Towards topological quantum computing with Kitaev materials Nature 559, 227 (2018) Jason Alicea, Caltech Topological quantum computation overview Non-Abelian anyons Locally Inherently indistinguishable fault-tolerant ground states qubits! 000 , 001 ,... Anyons carry exotic | i | i zero-energy modes Alexei Kitaev Non-Abelian anyons Locally Inherently indistinguishable fault-tolerant ground states qubits! 000 , 001 ,... Anyons carry exotic | i | i zero-energy modes Non-Abelian Inherently statistics fault-tolerant gates! U i ! ij j How to build the hardware? Alexei Kitaev “Designer” realizations 1D p-wave 2D p+ip superconductor superconductor “Designer” realizations Mourik et al. Science 336, 1003 (2012) Albrecht et al. Nature 531, 206 (2016) ..and many more Nadj-Perge et al. Science 346, 602 (2014) Ren et al. arXiv:1809.03076 “Intrinsic” realizations Fractional quantum Hall Quantum spin liquids http://manfragroup.org/non-abelian-phases-in-the-fractional-quantum-hall-regime/ https://neutronsources.org/news/scientific-highlights/neutrons-exotic-particles.html Non-Abelian spin liquids & “Kitaev materials” h Frustration x x SySy SzSz H = − J ∑ Sr Sr′" + ∑ r r′" + ∑ r r′" (⟨rr′⟩" ⟨rr′⟩" ⟨rr′⟩" ) −∑ h ⋅ Sr r 0 Sr = iγr γr⃗ → ~free Majorana fermions Gapless Gapped, non-Abelian spin liquid spin liquid! Chiral Majorana edge state Fermion x x SySy SzSz ψ H = − J ∑ Sr Sr′" + ∑ r r′" + ∑ r r′" (⟨rr′⟩" ⟨rr′⟩" ⟨rr′⟩" ) σ − h ⋅ S ‘Ising’ non-Abelian anyons ∑ r r σ 0 Sr = iγr γr⃗ → ~free Majorana fermions Gapless Gapped, non-Abelian spin liquid spin liquid! Chiral Majorana edge state Chiral Majorana edge state Fermion Fermion Emergent fermions Built from physical in a bosonic system microscopic fermions ψ ψ σ ≠ σ ‘Ising’ non-Abelian anyons ‘Ising’ non-Abelian anyons σ σ non-Abelian spin liquid p+ip superconductor Towards experimental reality h x x SySy SzSz H = − J ∑ Sr Sr′" + ∑ r r′" + ∑ r r′" (⟨rr′⟩" ⟨rr′⟩" ⟨rr′⟩" ) −∑ h ⋅ Sr +⋯ r Focus here: ⍺-RuCl3 Plumb et al., PRB 90, 041112 (2014) Yields quantized thermal Hall conductance Chiral Majorana edge state Quantized value Fermion !!! ψ σ ‘Ising’ non-Abelian anyons conductivity Thermal Hall σ Magnetic field [T] First experimental evidence of a non-Abelian spin liquid! Contender for topological quantum computing hardware? Virtues: energy scales, fabrication flexibility Challenges: manipulation & detection of anyons Nature 559, 227 (2018) So far withstood theoretical scrutiny! Vinkler-Aviv & Rosch PRX; Ye, Halasz, Savary, Balents PRL Punchline I Detectable with electrical Chiral Majorana edge state probes—even though RuCl3 is an electrical insulator! Fermion ψ σ ‘Ising’ non-Abelian anyons Dave Roger Ben David σ Aasen Mong Hunt Mandrus Phys. Rev. X 10, 031014 (2020) Electrical probes of non-Abelian spin liquids Converting physical and emergent fermions Integer quantum RuCl3 Hall Superconductor Turn on interactions… arXiv:1709.01941 Circuit designs V Edge-state detection e2 G(V → 0,T → 0) = 2h IQH RuCl3 IQH Drain Source (Floating) superconductor Ising-anyon detection Ising-anyon/fermion detection V Gσ ≠ GI = Gψ V GI,σ,ψ all differ ψ IQH RuCl3 IQH RuCl3 Source Source σ σ (Grounded) (Grounded) superconductor superconductor Experimental status Punchline II Detectable with simpler time-domain probes— no QH/SC needed! Chiral Majorana edge state Fermion ψ σ ‘Ising’ non-Abelian anyons Kai Dave Roger Eugene σ Klocke Aasen Mong Demler Available soon… Majorana edge-mode detection Excited Ground state state RuCl3 Probes chiral edge state velocity x H(t) = − iv γ∂xγ + h ⋅ sj + iλj(t)s γ∂xγ| ∫ ∑ [ j xj ] x j=1,2 time Time-domain anyon interferometry Excited state Fermion RuCl3 ψ σ Ground state Time-domain anyon interferometry Excited state Fermion RuCl3 ψ σ Ground state Time-domain anyon interferometry Fermion RuCl3 ψ σ Time-domain anyon interferometry Paths can interfere in a way that depends on enclosed quasiparticle type Fermion RuCl3 ψ σ -0.84 1 ∆ = v⌧ = La/2 a = | | sz(t) h 2 i a = σ a = Detects individual -1 bulk anyons and their exotic statistics! λj(t) λ1(t) λ2(t) 2⌧ energy arrives 0 t1 t t2 time Summary & Outlook Non-Abelian spin liquids in “Kitaev materials” appear to be experimental reality, providing new horizon in topological quantum computation. Key features detectable with Chiral Majorana edge state electrical probes and simpler time-domain measurements! RuCl3 Fermion ψ σ ‘Ising’ non-Abelian anyons σ Practical anyon manipulation schemes needed (Ising anyons particularly subtle). Warm-up: stitching p+ip superconductors Chiral Majorana edge state p+ip p+ip superconductor superconductor ℋ = − ivγR∂xγR + ivγL∂xγL +imγRγL Warm-up: stitching p+ip superconductors Chiral Majorana edge state p+ip SC’s p+ip superconductor sewn up superconductor ℋ = − ivγR∂xγR + ivγL∂xγL +imγRγL Warm-up: stitching non-Abelian spin liquids Chiral Majorana edge state RuCl3 RuCl3 ℋ = − ivγR∂xγR + ivγL∂xγL +imγRγL Unphysical! (because fermions are emergent) Warm-up: stitching non-Abelian spin liquids Chiral Majorana edge state Spin liquids RuCl3 sewn up RuCl3 Assume “large”… ℋ = − ivγR∂xγR + ivγL∂xγL +λγRγRγLγL → λ⟨iγRγL⟩iγRγL ≡ imeffγRγL …mass generated spontaneously!.