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
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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...
strong spin-orbit interactions in atomic p-orbitals create an inverted band gap (p-band on top of s-band)
supports a single Kramers pair of helical edge states inside the inverted gap
Kramers degeneracy at k=0 protects the stability of the edge states 1
ballistic transport
2 2e ikxn G = α(x) = α(x) + αˆ(kn)e B. A. Bernevig et al., PRL 95, 066601 (2005) 4-terminal measurement, h n equilibration in contacts
ik x L 1µm Ψ (x)=ψ (x)e± F ∼ ↑↓ ↑↓
L 20µm ikF x Ψ (x)=ψ (x)e− ∼ ↓ ↓
g H = α(x)(z k) σ R × ·
g 1/2v ∼ B = A ∇ ×
ξ < L E = E(x, y, 0)
αˆ( 2k ) ± F (E k) σ = Eσz(k x k y) × · y − x
2kF ± B A = (y, x, 0) 2 − z 2g√CK ⊥ ≡ A k eB(k x k y) · ∼ y − x z 4K 2 ≡ − e2 G = ν h γ = (αˆ(2k ) + αˆ( 2k ))/2 F − F
M 100 meV c ≈ ∂xθ D. Grundler, Phys. Rev. Lett. 84, 6074 (2000)
W. H¨ausler, L. Kecke, and A. H. MacDonald, Phys. ψ = η exp i√π[φ + θ] /√2πκ ↑ ↑ Rev. B 65, 085104 (2002) 11 α0 2 10− eV m ψ = η exp i√π[φ θ] /√2πκ × ↓ ↓ − − Λ 0.5 eV v 1 106 m/s F × K + K 1.8 H0 + Hd + Hf + HR c s
(αˆ2) H O R At the edge: A new kind of electron liquid
# of Kramers pair in a nontrivial (trivial) helical liquid 1
NK = 1 mod 2 = 0 mod 2
ν =0 , 1 is a ”Z2” topological invariant and can be calculated from the band structure of the bulk (”bulk-edge correspondence”) L. Fu and C. L. Kane, PRB 76, 045302 (2007)
J ,z ω T ⊥ 2D topological insulator (a.k.a. quantum spin Hall system)
()
K K →
c 2 = log( 1 2 ) + const. E 8 21 + 2
θ
21 2
2 2(√K λ/2)2 1 2 2(√K+λ/2)2 1 E 2 2K 1 E 2 γ (J + J ) T − − , γ (J J ) T − , γ J T − , γ˜ J T 0 ∼ x y 0 ∼ x − y 0 ∼ NC 0 ∼ NC
I = G V δI 0 −
T T K
K<1/4
2(1/4K 1) G (T/T ) − ∼ K
G =2e2/h
G =0 1 What if time-reversal symmetry is broken...?
... for example, by the presence of a magnetic impurity? 1 1 Mn2+ 1 1 1 Mn2+ 2+ z 2 Mn (caseS ) study: Mn2+ 2+ 2+ Mn Mn (Sz)2 z 2 large and positive single-ion anisotropy S =5/2 S =1/2 z 2 (S ) −→ eff (S ) S =5/2 Seff =1/2 (Sz)2 −→
low T S =5/2S =5S/eff2 =1/2 S =1/2 z 2 −→ −→ eff from M. König et al., Science 318, 766 (2007) (S ) low T T To model the edge electrons, we introduce the two-spinorsS =5Ψ/2 = ψSe,ffψ=1,where/2 ψ (ψ ) annihilates a right-moving (left-moving) electron with spin-up (spin-down) along the growth−→ direction↑ ↓ of the↑ quantum↓ well. Neglecting e-e anisotropic spin exchange with the edge electrons interactions, the edge Hamiltonian can then beR. written Zitko et al., asPRB 78, 224404 (2008)
+ + z z HK = Ψ†(0) J (σ S− + σ−Slow)+TJzσ Seff Ψ(0), ⊥ eff eff To model the edge electrons,S we=5 introduce /2 the two-spinorsSeΨffT ==1ψ,/ψ2,whereψ (ψ ) annihilates a right-moving (left-moving) electron with spin-up (spin-down)−→ along the growth direction↑ ↓ of the↑ quantum↓ well. Neglecting e-e interactions, the edge Hamiltonian can then be written as
+ + z z HK = Ψ†(0) J (σ S− + σ−S )+Jzσ S Ψ(0), ⊥ Adding a magnetic impurity...
The Kondo interaction is time-reversal invariant! Could it still cause a spontaneous breaking of time reversal invariance and collapse the QSH state? 1
∂J ⊥ = νJ Jz + ... 1 ∂D − ⊥ ∂J Adding a magnetic impurity...z = νJ 2 + ... (1) ∂D − ⊥ 1 ∂J Recall the Kondo effect⊥ = νJ Jz + ... ∂D − ⊥ J max(J ,J ) ∂J 0 z D=D0 z = νJ∂2J+ ...≡ ⊥ (1) One-loop RG equations:∂D ⊥ ⊥ = νJ Jz + ... − ∂D − ⊥ P. W. Anderson, J. Phys. C 3, 2436 (1970) ∂Jz 2 1 T = Dνexp(J + ...const./J ) (1) ∂DK −0 ⊥ − 0
J0 max(J ,Jz)D=D0 ≡ ⊥ ∂J ⊥ = νJ Jz + ... ∂D − ⊥ J0 max(J ,Jz)D=D0 strong-coupling physics≡ for ∂T< formation of impurity-electron2+ singlet (”Kondo screening”) Mn TKz=2D0 exp( const./J0) (S ) − (Sz)2 T< + + z z + HK = Ψ+†(0) J (σ Sze−ff +zσ−Seff)+Jzσ Seff Ψ(0), HK = Ψ†(0) J (σ Se−ff + σ−Seff)+⊥Jzσ Seff Ψ(0), ⊥ Adding a magnetic impurity... Pauli principle: punctured 1D lattice insulator! Adding a magnetic impurity... Does this really happen for the helical liquid? To find out, add e-e interactions.... important in 1D! bosonization, RG, and linear response J. Maciejko et al., PRL 102, 256803 (2009) 1 G =2e2/h ”weak” e-e interaction ”strong” e-e interaction G =0 1/4 8K 2 2K+2 (T/T ) − (T/T ) ∼ K ∼ K 2e2 G = δG h − T =0 K>1/4 T T K Jx = Jy = Jz = 10 meV Jx = Jy =5meV,Jz = 50 meV θ Jy ,Jz ,JNC displaymath α x x y y z z y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) Adding a magnetic impurity... 1 no correction to the edge conductance at T = 0 ! G =2e2/h ”weak” e-e interaction ”strong” e-e interaction G =0 1/4 8K 2 2K+2 (T/T ) − (T/T ) ∼ K ∼ K 2e2 G = δG h − T =0 K>1/4 T T K Jx = Jy = Jz = 10 meV Jx = Jy =5meV,Jz = 50 meV θ Jy ,Jz ,JNC displaymath α x x y y z z y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) Adding a magnetic impurity... Due to its topological nature, the QSH state follows the new shape of the edge. Weak coupling QSH states are robust against local breaking of time-reversal symmetry! J. Maciejko et al., PRL 102, 256803 (2009) But... one important thing is missing from the analysis Rashba spin-orbit interaction! z d 2DEG semiconductor heterostructure 1 y x 2 HR = α(kxσ kyσ ) J V /U Yu. A. Bychkov and− E. I. Rashba, ∝ J. Phys. C 17, 6039 (1984) Spatial asymmetry of d z band edges mimics an E-field in the z-direction H = λ ( V k) σ SO crystal ∇ × · d 2 6 λcrystal /4m∗Eg 10 λvac F ≈ ≈ 2 2 2 6 2 λvac = /4m c 3.7 10− A d +U 0 ≈ × H = λ ( V k) σ C (T/T ) ln(T /T ), S = k ln(√2), ... SO vac ∇ × · imp ∝ K K imp K 2.2T S = k ln(√2) c ≈ K imp 1 T 2 K(R)critical K(R) δ = π/2 R δ = 0 K(R) (J 2/D) cos(k R) ψ(r) ∝ F − T D exp( 1/ρ J) ψ(r) K ∝ − F K(R) > T K(R) K → ∞ V K(R) → −∞ U H = J S σ + J S σ + K(R)S S int 1 1 · 1 2 2 · 2 1 · 2 V J = J g 1 2 Rashba spin-orbit interaction! z d x semiconductor heterostructure 1 1 y HR = α kxσ x d z doesn’t conserve spin K y HR = α kxσ 2 2 K H =(v/2) dx (∂xϕ) +(∂xϑ) A B C + cos(√4πKϕ)+ H sin(=(√v/42)πKdϕx )+(∂ ϕ)2 +(∂∂xϑϑ)2 κ κ x √K x gU A √ B √ C + cos(√16πKϕ+) cos( 4πKϕ)+ sin( 4πKϕ)+ ∂xϑ 2(πκ)2 κ κ √K g g + U cos(√16πKϕ) ie 2 √ 2(πκ)2 + :(∂xϑ) cos( 4πKϕ): (1) 2√ gie 2π K + :(∂2ϑ) cos(√4πKϕ): (1) 2π2√K x kk k kkF kF F − F − ∂J ⊥ = νJ Jz + ... ∂J ∂D − ⊥ ⊥ = νJ Jz + ...∂Jz 2 ∂D − ⊥ = νJ + ... (2) ∂D − ⊥ ∂J z = νJ 2 + ... (2) ∂D − ⊥ J0 max(J ,Jz)D=D ≡ ⊥ 0 J0 max(J ,Jz)D=TDK = D0 exp( const./J0) ≡ ⊥ 0 − T< T< S =5/2 Seff =1/2 Mn2+ −→ low T To model the edge electrons, wez introduce2 the two-spinors ΨT = ψ , ψ ,whereψ (ψ ) annihilates a right-moving (left-moving) electron with spin-up(S ) (spin-down) along the growth direction↑ ↓ of the↑ quantum↓ well. Neglecting e-e interactions, the edge Hamiltonian can then be written as + + z z HK = Ψ†(0) J (σ S− + σ−S )+Jzσ Seff Ψ(0), ⊥ eff eff S =5/2 Seff =1/2 −→ low T To model the edge electrons, we introduce the two-spinors ΨT = ψ , ψ ,whereψ (ψ ) annihilates a right-moving (left-moving) electron with spin-up (spin-down) along the growth direction↑ ↓ of the↑ quantum↓ well. Neglecting e-e interactions, the edge Hamiltonian can then be written as + + z z HK = Ψ†(0) J (σ S− + σ−S )+Jzσ Seff Ψ(0), ⊥ eff eff Electrical control of the Kondo effect in a helical edge liquid Erik Eriksson,1 Anders Str¨om,1 Girish Sharma,2 and Henrik Johannesson1 1Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden 2Centre de Physique Th´eorique, Ecole Polytechnique, 91128 Palaiseau Cedex, France Magnetic impurities affect the transport properties of the helical edge states of quantum spin Hall insulators by causing single-electron backscattering. We study such a system in the presence of a Rashba spin-orbit interaction induced by an external electric field, showing that this can be used to control the Kondo temperature, as well as the correction to the conductance due to the impurity. Surprisingly, for a strongly anisotropic electron-impurity spin exchange, Kondo screening may get obstructed by the presence of a non-collinear spin interaction mediated by the Rashba coupling. This challenges the expectation that the Kondo effect is stable against time-reversal invariant perturbations. PACS numbers: 71.10.Pm, 72.10.Fk, 85.75.-d Introduction. The discovery that HgTe quantum wells states are still connected via a time-reversal transforma- support a quantum spin Hall (QSH) state? has set off tion (”Kramers pair”)? . Provided that the Rashba inter- an avalanche of studies addressing the properties of this action is spatially uniform and the e-e interaction is not novel phase of matter? . A key issue has been to deter- too strong, this ensures the robustness of the helical edge mine the conditions for stability of the current-carrying liquid? . states at the edge of the sample as this is the feature What is the physics with both Kondo and Rashba in- that most directly impacts prospects for future applica- teractions present? In this Letter we address this ques- tions in electronics/spintronics. In the simplest picture tion with a renormalization group (RG) analysis as well of a QSH system the edge states are helical, with counter- as a linear-response and rate-equation approach. Specif- propagating electrons carrying opposite spins. By time- ically, we predict that the Kondo temperature TK reversal invariance electron transport then becomes bal- which sets the scale below which the electrons screen the− listic, provided that the electron-electron (e-e) interaction impurity can be controlled by varying the strength is sufficiently well-screened so that higher-order scatter- of the Rashba− interaction. Surprisingly, for a strongly ing processesAdding do not the come intoRashba play??. interaction...anisotropic Kondo exchange, a non-collinear spin inter- The picture gets an added twist when including effects action mediated by the Rashba coupling becomes rel- from... magnetic breaks impurities, the locking contributed of spin by dopant to momentum. ions or evant (inHowever, the sense there of RG) is and still competes with the electrons a trapped single by Kramers potential inhomogeneities. pair on the SinceQSH an edge,Kondo and screening. this is This all challengesthat matters! the expectation that edge electron can backscatter from an impurity via spin the Kondo effect is stable against time-reversal invari- 1 exchange, time-reversal invariance no longer protects the ant perturbations? . Moreover, we show that the im- 1 helical states from mixing. In addition, correlated two- purity contribution to the dc conductance at tempera- ? ?? electron and inelastic single-electron processes must tures T>TK can be switched on and off by adjusting now also be accounted for. Askinetic a result, term at high temper- the RashbaRashba coupling. With the Rashba coupling being1 tunable by a gate voltage, this suggests a new inroad to atures T electron scattering off the impurityz leads to a y ln(T ) correctionH = vF? ofd thex conductanceΨ†(x)[ iσ at∂ lowx] frequenciesΨ(x)+α controldx Ψ charge†(x)[ transportiσ ∂x] atΨ the(x) edge of a QSH device. ? − Model. To− model the edge electrons, we introduce ω, which, however, vanishes in the dc limit ω 0. At 1 → 2 2 T low T , for weak e-e interactions, the quantizedv edgeα = con-v + αthe two-spinors Ψ = ψ , ψ ,whereψ (ψ ) annihi- F ↑ ↓ ↑ ↓ ductance G = e2/h is restored as T 0 with power lates a right-moving (left-moving) electron with spin-up 0 → iσxθ/2 1 laws distinctive of a helical edge liquid. For strongΨx inter-=e− (spin-down)Ψ along the growth direction of the quantum actions the edge liquid freezes into an insulator at T = 0, well. Neglecting e-e interactions, the edge Hamiltonian cos θ = v /v 2 2 with thermally induced transport via tunneling of frac-F αcanv thenα = bev writtenF + α as tionalized charge excitations through the impurity? . y z 1 A more complete description of edge transport in a H = vF dx Ψ†(x)[ iσ ∂x] Ψ(x)+ HzR = α kxσ 1 QSH system must include also the presence of a Rashba − H = vα dx Ψ†(x) iσ ∂x sinΨθ(x=)α/vα cos θ =z vF /vα spin-orbit interaction. This interaction,− which can be z’ y +α dx Ψ†(x)[ iσ ∂x] Ψ(x)+ (1) tuned by an external gate voltage, is a built-in feature − T of a quantum well? . In fact, HgTe quantumΨ wells= ψ ex-, ψ x x y y z z z y KT ↑ ↓ +Ψ†(0) [Jxσ S + Jyσ S + Jzσ S ] Ψ(0), Ψ = ψ , ψ sin θ =θ α/vα H = vF dx Ψ†(x)[ iσ ∂x] Ψ(x)+α dx Ψ†(x)[ iσ ∂x] Ψ(x) hibit some of the largest known Rashba couplings of↑ ↓ ? − − any semiconductor heterostructures The. As ”Rashba-rotated” a consequence, spinorwith vF the Fermi velocity parameterizing the linear ki- spin is no longer conserved, contrarystill to defines what is a assumed Kramers pairnetic energy. The second term encodes the Rashba inter- x ? √ 2 z T √ 2 iσ θ/2 in the minimal model of a QSHH system= v2α 2( .dKx However,Ψλ/†2)(x)1 iσ action∂zx Ψ of(x strength)2 2( K+αλ,/ with2) 1 theE third2 term2K being1 E an anti-2 γ0 (Jx + JHy )=Tvα −dx2 Ψ†−(x,)γ0 iσ2(J∂Ψxx ΨJ=y()xψ)T , ψ − , γ0 JNCT − , γ˜0 JNCT Ψ =e− Ψ since the RashbaH interaction=(v/∼ preserves2) dx time-reversal(∂xϕ) +( in-∂−xϑ−∼)ferromagnetic− ↑ Kondo↓ interaction∼ between electrons∼ (with i variance, Kramers’ theorem guarantees that the edge Pauli matrices σ ,i= x, y, z) and a spin-1/2 magnetic im- x A B C √ √ z + cos( 4πKϕ)+ sin(H =4vπKIϕdx=)+ΨG†(Vx) δ∂Iixσϑ∂ Ψ(x) κ κ α 0√ − x x K− x iσ θ/2 Ψ i=eσ θ−/2 Ψ g Ψ =e− Ψ + U cos(√16πKϕ) 2(πκ)2 H = α k σy z y R x H = v dx Ψ†(x)[ iσ ∂ ] Ψ(x)+α dx Ψ†(x)[ iσ ∂ ] Ψ(x) F iσxθx/2 gie 2 x Ψ =e− − + Ψ :(∂ ϑ) cos(√4π−Kϕ): (1) 2√ x 2π K z T TK y H = vF dx Ψ†(x)[ iσ ∂x] Ψ(x)+α dx Ψ†(x)[ iσ ∂x] Ψ(x) −z − y H = v dx Ψ†(x)[ iσ ∂ ] Ψ(x)+α dx Ψ†(x)[ iσ ∂ ] Ψ(x) F − x − x K x kkF kF x − z K<1/4 y H = v dx Ψ†(x)[ iσ ∂ ] Ψ(x)+α dx Ψ†(x)[ iσ ∂ ] Ψ(x) F − x − x x ∂J y y⊥ = HνRJ= αJkzx+σ ... x 2 2 HR = α kx∂σD − ⊥ H =(v/2) dx (∂xϕ) +(∂xϑ) G (T/T )2(1/4K 1) y K − ∂Jz HR = α2kxσ ∼ = νJ + ... A (2) B C ∂D − ⊥K y HR = α kxσ + cos(√4πKϕ)+ sin(√4πKϕ)+ ∂xϑ G =2e2/h κ κ √K K K gU 2 2 √ H =(J0 v/2)max(dx J(∂ ϕ,J) z+()D∂=ϑD)0 + cos( 16πKϕ) ≡ ⊥x x K 2(πκ)2 A B C + cos(√4πKϕ)+ sin(√4πKϕ)+ ∂ ϑ gie 2 κ 2 κ 2 √ x + :(∂ ϑ) cos(√4πKϕ): (1) H =(2 v/2) d2x (∂xϕ) +(∂xϑ) K 2 x H =(v/2) dx (∂ ϕ) T+(∂=gϑUD) exp( const./J ) 2 2 2π √K x +K x 0 cos(H√16=(πKv/ϕ2)) dx0 (∂xGϕ)=0+(∂xϑ) A 2 B C 2(√πκ) − √ + cos( 4πKϕ)+ sin(A 4πKϕ)+ B ∂xϑ C κ gie 2 κ √K A B + :(∂ ϑ+) cos(Ccos(√4√πK4πϕK):ϕ)+ sin(√4πKϕ)+ ∂xϑ (1) √ √2 x κ κ √ + cos( 4πKϕ)+ sin(gU2π 4√πKK√ϕ)+ ∂xϑ K κ +κ 2 cos( 16πK√ϕK) g 1/4 J0 max(J ,Jz)D+=D0 + z z HK ≡= Ψ†(0)⊥ J (σ S− + σ−S )+Jzσ Seff Ψ(0), ⊥ eff eff T< T< ∂J ⊥ = νJ Jz + ... ∂D − ⊥ ∂J z = νJ 2 + ... (2) ∂D − ⊥ J0 max(J ,Jz)D=D ≡ ⊥ 0 T = D exp( const./J ) K 0 − 0 T< Mn2+ (Sz)2 S =5/2 S =1/2 −→ eff 1 low T To model theAdding edge electrons, the Rashba we introduce interaction... the two-spinors ΨT = ψ , ψ ,whereψ (ψ ) annihilates a right-moving (left-moving) electron with spin-up (spin-down) along the growth direction↑ ↓ of the↑ quantum↓ well. Neglecting e-e 1 1 interactions, thee-e edgeinteraction Hamiltonian is invariant can then under be Ψ writtenΨ as → + + z z Kondo interaction HK = Ψ†(0) J (σ S− + σ−S )+Jzσ Seff Ψ(0), x x y y⊥ ze ff z eff y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) x 1 Ψ =e iσ θ/22Ψ 2 vα =− vF + α 1 iSxθ/2 iSxθ/2 S =e− Se x x y y z z Jy ,Jyz ,Jz NC z y HK = Ψ†(0)[Jxσ S + Jy σ S + JΨz σ ΨS + JNC(σ S + σ S )]Ψ(0) cos θ = vF /vα → z y displaymath H = vF dx Ψ†(x)[ iσ ∂x] Ψ(x)+α dx Ψ†(x)[ iσ ∂x] Ψ(x) − − XYZ Kondo Non-Collinear term x x iS θ/2 2 iS θ2/2 S =ev−α = vSF e+ α depend on the Rashba coupling α x controllablesin θ = by αa gate/vα voltage cos θ = v /v Ψ ΨF α → x x y y z z y z z y y H = Ψ†(0)[...bosonizationJ σ S + J andσ SRG+ J σ S + J (σ S HR+=σ αSkxσ)]Ψ(0) K Tx y z NC Ψ = ψ , ψ E. Eriksson, A. Ström, G. Sharma, H.J., PRBsin θ86=, 161103(R)α/vα (2012) 2 ↑ 2 ↓ vα = v + α F iSxθ/2 iSxθ/2 K T S =e− Se Ψ = ψ , ψ ↑ ↓ z H = vα cosdxθΨ=†v(Fx/v) α iσ ∂x Ψ(x) − 2 2 H =(v/2) dx (∂xϕ) +(∂xϑ) z H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x Ψ Ψ → A B C sin θ = α/vα + cos(√4πKϕ)+ sin(√4πKϕ)+ ∂xϑ κ κ √K g + U cos(√16πKϕ) x 2(πκ)2 iσ θ/2 x 2 2 Ψ =e Ψ iσ θ/2 T vα = v + α − Ψ =e− Ψ Ψ = ψ , ψ F gie 2 ↑ ↓ + :(∂ ϑ) cos(√4πKϕ): (1) 2π2√K x z H = vα dx Ψ†(x) iσ ∂xcosΨθ(x=) v /v F α kkF kF z − y − H = v dx Ψ†(x)[ iσ ∂ ] Ψ(x)+α dx Ψ†(x)[ iσ ∂ ] Ψ(x) F − x − x z y H = v dx Ψ†(x)[ iσ ∂ ] Ψ(x)+α dx Ψ†(x)[ iσ ∂ ] Ψ(x) F − x − x ∂J sin θ = α/vα ⊥ = νJ Jz + ... x ⊥ iσxθ/2 ∂D − Ψ =e− Ψ ∂J z = νJ 2 + ... (2) ∂D − ⊥ y HR =xα kxσ T Ψ = ψ , ψ ↑ ↓ J max(J ,J ) 0 z D=D0 z y ≡ ⊥ H = vF dx Ψ†(x)[ iσ ∂x] Ψ(x)+K α dx Ψ†(x)[ iσ ∂x] Ψ(x) − y − HR = α kxσ z H = vα dx Ψ†(x) iσ ∂x Ψ(x) − TK = D0 exp( const./J0) 2 2 − H =(v/2) dx (∂xxϕ) +(∂xϑ) A B C + cos(√4πKϕ)+ sin(√4πKϕ)+ ∂xϑ κ Kκ √K T< 1 Electrical control of the Kondo temperature via the ”Rashba angle” θ ~ gate voltage Jx = Jy = Jz = 10 meV 1 Jy ,Jz ,JNC Jx = Jy =5meV,Jz = 50 meV Jx = Jy = Jz = 10 meV displaymath easy-axis Kondo Jx = Jy =5meV,Jz = 50 meV θ α θ Jy ,Jz ,JNC RegionJ wherey ,Jz ,J NC dominates the RGdisplaymath flow x x obstructiony y of Kondoz screeningz ! y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) displaymath α x x y y z z y z z y H = Ψ†(0)[J σ S + J σ S + J σ S + J (σ S + σ S )]Ψ(0) α K x y z NC iSxθ/2 iSxθ/2 S =e− Se iSxθ/2 iSxθ/2 S =e− Se x x y y z z y z z y H = Ψ†(0)[J σ S + J σ S + J σ S + J (σ S + σ S )]Ψ(0) Ψ Ψ K x y z NC → Ψ Ψ v = v2 + α2 → α F iSxθ/2 iSxθ/2 S =e− Se cos θ = vF /vα v = v2 + α2 α F sin θ = α/vα Ψ Ψ → T Ψ = ψ , ψ ↑ ↓ z cos2 θ =2vF /vα H = v dx Ψ†(x) iσ ∂ Ψ(x) vα = vF + α α − x x Ψ =e iσ θ/2Ψ cos θ = v /v − sinF θ =α α/vα sin θ = α/vα T Ψ = ψ , ψ ↑ ↓ T Ψ = ψ , ψ ↑ ↓ z H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x z H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x x Ψ =e iσ θ/2Ψ − iσxθ/2 Ψ =e− Ψ z y H = v dx Ψ†(x)[ iσ ∂ ] Ψ(x)+α dx Ψ†(x)[ iσ ∂ ] Ψ(x) F − x − x x 1 1 Electrical control of the Kondo temperature via the ”Rashba angle” θ ~ gate voltage Jx = Jy = Jz = 10 meV 1 Jy ,Jz ,JNC Jx = Jy =5meV,Jz = 50 meV Jx = Jy = Jz = 10 meV displaymath easy-axis Kondo Jx = Jy =5meV,Jz = 50 meV θ α θ Jy ,Jz ,JNC RegionJ wherey ,Jz ,J NC dominates the RGdisplaymath flow x x obstructiony y of Kondoz screeningz ! y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) displaymath challenges the ‘folklore’ that the α Kondo effect is blind to time- reversal invariant perturbations! x x y y z z y z z y H = Ψ†(0)[J σ S + J σ S + J σ S + J (σ S + σ S )]Ψ(0) α K x y z NC iSxθ/2 iSxθ/2 drawn from Y. Meir and N.S. Wingreen, PRB 50, 4947 (1994) S =e− Se iSxθ/2 iSxθ/2 S =e− Se x x y y z z y z z y H = Ψ†(0)[J σ S + J σ S + J σ S + J (σ S + σ S )]Ψ(0) Ψ Ψ K x y z NC → Ψ Ψ v = v2 + α2 → α F iSxθ/2 iSxθ/2 S =e− Se cos θ = vF /vα v = v2 + α2 α F sin θ = α/vα Ψ Ψ → T Ψ = ψ , ψ ↑ ↓ z cos2 θ =2vF /vα H = v dx Ψ†(x) iσ ∂ Ψ(x) vα = vF + α α − x x Ψ =e iσ θ/2Ψ cos θ = v /v − sinF θ =α α/vα sin θ = α/vα T Ψ = ψ , ψ ↑ ↓ T Ψ = ψ , ψ ↑ ↓ z H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x z H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x x Ψ =e iσ θ/2Ψ − iσxθ/2 Ψ =e− Ψ z y H = v dx Ψ†(x)[ iσ ∂ ] Ψ(x)+α dx Ψ†(x)[ iσ ∂ ] Ψ(x) F − x − x x Conclusion E. Eriksson et al., PRB 86, 161103(R) (2012) The Kondo effect in a 2D topological insulator can be ”controlled” via a tunable gate voltage (given the ”right” setup: weak e-e screening, easy-axis Kondo exchange,...) Epilogue Is this kind of thing of any use? Possible applications in quantum information/simulation/ control/...? Why bother about the Kondo effect? An intriguing scenario: Generating long-distance qubit entanglement via the Kondo effect P. Sodano, A. Bayat, S. Bose, Phys. Rev. B 81, 100412(R) (2010) Left Bulk Right Bulk (a) ’Kondo spin chain’ N N-1 . . . 3 2 J ' 2 3 . . . N-1 N L J I J 'R 1 1 2 3 ...... N -1 N (b) M N 1 N 2 − − H = J (J σ σ + J σ σ )+J σ σ + J σ σ 1 1 · 2 2 1 · 3 1 i · i+1 2 i · i+2 i=2 i=2 N 2 N 3 M-2same low-energy physics as the spin sector of− the Kondo model− (c) H = J (J1σ1 σ2 + J2σ1 σ3 + J1σN 1 σN + J2σN 2S. EggertσN )+ andJ1 I. Affleck,σi PRBσi+1 46,+ 10866J2 (1992)σi σi +2 · · − · − · · · eff i=2 i=2 H I Ψ o | (d) ξK Traced out spins Traced out spins SA(r)= Tr(ρ log ρ ) (1) − [0,r] [0,r] ρ[0,r] =Tr[r, ]ρ =Tr[r, ] Ψ Ψ (2) ∞ ∞ | | H = H∗ + λ + ... 1O1 A ∂ n ∂ Z n S (r)= lim Trρr = lim R n 1 n 1 n − → ∂n − → ∂n Z Z n = Z∗ + ∂Z n R Rn R ∞ ∂Z = λ1 dτ 1(0, τ) + ... Rn O Rn −∞ A πξK vF c = 0 + + ... r ξK = 12r TK β r πξ cA =ln√2+ K + ... r ξ 12r K r c r SA(r)= log( )+cA + nonuniversal constant 6 a An intriguing scenario: Generating long-distance qubit entanglement via the Kondo effect P. Sodano, A. Bayat, S. Bose, Phys. Rev. B 81, 100412(R) (2010) 1 (a) Left Bulk Right Bulk ”Kondo cloud”, tunable via J’ N N-1 . . . 3 2 J ' 2 3ξ K . . . N-1 N L J I J 'R 1 (b) M A S (r)= Tr(ρ[0,r] log ρ[0N,r]1)N 2 (1) − − − H = J (J σ σ + J σ σ )+J σ σ + J σ σ 1 1 · 2 2 1 · 3 1 i · i+1 2 i · i+2 i=2 i=2 N 2 N 3 ρM-2 =Tr ρ =Tr Ψ Ψ − − (2) (c) H = J (J1σ1[0,rσ2]+ J2σ1 [σr,3 +]J1σN 1 σ[Nr,+ J]2σN 2 σN )+J1 σi σi+1 + J2 σi σi +2 · · ∞ − · ∞ | −· | · · eff i=2 i=2 H I Ψ o | H = H∗ + λ1 1 + ... (d) O ξK A ∂ n ∂ Z n Traced out spinsS (r)= Tracedlim outTr spinsρAr = lim R n 1 S (r)= Tr(nρ[0,r1] log ρ[0,r])n (1) − → ∂n − → ∂n Z ρ[0,r] =Tr[r, ]ρ =Tr[r, ] Ψ Ψ (2) ∞ ∞ | | Z n = Z∗ + ∂Z n R Rn R H = H∗ + λ + ... 1O1 ∞ ∂ ∂ Z ∂Z = λ A dτ (0, τ) n + ... n n 1 S (r)= lim1 Trρr =n lim R R n 1 R n 1 n O− → ∂n − → ∂n Z −∞ Z n = Z∗ + ∂Z n R Rn R A πξK vF c = 0 + + ... r ξK = 12r TK ∞ ∂Z = λ1 dτ 1(0, τ) + ... Rn O Rn −∞ βA r πξK vF c = 0 + + ... r ξK = 12r TK β r πξ cA =ln√2+ K + ... r ξ 12r K πξ cA =ln√2+ K + ... r ξ 12r K r r c r c r SA(r)= log( )+cA + nonuniversal constant SA(r)= log( )+cA +6 nonuniversala constant 6 a U(1) U(1) SU(2) SU(2) U(1) U(1) SU(2) Ising ⊗ ⊗ 1 ⊗ 1 −→ ⊗ ⊗ 2 ⊗ gA = √2 Simp =ln√2 1/2 G G (1 λ T ),T>T∗ ≈ 0 − 1 1 1 An intriguing scenario: t>t Generating0 long-distance qubit entanglement via the Kondo effect P. Sodano, A. Bayat, S. Bose, Phys. Rev. B 81, 100412(R) (2010) 1 iH t Ψ (t) e− q Ψ → | q ≡ | o (a) Left Bulk Right Bulk N 1 N 2 N N-1 . . . 3 2 J ' − 2 3 ξ. K. . − N-1 N H = J (J1σ1 σ2 + J2Lσ1 σJ3I)+J1J 'R σi σi+1 + J2 σi σi+2 · · · · i=2 i=2 1 (b) M A N 2 N 3 S (r)= Tr(ρ[0,r− ] log ρ[0,r])− (1) Hq = J (J1σ1 σ2 + J2σ1 σ3)+J (J1σN 1 σN + J2σN 2 σN )+J1 σiN σ1 i+1 + J2 Nσi2 σi+2 − − − − − → · · · H = J (J σ σ· + J σ σ )+J · σ σ + J σ· σ 1 1 · 2 2 1 · 3i=2 1 i · i+1 i=22 i · i+2 i=2 i=2 Ψ N 2 N 3 M-2|ρ o =Tr ρ =Tr Ψ Ψ − − (2) (c) H = J (J1σ1 σ2 +[0J2,rσ]1 σ3 + J[1r,σN ]1 σN + J2[σr,N 2] σN )+J1 σi σi+1 + J2 σi σi +2 · ground state· with ”optimal”∞− · Kondo cloud∞− ·| | · · eff i=2 i=2 H I ξ K Ψ o | H = H∗ + λ1 1 + ... (d) O SA(r)= Tr(ρ log ρ )ξ (1) − [0,r] [0,r] K A ∂ n ∂ Z n Traced out spinsS (r)= Tracedlim out spinsA Trρ = lim R S (r)=rTr(ρ[0,r] log ρ[0,r])n (1) ρ[0,r] =Tr[r, ]ρ =Tr[−r, n] Ψ1 ∂Ψn − − n 1 ∂n Z (2) ∞ ∞ →| | → ρ[0,r] =Tr[r, ]ρ =Tr[r, ] Ψ Ψ (2) ∞ ∞ | | H = H∗ + λ1 1 + ... O Z n = Z∗ + ∂Z n R Rn R H = H∗ + λ + ... 1O1 A ∂ n ∂ Z n R S (r)= lim Trρr = lim n − n 1 ∂n − n 1 ∂n∞Z → → ∂ ∂ Z ∂Z = λ A dτ (0n, τ) + ...n n S1 (r)= lim 1Trρr = limn R R n 1 nR1 n − → ∂On − → ∂n Z −∞ Z n = Z∗ + ∂Z n R Rn R Z n = Z∗ + ∂Z n R Rn R A πξK vF c = + + ... r ξK = ∞ 0 ∂Z = λ1 dτ 1(0, τ12) r + ... TK Rn Rn ∞ O ∂Z = λ1 dτ 1(0, τ) + ... −∞ Rn O Rn −∞ A πξK vF c = + + ... r ξ A = β πξK r vF 0 cK = 0 + + ... r ξK = 12r TK 12r TK β r β r πξ cA=ln√2+ K + ... r ξ 12r K A πξK πξ c =ln√2+ + ... r ξK cA =ln√2+ K + ... r ξ 12r 12r K r r c r c SA(rr)= log( )+cA + nonuniversal constant SA(r)= log( )+6 cA a+ nonuniversal constant 6 a U(1) U(1) SU(2) SU(2) U(1) U(1) SU(2) Ising ⊗ ⊗ 1 ⊗ 1 −→ ⊗ ⊗ 2 ⊗ gA = √2 Simp =ln√2 1/2 G G (1 λ T ),T>T∗ ≈ 0 − 1 1 1 An intriguing scenario: Generating long-distance qubit entanglement via the Kondo effect P. Sodano, A. Bayat, S. Bose, Phys. Rev. B 81, 100412(R) (2010) 1 t>t0 Left Bulk Right Bulk (a) iH t local quantum quench at t = t0 Ψ (t) e− q Ψ → | q ≡ | o N N-1 . . . 3 2 J ' 2 3 ξ. K. . N-1 N L J I J 'R 1 N 1 N 2 1 − − H = J (J1σ1 σ2 + J2σ1 σ3)+J1 σi σi+1 + J2 σi σi+2 · · · · N 1 N 2 1 (b) M i=2A i=2 − − S (r)=H = JTr((J1ρσ1 σ2 log+ J2σρ1 σ3)+)J1 σi σi+1 + J2 σi σi+2 (1) [0·,r] N 1 [0· ,r] N 2 · · − − i=2− i=2 H = J (J1σ1 σ2 + J2σ1 σ3)+J1 σi σi+1 + J2 σi σi+2 · · N 2 · N 3 · i=2 i=2 − − 1 Hq = J (J1σ1 σ2 + J2σ1 σ3)+J (J1σN 1 σN + J2σN 2 σNt>t)+J0 1 σi σi+1 + J2 σi σi+2 N 2 N 3 → · · − · − · · · − − Hq = J (J1σ1 σ2 + J2σ1 σi3=2)+J (J1σN 1 σNi+=2J2σN 2 σN )+J1 σi σi+1 + J2 σi σi+2 M-2→ · · − · N 2 − · N 3 · · ρ =Tr ρ =Tr Ψ Ψ − − i=2 i=2 (2) (c) H = J (J1σ1 σ2 +[0J2,rσ]1 σ3 + J[1r,σN ]1 σN + J2[σr,N 2] σN )+J1 σi σi+1 + J2 σi σi +2 · · ∞− · ∞− ·| | · · eff i=2 i=2 iHq t H I Ψo Ψq(t) e− Ψo t>t0 | Ψo → | ≡ | | Ψ o | H = H∗ + λ1 1 + ... N 1 N 2 ξ O iHq t (d) K − Ψq(t) e−ξ−K Ψo H = J (J1σ1 σ2 + J2σ1 σ3)+J1 σ→i σi+1 +|≡J2 σi σ|i+2 · · ξK· · i=2 i=2 A A ∂ S (r)= ∂Tr(ρZ[0,r] log ρ[0,r]) (1) S (r)= ATr(ρ[0,r] log ρ[0,r])n − N 1 n (1) N 2 Traced out spinsS− (r)= Tracedlim out spinsA Trρ = lim N− 2R N−3 S (r)=rTr(ρ[0,r] log ρ[0,r])n (1) H =−Jn(J11σ∂1 nσ2 + −J2σ1 −σn3)+1J∂1n −Zσi σi+1 + J2 − σi σi+2 Hq = J (J1σ1 σ2 + J2σ1 σ3)+J (J1σN →1 σN· + J2σN 2 · σN→)+J1 σi · σi+1 + J2 σi · σi+2 → · · − · − · i=2 · i=2 · ρ[0,r] =Tr[r, ]ρi=2=Tr[r, ] Ψ Ψ i=2 (2) ∞ ∞ | | ρ[0,r] =Tr[r, ]ρ =Tr[r, ] Ψ Ψ (2) ∞ ∞ | ρ[0,r|] =Tr[r, ]ρ =Tr[r, ] Ψ Ψ (2) ∞ ∞ | | N 2 N 3 − − Z n = Z∗ n + ∂Z n Hq = J (J1σ1 σ2 + J2σR1 σ3Ψ)+o RJ (J1σN 1RHσ=NH+ J+2λσN +2 ...σN )+J1 σi σi+1 + J2 σi σi+2 → · · | − · ∗ 1 1− · · · O i=2 i=2 H = H∗ + λ + ... H = H∗ + λ1 1 + ... 1O1 O A ∂ n ∂ Z n ∞ξK S (r)= lim Trρr = lim R ∂ n 1Ψ∂on∂ Z n 1 ∂n Zn ∂Z = λ A dτ (0n, τ−) → + ...n − → ∂ n S1 (r∂)=Z lim 1Trρr = lim n | R A Rn − n 1 ∂n − nR1 ∂n Zn S (r)= lim Trρr = lim R → O → n 1 n 1 n − → ∂n − → ∂−∞n Z A Z n = Z∗ + ∂Z n S (r)= Tr(ρ log ρ )R Rn R (1) Z[0,r] = Z∗ [0,r+]∂Z n n n ξ − R R R K πξK vF Z n =AZ∗ + ∂Z n R c R=n 0 +R + ... r ξK = ∞ 12r ∂Z = λ1 dTτ K1(0, τ) + ... ∞ n n ∂Z = λ dτ R (0, τ) + ...O R ρ[0,r] =Tr[r, ]ρn =Tr1 [r, A] Ψ 1 Ψ −∞n (2) ∞R S∞(r|)=O |Tr(Rρ log ρ ) (1) ∞ −∞ − [0,r] [0,r] ∂Z n = λ1 dτ 1(0, τ) n + ... R O R A πξK vF −∞ πξ c = 0 + + ...v r ξK = A β K r 12r F TK c = 0 + + ... r ξK = H = H∗ + λ112r1 + ... TK ρO[0,r] =Tr[r, ]ρ =Tr[r, ] Ψ Ψ (2) A πξK vF ∞ ∞ | | c = 0 + + ... r ξK = 12r T β r K β r A A ∂ πξnK ∂ Z n √ R S c(r)==lnlim2+Trρr =+ ...lim r n ξK − n 1 ∂n − nH 1=∂Hn ∗Z+ λ1 1 + ... → 12r →A πξK β r πξc =ln√2+ O + ... r ξK cA =ln√2+ K + ... r 12ξ r 12r K Z = Z∗ + ∂Z n n n ∂ ∂ Z πξ R RA R n r n A K S (rr)= lim Trρr = lim R c =ln√2+ + ... r ξK −rn 1 ∂n − n 1 ∂n Zn 12r → → ∞ c r ∂Z = λ cdτ rS(0A(,r)=τ) log(+ ...)+cA + nonuniversal constant n SA1(r)= log( 1)+cA + nonuniversal6 n a constant R O ZR = Z∗ + ∂Z A c r−∞6 A a n n n S (r)= log( )+c + nonuniversalR R constantR 6 a πξK vF cA = + + ... r ξ = 0 K ∞ 12r TK ∂Z = λ1 dτ 1(0, τ) + ... Rn O Rn U(1) U(1) SU(2) SU(2) U(1)−∞ U(1) SU(2) Ising ⊗ ⊗ 1 ⊗ 1 −→ ⊗ ⊗ 2 ⊗ β r A πξK vF c = 0 + + ... r ξK = 12r TK A πξg = √2 cA =ln√2+ K + ... r ξ 12r K β r Simp =ln√2 πξ cA =ln√2+ K + ... r ξ 12r K 1/2 G G (1 λ T ),T>T∗ ≈ 0 − 1 1 An intriguing scenario: Generating long-distance qubit entanglement via the Kondo effect P. Sodano, A. Bayat, S. Bose, Phys. Rev. B 81, 100412(R) (2010) 1 (a) Left Bulk Right Bulk N N-1 . . . 3 2 J ' 2 3 . . . N-1 N L J I J 'R t>t0 (b) M iH t Entanglement dynamics in Ψ ( t ) : e− q Ψ → q |≡ o | fast oscillatory and long-lived entanglement between the end spins! N 1 N 2 (c) M-2 − − H = J (J σ σ + J σ σ )+J σ σ + J σ σ eff 1 1 · 2 2 1 · 3 1 i · i+1 2 i · i+2 H i=2 i=2 I N 2 N 3 − − (d) Hq = J (J1σ1 σ2 + J2σ1 σ3)+J (J1σN 1 σN + J2σN 2 σN )+J1 σi σi+1 + J2 σi σi+2 → · · − · − · · · i=2 i=2 Traced out spins Traced out spins Ψ o | ξK SA(r)= Tr(ρ log ρ ) (1) − [0,r] [0,r] ρ[0,r] =Tr[r, ]ρ =Tr[r, ] Ψ Ψ (2) ∞ ∞ | | H = H∗ + λ + ... 1O1 A ∂ n ∂ Z n S (r)= lim Trρr = lim R n 1 n 1 n − → ∂n − → ∂n Z Z n = Z∗ + ∂Z n R Rn R ∞ ∂Z = λ1 dτ 1(0, τ) + ... Rn O Rn −∞ A πξK vF c = 0 + + ... r ξK = 12r TK β r πξ cA =ln√2+ K + ... r ξ 12r K Appendix 1 1 1 Low-temperature transport, T T (away from the ”dome” on slide 54) K 1 ”weak” e-e interaction K>1/4 T =0 2 Jx = Jy = Jz = 10 meV 2e α(x) = α(x) + αˆ(k )eikxn G = n h T T n 1/4 K K> L 1µm ikF x TK Jx = Jy =5meV,Jz =Ψ 50(x) meV=ψ (x)e± ∼ J = J = J = 10 meV ↑↓ ↑↓ T x y z = 10 meV Jz L 20µm= ikF x ∼ = Jy J = J =5meV,J = 50 meV Ψ (x)=ψ (x)e− Jx x y z θ ↓ ↓ = 50 meV ,Jz =5meV g= Jy Jx θ HR = α(x)(z k) σ × · θ Jy ,Jz ,JNC g 1/2v ∼ NCJy ,Jz ,JNC B = A z,J displaymath Jy,J ∇ × displaymath ξ < L (0) y )]Ψ α z S E = E(x, y, 0) α z + σ α y S (σ NC z + J z S αˆ( 2kF ) σ y + Jz displaymath ± y S z x x σy y z z y z (Ez yk ) σ = Eσ (kyx kxy) x + Jy HK = Ψ†(0)[Jxσ Sx + Jy σ S + Jz σ S x /+2 JNC(σ S + σ ×S )]Ψ· (0) − S x x S θy y z z y z z y Jxσ i H =(0)[Ψ†(0)[Jxσ S +x θ/2JSeσ S + J σ S + JNC(σ S + σ S )]Ψ(0) KΨ† iS y z = − HK 2k =e ± F S iSxθ/2 iSxθ/2 B S =e− Se A = (y, x, 0) Ψ 2 − Ψ → iSxθ/2 iSxθ/2 z 2g√CK S =e− Se ⊥ ≡ 2 2 + α Ψ Ψ v F = A k eB(kyx kxy) →vα · ∼ − z 4K 2 /vα ≡ − vF θ = 2 cos 2 Ψ Ψ 2 vα = v + α e F → G = ν /vα h γ = (αˆ(2k ) + αˆ( 2k ))/2 = α F − F sin θ cos θ = vF /vα , ψ 2 2 vα =ψ ↓v + α Mc 100 meV T = ↑ F ≈ ∂xθ Ψ D. Grundler, Phys. Rev. Lett. 84, 6074 (2000) sin θ = α/vα W. H¨ausler, L. Kecke, and A. H. MacDonald, Phys. ψ = η exp i√π[φ + θ] /√2πκ cos θ = v /v ↑ ↑ Rev. B 65,F085104α (2002) T Ψ = ψ , ψ ↑ ↓ 11 α0 2 10− eV m ψ = η exp i√π[φ θ] /√2πκ × ↓ ↓ − − Λ 0.5 eV z sin θ = α/vα H = vα dx Ψ†(x) iσ ∂x Ψ(x) 6 − vF 1 10 m/s × K + K 1.8 H0 + Hd + Hf + HR c s T Ψ = ψ , ψ 2 ↑ ↓ (αˆ ) HR O z H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x 1 1 Low-temperature transport, T T (away from the ”dome” on slide 54) K 1 ”weak” e-e interaction K>1/4 J = J = J = 10 meV 2e2 x y z G = δG h − T T K T =0 Jx = Jy =5meV,Jz = 50 meV Jx = Jy = Jz = 10 meV K>1/J4 = J =5meV,J = 50 meV x y z θ T T θ K Jy ,Jz ,JNC J ,J ,J displaymath Jx = Jy = Jz = 10 meV y z NC displaymath J = J =5meV,J = 50 meV x y z α α θx x y y z z y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) x x y y z z y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) iSxθ/2 iSxθ/2 Jy ,Jz ,JNC S =e− Se iSxθ/2 iSxθ/2 displaymath S =e− Se Ψ Ψ → α v = v2 + α2 Ψ Ψ α F → x x y y z z y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σS + σ S )]Ψ(0) cos θ = vF /vα 2 2 iSxθ/2 iSxθ/2 vα = v + α S =e− Se F sin θ = α/vα Ψ Ψ → cos θ = vF /vα T Ψ = ψ , ψ ↑ ↓ 2 2 vα = vF + α z sin θ = α/vα H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x cos θ = vF /vα T Ψ = ψ , ψ ↑ ↓ sin θ = α/vα z H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x 1 1 1 G =0 G =0 Low-temperature transport,1/4 1/4 J ,J ,J θ y z NC Jx = Jy =5meV,Jz = 50 meV displaymath θ x x y y z z y z z y Jx = Jy =5meV,Jz = 50 meV HK = Ψ†(0)[Jy ,JJz ,JxNCσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) α displaymath θ Jy ,Jz ,JNC x x y y z z y z z y HK = Ψ†(0)[Jxσ S α+ Jy σ S + Jzθσ S + JNC(σ S + σ S )]Ψ(0) iSxθ/2 iSxθ/2 displaymath S =e− Se iSxθ/2 iSxθ/2 J ,J ,J x x y y z Sz =e− ySez z y y z NC HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) displaymath α Jy ,Jz ,JNC x x iS θ/2 iS θ/2 Ψ Ψ S =e− Se → displaymath Ψ Ψ α 2 2 → x x y y z zΨ Ψ vα =y v z+ α z y H = Ψ†(0)[J σ S + J σ S + J σ S →+ J (σ SF + σ S )]Ψ(0) K x y z NC α 2 2 vα = v + α F x x y y z z y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + J2z σ S 2+ JNC(σ S + σ S )]Ψ(0) iSxθ/2 iSxθ/2 vα = v + α S =e− Se F x x y y z z y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) iSxθ/2 iSxθ/2 S =e− Se Ψ Ψ → iSxθ/2 iSxθ/cos2 θ = vF /vα S =e− Se Ψ Ψ 2 2 → vα = vF + α Ψ Ψ → sin θ = α/vα cos θ = vF /vα T Ψ = ψ , ψ ↑ ↓ sin θ = α/vα z H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x 1 1 1 G =0 G =0 Low-temperature transport,1/4 1/4 Jx = Jy =5meV,Jz = 50T meV α Ψ = ψ , ψ ↑ ↓ sin θ = α/vα θ x x y y z z y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) z H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x Jy ,Jz ,JNC iSxθ/2 iSxθ/2 S =e− Se displaymath Ψ Ψ → 1 1 1 G =0 G =0 Low-temperature transport,1/4 1/4 2 2 vα = v + α T =0 F x x y y z z y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + J2z σ S 2+ JNC(σ S + σ S )]Ψ(0) iSxθ/2 iSxθ/2 vα = v + α S =e− Se F K>1/4 x x y y z z y z z y HK = Ψ†(0)[Jxσ S + Jy σ S + Jz σ S + JNC(σ S + σ S )]Ψ(0) K>1/4 iSxθ/2 iSxθ/2 S =e− Se Ψ Ψ T TK → iSxθ/2 iSxθ/cos2 θ = vF /vα S =e− Se T TK Ψ Ψ → 2 2 Jx = Jy = Jz = 10 meV vα = vF + α Ψ Ψ sin θ = α/vα Jx = Jy =→Jz = 10 meV J = J =5meV,J = 50 meV cos θ = vF /vα x y z Jx = Jy =5meV,Jz = 50T meV Ψ = ψ , ψ ↑ ↓ θ sin θ = α/vα θ z H = v dx Ψ†(x) iσ ∂ Ψ(x) α − x Jy ,Jz ,JNC displaymath 1 1 ”High-temperature” transport, T T 1 K I = G0V δI − J ,z ω T ⊥ K< 1/4 () 2(1/4K 1) T TK G (T/Tbs) − K K ∼ → 2 c 2G12=2e /h K<1/4 = log( ) + const. E 8 21 + 2 θ E. Eriksson, A. Ström, G. Sharma, H.J., PRB 86, 161103(R) (2012); erratum, to be published 2(1/4K 1) G (T/TK ) − ∼ 21 G2 =0 2 2G2(=2√K eλ//h2)2 1 2 2(√K+λ/2)2 1 E 2 2K 1 E 2 γ0 (Jx + Jy ) T − − , γ0 (Jx Jy ) T − , γ0 JNCT − , γ˜0 JNCT ∼ ∼ 1/4− I = G V δI 0 − 8K 2 2K+2 (T/T ) − (T/T ) G =0 ∼ K ∼ K T TK 2e2 1/4 T =0 8K 2 2K+2 (T/T ) − (T/T ) ∼ K ∼ K 2(1/4K 1) G (T/T ) − ∼ K K>1/4 2e2 2 G = δG G =2e /h h − T T K T =0 G =0 Jx = Jy = Jz = 10 meV K>1/4 1/4 Jx = Jy =5meV,Jz = 50 meV T T K θ Jx = Jy = Jz = 10 meV Jx = Jy =5meV,Jz = 50 meV