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What Is the Quantum Spin Hall Effect? What Is the Quantum Spin Hall Effect? What is the quantum spin Hall effect? What is the quantum spin Hall effect? Fir! paper on "in Hal e$ect: D’yakonov and Perel ’, 1971 What is the quantum spin Hall effect? Recall the quantum Hall effect... Recall the quantum Hall effect... impurity scattering broadening of Landau levels into bands of localized states bulk insulator Recall the quantum Hall effect... What carries the current?! Recall the quantum Hall effect... What carries the current?! “skipping currents”! Recall the quantum Hall effect... 1 “skipping currents” conducting edge states quantization Halperin (1982) e2 No channel for backscattering ballistic transport = iv :R† (x)∂ R (x)::L† (x)∂ L (x): G = ν Hτ − F τ x τ τ x τ h 2ik0 (x+a/2) $ % 2∆R cos(Qx) e− F R† (x)L (x)+H.c.(6), − τ τ A QH state is stable against local perturbations $ % M 100 meV c ≈ τ = D. Grundler, Phys. Rev. Lett. 84, 6074 (2000) ± W. H¨ausler, L. Kecke, and A. H. MacDonald, Phys. Rev. B 65, 085104 (2002) q0 11 !α0 2 10− eV m " × y H = i (γ + γ cos (Qna)) c† σ c H.c. Λ 0.5 eV R − 0 1 n,µ µν n+1,ν − " n,µ,ν v 1 106 m/s & ' ( F " × Kc + Ks 1.8 " 1 γj = αja− (j = 0, 1) HR ϕc (ϕ+ + ϕ )/√2 (7) ≡ − H = dx [ + ] Hc Hs ϕs (ϕ+ ϕ )/√2 (8) ! ≡ − − v m = i [(∂ ϕ )2 +(∂ ϑ )2] i cos( 2πK ϕ ), (1) Hi 2 x i x i − πa i i int = g1 :Rτ† Lτ L† τ R τ : + g˜2τ :R+† R+Lτ† Lτ : " H − − − g4τ + (:R+† R+Rτ† Rτ : + R L) (9) ∆R = γ1 sin(q0a) 2 ↔ with vi and Ki functions of g1τ , g˜2τ and g4τ g˜ g δ g (10) 2τ ≡ 2τ − τ+ 1τ Λ bandwidth (2) ∼ with vi and K functions of g1τ , g2τ , g3τ (11) ϕc (ϕ+ + ϕ )/√2 (3) ≡ − (+, ) ( , ) in ”g-ology” notation (12) − ↔ ) ⊥ ϕs (ϕ+ ϕ )/√2 (4) ≡ − − ∂tϕτ = vF ∂xϑτ Rτ† and Lτ† create excitations at the Fermi points of the right- and left-moving branches with spin projection τ R (x) = η exp i√π[ϕ (x) + ϑ (x)] /√2πa (5) τ τ τ τ $ % L+ L (x) = η¯ exp i√π[ϕ (x) ϑ (x)] /√2πa τ τ − τ − τ $ % 2 2 vF = 2a t + γ0 and ∆R = γ1 sin(q0a) ητ , η¯τ Klein factors # Can a system be stable against local perturbations without breaking time-reversal invariance? Consider a Gedanken experiment... Bernevig & Zhang (2006) 1 1 1 1 uniformlyE char= Eged(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 (5) × · − righRτ† t-andandLτ† left-mocreate excvingitationsbranchesat thewithFermispinpoinprotsjectionof theτ 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 QHE2 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· for∼ce − 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 + γ and ∆R = γ1 sin(q0a) · ∼· ∼ y − x− 0 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)# # = iv :R† (x)∂ R (x)::L† (x)∂ L (x): B G =Gνh= ν Hτ − F τ x τ τ x τ A = (y, x, 0) h h 0 2 − 2ikF (x+a/2) τ = ivF :Rτ† (x$)∂xRτ (x)::Lτ† (x)∂xLτ (x): % 2∆R cos(Qx) e− Rτ† (x)Lτ (x)+H.c.(6), Hτ = −ivτ F= :Rivτ† (Fx):∂RxR† (τx(x)∂)x::RLτ†(x)∂::xLL†τ((xx))∂: xL (x): Mc 100 meV L+ H − τ τ τ τ − 0 ≈ H − 0 2ikF (x+0a/2) $ % 2∆R cos(Qx2)ikFe−(x+a/$22)ik (x+Ra/τ†2()x)Lτ (x)+H.c.(6), % $ % 2∆R cos(2Qx∆) cos(e− Qx) e− RFτ† (x)Lτ (xR)+† (Hx).c.L(6),(x)+H.c.(6), A k eB(kyx kxy) MMc 100100meVmeV $ − − % R τ τ · ∼ D. Grundler,− Phys.c≈MRev.c 100Lett.meV84, 6074 (2000) − ≈ ≈ $ % 2 2 $ $ % % D. Grundler, Phys. Rev. Lett. 84, 6074v(2000)F = 2a t + γ0 and ∆R = γ1 sin(q0a) τ = D. D.Grundler,Grundler,Phys.PhRev.ys. Rev.Lett.Lett.84, 607484, 6074(2000)(2000) ± W. e2H¨ausler, L. Kecke, and A. H. MacDonald,# Phys. G = ν τ =τ = Rev.h B 65, 085104 (2002) ±τ = W.W.H¨ausler,H¨ausler,L.L.KecKecke,ke, andand A.A. H.H. MacDonald,MacDonald, PhPhys.ys. ± ± 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 Rev. B 65, 085104 (2002) 2ikF (x+a/2) $ % 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)α 2Λ 10 011.115 eVeV m q0 !α! 0 0 2 10−− eV11m !%%α0 ××%2 10− 6 eV m τ = W. H¨ausler, L. Kecke, and A. H. MacDonald,Λ Phv0F%ys..5 eV×1 10 m/s ± y Λ % 0.5 eV HR = i (γ0 + γ1 cos (Qna)) cn,µ† σµν cn+1,ν H.c. Rev. B 65, 085104 (2002) %Λ %0.56 eV× − − vKF + K1% 1061.m/s8 n,µ,ν y vF c 1 s 10 m/s6 HR = i (γ0 + γ1 cos (Qna)) c† σ c y H.c. v% ×%1 10 m/s HR = i (γ0 &+ γ1 cos (Qna))n,µc†µν σn'+1c,νy H.c. ( % F × −H = i (γ + γ cos (Qnan,µ)) cµ†ν σn+1− c,ν H.c. Kc + Ks 1%.8 × q0 −Rn,µ,ν 0 1 n,µ µν n−+1,ν K11c + Ks 1.8 &n,µ,−ν ' ( − !α0 2 10− eV Km + K% 1.8 n,µ,ν ' ( & × c % s & ' ( 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)) c† σ c H.c. γj = αja− (j = 0, 1) & × − n,µ µν n+1,ν − 1 K + K 1.8 HR n,µ,ν γj = αja− (j1= 0, 1) c s ' ( γj = αja− (j = 0, 1) & HR & HR 1 ϕc (ϕ+ + ϕ )/√2 (7) γj = αja− (j = 0, 1) ≡ − HR H = dx [ c + s] ϕc (ϕ+ + ϕ )/√2 (7) H H ≡ − √ H = dx [ c + s] ϕc (ϕ+ + ϕ )/ 2 √ (7) H! H ≡ϕc (ϕ+−+ ϕ )/ 2 (7) H = H! =dx [ dcx+[ s+] ] ≡ − H Hc s ϕc (ϕ+ + ϕ )/√2 (7) H = dx [ + ] ! H H ≡ − ϕs (ϕ+ ϕ )/√2 (8) Hc Hvs i 2 ! 2 mi ≡ √ − − ! v = [(∂ ϕ ) +(∂ ϑm) ] cos( 2πK ϕ ), (1) ϕs (ϕ+ ϕ )/ 2 (8) i i 2 2 x i 2 x i i πa i i ≡ − − i = H[(∂xϕi) +(∂xϑi) ] cos(− 2πKiϕi), (1) ϕs (ϕ+ ϕ )/√2 (8) H vi 2 −mπai ϕs (ϕ+ ϕ )/√2 (8) vi 2 2 mi "ϕs (ϕ+ ϕ )/√2 (8) ≡ − − vi 2 i = 2 [(m∂xi ϕi) +(∂2xϑi) ] 2 cos( 2πKiϕi),≡ (1)− − ≡ − − i = [(∂xϕi) +(∂xϑi) ] i = cos([(∂x2ϕπiK) iϕ+i)(, ∂x(1)ϑi) ] "cos( 2πKiϕi), (1) H 2 H H2 − πa2 − πa− 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 − 4†τ + (:R† R† R† R : + R L) (9) ∆R = γ1 sin(q0a) H − − int = g1 :Rτ† L+τ L τ(:RR††τR: +Rg˜†2Rτ :+R: +++R+τ†Lτ†LτL)τ :(9) ∆ = γ sin(q a) g4τ H int −= g1 :Rτ†−Lτ L−+τ R+2 ττ: +τ g˜2τ :R+R+Lτ† Lτ↔: with vi and RKi functions1 0 of g1τ , g˜2τ and+g4τ (:R+† R+Rτ† Rτ : + RH L) (9) − 2 − − ↔ with vi and Ki functions∆R =of γg1τsin(, g˜2τq0anda) g4τ 2 ↔ g4τ with vi and Ki functions of g1τ , g˜2τ and g4τ g4τ† + +(:R+(:RR+†RRτ† RRτ :† R+ R: + RL) L(9)) (9) with v and K functions of g , g˜ and g 2 2 + + τ τ ↔ ↔ withi vi andi Ki functions1τof g21ττ , g˜2τ and4τ g4τ Λ bandwidth Λ bandwidthΛ(2) bandwidth g˜2τ g(2)2τ δτ+g(2)1τ (10) g˜2τ g2τ g˜2δττ+gg12ττ δτ+g1τ (10) (10) ∼ ∼ ∼ ≡ − ≡ − ≡ − g˜ g δ g (10) Λ Λbandwidthbandwidth (2) (2) 2τ g˜2τ2τ g2ττ+ δ1ττ+g1τ (10) ∼ ∼ ≡ ≡− − ϕc (ϕ+c + (ϕϕ+)/+√ϕ2 )/√2 (3) (3) with vi andwithKvfunctionsi and K functionsof g1τ , g2τ ,ofg3gτ1τ , g2(11)τ , g3τ (11) ≡ ≡ − − √ ϕc ϕ(ϕc + +(ϕϕ+ +)/ϕ 2)/√2 (3) (3) withwithvi andvi KandfunctionsK functionsof g1ofτ , gg21ττ,,gg32ττ , g3τ (11)(11) ≡ ≡ − − Consider a Gedanken experiment..
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