1. Journal Club of Topological Materials (2014)

2. If you raised your hand you’re in the wrong place!! Show of hands, who here is familiar with the concept of topological insulators?

3. The Quantum Spin Hall Effect Tejas Deshpande Joseph Maciejko, Taylor L. Hughes, and Shou-Cheng Zhang. “The Quantum Spin Hall Effect.” Annual Reviews of Condensed Matter Physics 2, no. 1 (2011): 31-53.The Quantum Spin Hall Effect.

4. Introduction Ginzburg-Landau Theory of Phase Transitions Classify phases based on which symmetries they break Rigorous definition of “symmetry breaking”: ground state does not possess symmetries of the Hamiltonian Example: classical Heisenberg model Ordered phase characterized by local order parameter Phases Defined by Symmetry Breaking Rotational and Translational: Crystalline Solids (continuous to discrete) Spin Rotation Symmetry: Ferromagnets and Antiferromagnets U(1) gauge symmetry: Superconductors

5. Introduction “Topological” Phases Integer Quantum Hall Effect (IQHE) discovered in 1980 Topological or “global” order parameter  Hall conductance quantized in integral units of e 2 /h Fractional Quantum Hall Effect (FQHE) discovered in 1982 Phase transitions do not involve symmetry breaking Experimental implications of “topological order” Number of edge states equal to topological order parameter (Chern number) Edge states robust to all perturbations due to “topological protection” Current = 1 μA Magnetic Field = 18 T Temperature = 1.5 K

6. Introduction Topological Protection Current carried only by chiral edge states Chiral edge states robust to impurities No tunneling between opposite edges FQHE FQHE with (1/m)e 2 /h (m odd) Hall conductance gives rise to bosonic quasiparticles Example: FQHE with m = 3 has quasiparticles with 3 flux quanta attached Chern-Simons theory is the low energy effective field theory

7. Introduction Road to Topological Insulators (TIs) IQHE without a magnetic field: Haldane model Observation of the “spin Hall effect” Occupations of Light-Hole (LH) and Heavy-Hole (HH) bands Spin conductance

8. Phenomenology of the Quantum Spin Hall Effect Classical spin vs. charge Hall effect Charge Hall effect disappears in the presence of time-reversal symmetry Odd under time reversal Constant Even under time reversal Non-zero spin Hall conductance in the presence of time-reversal symmetry Even under time reversal Constant Even under time reversal Does the quantum version of the spin Hall effect exist? Yes! Kane and Mele proposed the quantum spin Hall effect (QSHE) in graphene and postulated the Z 2 classification of band insulators

9. Phenomenology of the Quantum Spin Hall Effect QSHE as a “topologically” distinct phase “Fractionalization” at the boundary “Topological” in the sense that the electron degrees of freedom are spatially separated Mechanism of spatial separation: QHE  External magnetic field (time-reversal breaking) QSHE  intrinsic spin-orbit coupling (time-reversal symmetric)

10. The QSHE in HgTe Quantum Wells Review of basic solid state physics What does spin-orbit coupling do? What does time-reversal symmetry imply? Kramers pair states What does inversion symmetry imply? Kramers pairs well defined even when spin is not conserved What do both time-reversal and inversion symmetries imply?

11. The QSHE in HgTe Quantum Wells Banstructure of bulk CdTe s-like (conduction) band Γ 6 and p-like (valence) bands Γ 7 and Γ 8 with (right) and without (left) turning on spin-orbit interaction With spin-orbit interaction Γ 8 splits into the Light Hole (LH) and Heavy Hole (HH) bands away from the Γ point The split-off band Γ 7 shifts downward

12. The QSHE in HgTe Quantum Wells Banstructure of bulk HgTe s-like (conduction) band Γ 6 and p-like (valence) bands Γ 7 and Γ 8 with (right) and without (left) turning on spin-orbit interaction The Γ 8 splits into LH and HH like CdTe except the LH band is inverted The ordering of LH band in Γ 8 and Γ 6 bands are switched

13. The QSHE in HgTe Quantum Wells Quantum Well (QW) fabrication Molecular Beam Epitaxy (MBE) grown HgTe/CdTe quantum well structure Confinement in (say) the z-direction Transport in the x-y plane L = 600 μm and W = 200 μm Gate voltage (V G ) used to tune the Fermi level (E F ) in HgTe quantum well E z Band gap of QW Band gap of barrier EFEF

14. The QSHE in HgTe Quantum Wells Topological phase transition QW sub-bands invert for well thickness d > 6.3 nm Intersection of the first electron sub-band with hole sub-bands

15. The QSHE in HgTe Quantum Wells The Bernevig-Hughes-Zhang Model Hamiltonian with QW symmetries Components Elegant Hamiltonian form Break translational symmetry in the y-direction

16. BHZ Model

17. Since The QSHE in HgTe Quantum Wells The BHZ Model Numerical diagonalization? Try ansatz Writing Plugging in explicit expressions and multiplying by Γ 5 we get

18. The QSHE in HgTe Quantum Wells The BHZ Model Solutions Normalization condition Bulk dispersion Surface dispersion where s labels Kramers pairs

19. Using Landauer-Büttiker formalism for an n- terminal device For the helical edge channels we expect For a 2-point transport measurement between terminals 1 and 4 The QSHE in HgTe Quantum Wells The BHZ Model

20. If the transport is dissipationless where is the resistance coming from? In QSHE don’t we have spin currents of e 2 /h + e 2 /h = 2e 2 /h and charge currents of e 2 /h – e 2 /h = 0? Answer 1: dissipation comes from the contacts. Note that transport is dissipationless only inside the HgTe QW Answer 2: We do measure charge conductance! The existence of helical edge channels is inferred from charge transport measurements The QSHE in HgTe Quantum Wells The BHZ Model

21. For normal ordering of bands the Landau levels will get further apart as B increases For inverted bandstructures Landau levels will cross at a certain B Only inverted bandstruc- tures will reenter the quan- tum Hall states when B field increases The QSHE in HgTe Quantum Wells The BHZ Model

22. The concept of “helical” edge state  states with opposite spin counter-propagate at a given edge QH protected by “chiral” edge states; QSH edge states protected due to destructive interference between all possible back- scattering paths Clockwise and anticlockwise rot- ation of spin pick up ±π phase leading to destructive inter- ference Theory of the Helical Edge State

23. The physical description of edge state protection works only for single pair of edge states With (say) two forward- movers and two back- ward-movers backscattering is possible without spin flip Robust or non-dissipative edge transport requires odd number of edge states Theory of the Helical Edge State

24. Only two TR invariant non-chiral interactions can be added Combined with Umklapp term we get (opens a gap at k F = π/2) We can “bosonize” the Hamiltonian The forward scattering term simply renormalizes the parameters K and v F forward scattering term Two-particle backscattering or “Umklapp” term Boson to fermion field operators Stability of the Helical Liquid: Disorder and Interactions

25. Total Hamiltonian RG analysis  Umklapp term relevant for K < 1/2 with a gap: Interactions can spontaneously break time-reversal symmetry TR odd single-particle backscattering: Bosonize N x and N y. For g u < 0 fixed points at Umklapp term Due to thermal fluctuations TRS is restored for T > 0 For mass order parameter N y is disordered + TR is preserved with a gap For g u < 0, N y is the (Ising-like) ordered quantity at T = 0 Stability of the Helical Liquid: Disorder and Interactions

26. Total Hamiltonian Umklapp term Two-particle backscattering due to quenched disorder Gaussian random variables The “replica trick” in disordered systems shows disorder relevant for K < 3/8 N x and N y show glassy behavior at T = 0 with TRS breaking; TRS again restored at T > 0 Where would all these interactions come from? locally doped regions? Band bending? But edge states are immune to electrostatic potential scattering Potential inhomogeneities can trap bulk electrons which may then interact with the edge electrons K < 1 Stability of the Helical Liquid: Disorder and Interactions

27. Static magnetic impurity breaks local TRS and opens a gap Quantum impurity  Kondo effect: Doing the “standard” RG procedure we get flow equations Stability of the Helical Liquid: Disorder and Interactions

28. Static magnetic impurity breaks local TRS and opens a gap Quantum impurity  Kondo effect 1.At high temperature (T) conductance (G) is log 2.For weak Coulomb interaction (K > 1/4) conductance back to 2e 2 /h. At intermediate T the G ~ T 2(4K-1) due to Umklapp term 3.For strong Coulomb interaction (K < 1/4) G = 0 at T = 0 due to Umklapp. At intermediate T the G ~ T 2(1/4K–1) due to tunneling of e/2 charge Stability of the Helical Liquid: Disorder and Interactions

29. Quantized charge at the edge of domain wall o Jackiw-Rebbi (1976) o Su-Schrieffer-Heeger (1979) Helical liquid has half DOF as normal liquid  e/2 charge at domain walls Mass term ∝ Pauli matrices  external TRS breaking field Mass term to leading order Current due to the mass field For m 1 = m cos(θ), m 2 = m sin(θ), and m 3 = 0 Topological response  net charge Q in a region [x 1,x 2 ] at time t = difference in θ(x,t) at the boundaries Charge pumped in the time interval [t 1, t 2 ] Fractional-Charge Effect and Spin-Charge Separation

30. Two magnetic islands trap the electrons between them like a quantum wire between potential barriers Conductance oscillations can be observed as in usual Coulomb blockade measurements Background charge in the confined region Q (total charge) = Q c (nuclei, etc.) + Q e (lowest subband) Flip relative magnetization  pump e/2 charge Continuous shift of peaks with θ(B) AC magnetic field drives current Fractional-Charge Effect and Spin-Charge Separation

31. Simplified analysis: o Assume S z is preserved o QSHE as two copies of QHE Thread a π (units of ℏ = c = e = 1) flux ϕ TRS preserved at ϕ = 0 and π; also, π = –π Four possible paths for ϕ ↑ and ϕ ↓ : Current density from E || : Net charge flow: Fractional-Charge Effect and Spin-Charge Separation

32. 3D Topological Insulators Introduction 2D topological insulator  1D edge states Dirac-like edge state dispersion What happens in 3D? 3D topological insulator  2D surface states Surface dispersion is a Dirac cone, like graphene What happens in 1D? Nothing!

33. Topological band theory Difficult to evaluate ℤ 2 invariants for a generic band structure Consider the matrix At the TRIM B(Γ i ) is antisymmetric; we can define Trivial: (–1) ν 2D = +1 and Non-trivial: (–1) ν 2D = –1 3D Topological Insulators Topological invariant “Dimensional increase” to 3D Weak TI: (–1) ν 3D = +1 and Strong TI: (–1) ν 3D = –1

34. With inversion symmetry rewrite δ i as where ξ 2m (Γ i ) = ±1 is the parity eigenvalue of the 2m th band at Γ i ) and ξ 2m = ξ 2m–1 are Kramers pairs Gap closing (phase transition) k = (0, 0)  M = 0 k = (π, 0) and (0, π)  M = 4B k = (π, π)  M = 8B 3D Topological Insulators Simplified topological invariant expression Recall BHZ model

35. Conclusion and Outlook The quantum spin Hall effect (QSHE) Phenomenology Design of quantum wells in the QSHE regime Explicit solution of Bernevig-Hughes-Zhang (BHZ) model Experimental verification using transport Properties of the “2D topological insulator” Theory of helical edge states Effects of interactions and disorder Fractionalization and spin-charge separation Introduction to 3D topological insulators Topological Band Theory (TBT) Topological Invariant of the QSHE