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 eﬀect:

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...

“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. e2Häusler, L. Kecke, and A. H. MacDonald,# Phys. G = ν τ =τ = Rev.h B 65, 085104 (2002) ±τ = W.W.Häusler,Häusler,L.L.KecKecke,ke, andand A.A. H.H. MacDonald,MacDonald, PhPhys.ys. ± ± W. Häusler, 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. Häusler, 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... Bernevig & Zhang (2006) 1 1 1 1

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) ≡ ≡ − − Two copies of a QH system 1 bulk insulator with 1 time-reversed edge modes 1 1

electric Efield= E(x, y, 0) 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 tem × · 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) Quantum spin Hall (QSH) sys × · − 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· ∼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 + 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. e2Häusler, L. Kecke, and A. H. MacDonald,# Phys. G = ν τ =τ = Rev.h B 65, 085104 (2002) ±τ = W.W.Häusler,Häusler,L.L.KecKecke,ke, andand A.A. H.H. MacDonald,MacDonald, PhPhys.ys. ± ± W. Häusler, 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. Häusler, 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) ≡ ≡ − − Quantum spin Hall (QSH) system 2D bulk insulator with time-reversed edge modes

No breaking of time reversal!

Is the QSH state stable against local perturbations?

+ = 0

destructive interference from time-reversed path Is the QSH state stable against local perturbations?

Yes! The QSH state is protected by time reversal invariance. 1 Is the QSH state stable against local perturbations?

2 2e Yes! The QSH state is protected α(x) = α(x) + αˆ(k )eikxn G = by time reversal invariance. % & n h n # ballistic transport

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 Can QSH physics be realized in in the lab?

early proposal by Kane and Mele (2005)

But, probably doesn’t work... too small band gap in the bulk... QSH physics may be realized in HgTe quantum wells proposal: Bernevig, Hughes and Zhang (2006) supporting experiments: König et al. (2007) QSH physics may be realized in HgTe quantum wells proposal: Bernevig, Hughes and Zhang (2006) supporting experiments: by König et al. (2007)

from Science 318, 766 (2007) strong spin-orbit interactions in atomic p-orbitals create an inverted band gap (p-band on top of s-band) strong spin-orbit interactions in atomic p-orbitals create an inverted band gap (p-band on top of s-band)

supports a single pair of time-reversed edge modes inside the inverted gap

Kramer’s degeneracy at k=0 protects the stability of the edge modes single pair of Kramer’s pair (QSH insulator)

two Kramer’s pair (ordinary band insulator) strong spin-orbit interactions in atomic p-orbitals create an inverted band gap (p-band on top of s-band)

supports a single pair of time-reversed edge modes inside the inverted gap 1 Kramer’s degeneracy at k=0 protects the stability of the edge modes

2 2e α(x) = α(x) + αˆ(k )eikxn G = % & n h n # ballistic transport

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)

(αˆ2) H O R 1

2 2e ikxn König et al. (2007) α(x) = α(x) + αˆ(k )e G = % & n h n # ballistic transport!

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)

(αˆ2) H O R At the edge: A new kind of electron liquid At the edge: A new kind of electron liquid

A single pair of time-reversed states can’t exist in 1D At the edge: A new kind of electron liquid

? At the edge: A different kind of electron liquid

A single pair of time reversed edge states can exist at the boundary of a 2D system The QSH edge forms a “holographic liquid”

new physics...!