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Towards topological with Kitaev materials

Nature 559, 227 (2018)

Jason Alicea, Caltech Topological quantum computation overview Non-Abelian anyons

Locally Inherently indistinguishable fault-tolerant ground states !

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. 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 liquids

http://manfragroup.org/non-abelian-phases-in-the-fractional-quantum-hall-regime/ https://neutronsources.org/news/scientific-highlights/neutrons-exotic-.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 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 type

Fermion

RuCl3 ψ σ

-0.84 1 = v⌧ = La/2 a = | | sz(t) h 2 i a = a = Detects individual -1 bulk anyons and their exotic statistics! j(t) 1(t) 2(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!