Control of Complex Quantum Systems KITP, Santa Barbara, January 24, 2013 Electrical Control of the Kondo Effect at the Edge of a Quantum Spin Hall System Erik Eriksson (University of Gothenburg) Anders Ström (University of Gothenburg) Girish Sharma (École Polytechnique) Henrik Johannesson (University of Gothenburg) supported by the Swedish Research Council Control of Complex Quantum Systems KITP, Santa Barbara, January 24, 2013 Nota Bene! No closed-loop learning, feedback control, ”quantum circuits”, or anything of that kind... Electrical ”Control” of the Kondo Effect at the Edge of a Quantum Spin Hall System Erik Eriksson (University of Gothenburg) Anders Ström (University of Gothenburg) Girish Sharma (École Polytechnique) Henrik Johannesson (University of Gothenburg) supported by the Swedish Research Council Outline Quantum spin Hall system... some basics At the edge: A new kind of electron liquid Adding a magnetic impurity... ... and a Rashba spin-orbit interaction Electrical control of the Kondo effect! Why should people in quantum information/control/simulation bother? Long-distance qubit entanglement using minimal control! Quantum spin Hall system... some basics Quantum spin Hall system Integer Quantum Hall system B Quantum Hall system “skipping currents” B quantization chiral edge states no channel for backscattering ballistic transport along the edge, accounts for the quantization of the Hall conductance M. Büttiker, PRB 38, 9375 (1988) Can a many-body electronic state be stable against local perturbations without breaking time-reversal invariance? Consider a Gedanken experiment... B. A. Bernevig and S.-C. Zhang, PRL 96, 106802 (2006) 1 1 1 1 uniformlyE charged= E(x, ycylinder, 0) with electric field ϕs (ϕ+ ϕ )/√2 (4) ≡ − − E = E(x, y, 0) ϕs (ϕ+ ϕ )/√2 (4) E = E(x, y, 0) ≡ϕs −(ϕ+− ϕ )/√2 (4) − | > spin-orbitE = E(x, interactiony, 0) ϕs (ϕ+ ≡ϕ )/√−2 (4) z ≡ − − (E k) σ = Eσ (kyx kxy) Rτ† and Lτ† create excitations at the Fermi points of the × · z − (E k) σ = Eσ (kyx kxy) × · z − Rτ† andrighLt-τ† andcreateleft-moexcitationsving brancat thehesFwithermispinpoinprots ofjectionthe τ (E k) σ =z Eσ (kyx kxy) R† and L† create excitations at the Fermi points of the | > (E k)× σ =· Eσ (kyx kxy−) τ 1τ righRτ† t-andandLτ† left-mocreate excvingitationsbranchesat thewithFermispinpoinprotsjectionof theτ (5) × · B − right- and left-moving branches with spin projection τ A = (y, x, 0) right- and left-moving branches with spin projection τ (5) cf. withB the IQHE2 −in a symmetric gauge (5) A = (y, x, 0) (5) B = A B ϕc (ϕ+ + ϕ )/√2 (3) ∇ × A2B= −(y, x, 0) ≡ − L+ A = (y,2 x, 0)− 2 − L + L A k eB(kyx kxy) L+ + E = E(x, y, 0) Lorentz· force∼ − ϕs (ϕ+ ϕ )/√2 (4) A k eB(k x k y) ≡ − − y x 2 2 A· kA∼ keB(keBx(−kykx y) kxy) vF = 2a t + γ0 and ∆R = γ1 sin(q0a) · ∼· ∼ y − x− 2 2 (E k) σ = Eσz(k x k y) 2 vF = 2a t +γ and ∆R = γ1 sin(q0a) | > y x R†eand L† create excitations at the Fermi points of the 2 2 20 2 × · − G = ντ τ vF = 2vaF =t 2+aγ0 tand+ γ∆0Rand= γ1∆sin(R =q0γa1) sin(q0a) 2 right-hand left-moving branches with spin projection τ e 2 2 G = ν e e (5) B G =Gνh= ν τ = ivF :Rτ† (x)∂xRτ (x)::Lτ† (x)∂xLτ (x): A = (y, x, 0) h H − 2 − h 2ik0 (x+a/2) τ = ivF :Rτ† (x)∂xRτ (x)::Lτ† (x)∂xLτ (x): 2∆ cos(Qx) e− F R† (x)L (x)+H.c.(6), τ = ivF :R† (x)∂xR (x)::L† (x)∂xL (x): R τ τ Mc 100 meV L+ H −τ = ivτ F :Rτ† (τx)∂xRττ(x)::Lτ†τ(x)∂xLτ (x): − H − 2ik0 (x+a/2) ≈ H − 0 F 0 2∆R cos(Qx2)ikFe−(x+a/2) R† (x)L (x)+H.c.(6), 2∆ cos(Qx) e− 2ikRF† (x+)La/τ2()x)+Hτ .c.(6), A k eB(k x k y) Mc 100 meV −R 2∆R cos(Qx) e− τ τ Rτ† (x)Lτ (x)+H.c.(6), · ∼ D. Grundler,y − x PhMys.c≈MRev.100 meV100Lett.meV84, 6074 (2000) − − ≈ c ≈ 2 2 D. Grundler, Phys. Rev. Lett. 84, 6074v(2000)F = 2a t + γ0 and ∆R = γ1 sin(q0a) τ = D. Grundler, Phys. Rev. Lett. 84, 6074 (2000) ± D.W.Grundler,e2H¨ausler,PhL.ys. KecRev.ke,Lett.and 84A., 6074H. MacDonald,(2000) Phys. GRev.= ν B 65, 085104 (2002) τ =τ = W.W.H¨ausler,H¨ausler,h L.L.KecKecke,ke, andand A.A. H.H. MacDonald,MacDonald, PhPhys.ys. ± ±τ = W. H¨ausler, L. Kecke, and A.τ = H.ivFMacDonald,:Rτ† (x)∂xRτ (x)::PhLτ† (ys.x)∂xLτ (x): ± Rev.Rev.B B6565, 085104, 085104(2002)(2002) H − 0 2ikF (x+a/2) Rev. B 65, 085104 (2002) 2∆R cos(Qx) e− Rτ† (x)qL0τ (x)+H.c.(6), Mc 100 meV 11 − ≈ α0 2 10− eV m × q0 q0 D. Grundler, Phys. Rev. Lett. 84, 6074 (2000) Λ 011.115 eV q0 αα0 0 22 1010−− eVeV11m α0 ××2 10− 6 eV m τ = W. H¨ausler, L. Kecke, and A. H. MacDonald,ΛΛ Ph0v0.Fys.5.5eVeV×1 10 m/s ± y HR = i (γ0 + γ1 cos (Qna)) cn,µ† σµν cn+1,ν H.c. Rev. B 65, 085104 (2002) Λ 0.56 eV× − − vKFc + K1s 1061m/s.8 n,µ,ν y vF 1 ×10 m/s6 HR = i (γ0 + γ1 cos (Qna)) cn,µ† σµν cny+1,ν H.c. vF × 1 10 m/s HR =− i (γ0 + γ1 cos (Qna)) cn,µ† σµν cn+1y− ,ν H.c. Kc + Ks 1.8 q0 H−Rn,µ,=ν i (γ0 + γ1 cos (Qna)) cn,µ† σµν cn−+1,ν H.c. K11c + Ks 1.8 × n,µ,ν α0 2 10− eV m − − × Kc +Ks 1.8 n,µ,ν Λ 0.5 eV 1 γj = αja− (j = 0, 1) 6 vF 1 10 m/s HR y 1 × HR = i (γ0 + γ1 cos (Qna)) cn,µ† σµν cn+1,ν H.c. γj = αja− (1j = 0, 1) − − γ = α a− (j = 0, 1) Kc + Ks 1.8 HR n,µ,ν j γ j= α a 1 (j = 0, 1) HR j j − HR √ γ = α a 1 (j = 0, 1) ϕc (ϕ+ + ϕ )/ 2 (7) j j − ≡ − HR H = dx [ c + s] ϕc (ϕ+ + ϕ )/√2 (7) ≡ − √ H = dx [ c + Hs] H ϕc (ϕ+ + ϕ )/ 2 (7) H H ≡ϕc (ϕ+−+ ϕ )/√2 (7) H = dx [ c + s] ≡ − H = Hdx [ Hc + s] ϕc (ϕ+ + ϕ )/√2 (7) H H ≡ − H = dx [ c + s] ϕs (ϕ+ ϕ )/√2 (8) H Hvi 2 2 mi √ − v = [(∂ ϕ ) +(∂ ϑm) ] cos( 2πK ϕ ), (1) ϕs (ϕ+ ϕ≡ )/ 2− (8) Hi i 2 2 x i 2 x i i − πa i i ≡ − − √ i =v [(∂xϕi) +(∂xϑi) ] m cos( 2πKiϕi), (1) ϕs (ϕ+ ϕ )/ 2 √ (8) H i 2 2 2 − πai ϕs (ϕ+ ϕ )/√2 (8) ≡ϕs −(ϕ+− ϕ )/ 2 (8) vi 2 i = 2 [(m∂xvi ϕi i) +(∂2xϑi) ] 2 cos(mi 2πKiϕi),≡ (1)− − ≡ − − i = [(∂xϕi) +(∂xϑi) 2] i = cos([(∂x2ϕπiK) iϕ+i)(, ∂x(1)ϑi)π]a cos( 2πKiϕi), (1) H 2 H H− πa2 − − πa int = g1 :Rτ† Lτ L† τ R τ : + g˜2τ :R+† R+Lτ† Lτ : ∆R = γ1 sin(q0a) int = g1H :Rτ† Lτ L−† τ R τ : +−g˜2τ :−R+† R+Lτ† Lτ : ∆R = γ1 sin(q0a) H − − − g4τ int = g1 :Rτ† Lτ L† τ R τ : + g˜2τ :R+† R+Lτ† Lτ : g ∆R = γ1 sin(q0a) H − − − int = g1 :R† L L4†τ R + : + g˜(:2Rτ :+†RR†+RRτ†LR†τL: +: R L) (9) g4τ τ +τ τ(:R††τR R† R : ++ R+† τL)τ (9) ∆R = γ1 sin(q0a) H int −= g1 :Rτ†−Lτ L−+τ R+2 ττ: +τ g˜2τ :R+R+Lτ† Lτ↔: with vi and Ki∆functions= γ sin(of qg1aτ), g˜2τ and+g4τ(:R+† R+Rτ† Rτ : + RH L) (9) − 2 g − − ↔ with v and K functionswith ofvigand, g˜ Kandi functionsg R of g1τ , g˜2τ 0and g4τ 2 ↔ 4τ g i i 1τ 2τ 4τ + (:R4τ+† R+Rτ† Rτ : + R L) (9) 2+ (:R+† R+Rτ† Rτ : +↔R L) (9) with vi and Ki functions of g1τ , g˜2τ and g4τ 2 ↔ with vi and Ki functions of g1τ , g˜2τ and g4τ Λ bandwidth (2) g˜ g δ g (10) Λ bandwidthΛ bandwidth 2τ (2)2τ τ+ (2)1τ g˜2τ g2τ g˜2δττ+gg12ττ δτ+g1τ (10) (10) ∼ ∼ ∼ ≡ − ≡ − ≡ − Λ bandwidth (2) g˜2τ g2τ δτ+g1τ (10) Λ bandwidth (2) g˜2τ g2τ δτ+g1τ (10) ∼ ∼ ≡ ≡− − ϕc (ϕ+c + (ϕϕ+)/+√2ϕ )/√2 (3) (3) with vi andwithKvfunctionsi and K functionsof g1τ , g2τ ,ofg3gτ 1τ , g2(11)τ , g3τ (11) ≡ ≡ − − ϕc (ϕ+ + ϕ )/√2 √ (3) with vi and K functions of g1τ , g2τ , g3τ (11) ≡ϕc (ϕ+−+ ϕ )/ 2 (3) with vi and K functions of g1τ , g2τ , g3τ (11) ≡ − Quantum spin Hall (QSH) system single Kramers pair Two copies of an IQH system, bulk insulator with helical edge states | > | > First proposed by Kane and Mele for graphene (2005) too weak spin-orbit interaction, doesn’t quite work... Bernevig et al. proposal for HgTe quantum wells (2006) Experimental observation by König et al. (2007) from M. König et al., Science 318, 766 (2007) Bernevig et al. proposal for HgTe quantum wells (2006) Experimental observation by König et al. (2007) Are the helical edge states stable against local perturbations? Are the helical edge states stable against local perturbations? Yes! As long as the perturbations are time-reversal invariant! Look at the band structure of an HgTe quantum well..