z 2 π ◮ Massless Chiral Fermions ◮ Nodal Fermions in 3D ◮ Edge Fermions in 2D

Massless Chiral Fermions in the 2D 3 He-A Films π ∆ | p x | ξ ∆ = � v f / 2∆ ≈ 10 2 ˚ ε + iγ − ε bs ( p || ) e − x/ξ ∆ Edge Fermions: G R edge ( p , ε ; x ) = A ≫ � /p f ◮ ε bs = − c p || with c = ∆ /p f ≪ v f ◮ Broken P & T � Edge Current Unoccupied Vacuum Occupied ◮ M. Stone, R. Roy, PRB 69, 184511 (2004) ◮ J. A. Sauls, Phys. Rev. B 84, 214509 (2011)

Chiral Edge Current Circulating a Hole or Defect in a Chiral Superfluid ∆ ~ (p + i p ) z x y ◮ R ≫ ξ 0 ≈ 100 nm y R ◮ Sheet Current : � ^ l J J ≡ dx J ϕ ( x ) x

Chiral Edge Current Circulating a Hole or Defect in a Chiral Superfluid ∆ ~ (p + i p ) z x y ◮ R ≫ ξ 0 ≈ 100 nm y R ◮ Sheet Current : � ^ l J J ≡ dx J ϕ ( x ) x 1 ( n = N/V = 3 He density) ◮ Quantized Sheet Current: 4 n � J = − 1 w.r.t. Chirality: ˆ ◮ Edge Current Counter -Circulates: 4 n � l = + z

Chiral Edge Current Circulating a Hole or Defect in a Chiral Superfluid ∆ ~ (p + i p ) z x y ◮ R ≫ ξ 0 ≈ 100 nm y R ◮ Sheet Current : � ^ l J J ≡ dx J ϕ ( x ) x 1 ( n = N/V = 3 He density) ◮ Quantized Sheet Current: 4 n � J = − 1 w.r.t. Chirality: ˆ ◮ Edge Current Counter -Circulates: 4 n � l = + z ◮ Angular Momentum: L z = 2 π h R 2 × ( − 1 4 n � ) = − ( N hole / 2) � N hole / 2 = Number of 3 He Cooper Pairs excluded from the Hole ∴ An object in 3 He-A inherits angular momentum from the Condensate of Chiral Pairs!

Electron bubbles in the Normal Fermi liquid phase of 3 He ◮ Bubble with R ≃ 1 . 5 nm, ◮ QPs mean free path l ≫ R 0 . 1 nm ≃ λ f ≪ R ≪ ξ 0 ≃ 80 nm ◮ Mobility of 3 He is independent of T for ◮ Effective mass M ≃ 100 m 3 T c < T < 50 mK ( m 3 – atomic mass of 3 He) B. Josephson and J. Leckner, PRL 23, 111 (1969)

Current bound to an electron bubble ( k f R = 11 . 17 ) ∆ ~ (p + i p ) z x y y R = ⇒ ^ l J x L ( T → 0) ≈ − � N bubble / 2ˆ l ≈ − 100 � ˆ l

Electron bubbles in chiral superfluid 3 He-A ∆(ˆ k ) = ∆(ˆ k x + i ˆ k y ) = ∆ e iφ k quasiparticle v AH v E � �� � ���� µ AH E × ˆ ◮ Current: v = µ ⊥ E + l R. Salmelin, M. Salomaa & V. Mineev, PRL 63 , 868 (1989) tan α = v AH /v E = | µ AH /µ ⊥ | ◮ Hall ratio:

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

Forces on the Electron bubble in 3 He-A: ◮ M d v dt = e E + F QP , F QP – force from quasiparticle collisions

Forces on the Electron bubble in 3 He-A: ◮ M d v dt = e E + F QP , F QP – force from quasiparticle collisions ↔ ↔ ◮ F QP = − η · v , η – generalized Stokes tensor   η ⊥ η AH 0 ◮ ↔ for broken PT symmetry with ˆ η = − η AH η ⊥ 0 l � e z   0 0 η �

Forces on the Electron bubble in 3 He-A: ◮ M d v dt = e E + F QP , F QP – force from quasiparticle collisions ↔ ↔ ◮ F QP = − η · v , η – generalized Stokes tensor   η ⊥ η AH 0 ◮ ↔ for broken PT symmetry with ˆ η = − η AH η ⊥ 0 l � e z   0 0 η � M d v dt = e E − η ⊥ v + e for E ⊥ ˆ c v × B eff , ◮ l B eff = − c B eff ≃ 10 3 − 10 4 T eη AH ˆ !!! ◮ l

Forces on the Electron bubble in 3 He-A: ◮ M d v dt = e E + F QP , F QP – force from quasiparticle collisions ↔ ↔ ◮ F QP = − η · v , η – generalized Stokes tensor   η ⊥ η AH 0 ◮ ↔ for broken PT symmetry with ˆ η = − η AH η ⊥ 0 l � e z   0 0 η � M d v dt = e E − η ⊥ v + e for E ⊥ ˆ c v × B eff , ◮ l B eff = − c B eff ≃ 10 3 − 10 4 T eη AH ˆ !!! ◮ l ◮ Mobility: d v ↔ ↔ ↔ − 1 dt = 0 � v = µ E , where µ = e η ◮ O. Shevtsov and JAS, Phys. Rev. B 96, 064511 (2016)