v. Tε n k =ε n k T r T = r, T v T = r, I v I = I r I = v. Iε n k =ε n k Berry curvature: Symmetry Consideration n k = n k

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2 Berry curvature: Symmetry Consideration Time reversal (i.e. motion reversal) 1 1 T r T = r, T v T = v. Tε n k =ε n k n k = n k Inversion Symmetry: 1 1 I r I = r, I v I = v. Iε n k =ε n k n k = n k

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5 θ k1, k2 x, y 1 p= dl 2π dl a

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7 Eq. (a) is gauge invariant although depends on gauge. This Eq. represents an important topological feature of the systems. Consider an arrow whose directional angles are given by the phase of the wave function. The arrow rotates p times as we go around the boun. This gives a topological constraint to the wave fn. Consider a zero of the wave function. If we go around clockwise a small circle which contains the 0, the corresponding arrow rotates once either clwise or anticlwise.

8 Therefore we can regard a 0 of a wvfn as a vortex which has vorticity either 1 or -1 corresponding clwise or antclwise rotns of arrow,resp. Cases where we have a multiple rotation are considered to be specail one of having several vortices at the same point. The magnetic fd. forces a wave function to have p vorticity in the magnetic unit cell. This is topological constraint because the total vorti city in the magnetic unit cell is independent of particular potential chosen.

9 Define vector potential (Berry Connection) Hall Conductivity as Curvature

10 Observation: The magnetic BZ is topologically a Torus T2. Application of Stoke s thm to Eq. would give cond. 0 if A(k1,k2) is uniquely defined on the entire torus. A possible non 0 value of cond. Is a consequences of a non-trivial topology of A. In order to understand non-trivial topology of A, let us first discuss a guage transformation of a special kind. Introduce a transformation

11 Non-trivial topology arises when the phase of wavefn can not determined uniquely over entire MBZ The previous transformation implies that over all pha se factor can be chosen arbitrary. The phase can be determined by demanding that a a component of the state vector u(x0,y0) is real. This convention is not enough to fix the phase on the entire MBZ, since u(x0,y0) vanishes for some value of (k1,k2) Consider a simple case when u(x0,y0) vanishes only at one point (k10,k20) in MBZ.

12 Phase can be fixed by deman ding that a component of the State vector u(x0,y0) is real. However, this is not enough to fix the phase over the MBZ, when u vanishes at some pt. Divide Torus in 2 pieces H1 & H2 such that H1 contains (k10,k20). Adopt different convention in H1 so that another component u(x1,y1) real. The overall phase Is uniquely determined.

13 The Chern no is topological in the sense that it is invariant under small deformation of the Hamiltonian Small changes of the Hamiltonian result a small change of the Berry Curvature (adiabatic curvature),one might think small change in chern no, but chern no is invaria nt. Therefore, we observe plateau. But how chern no change from one plateau to the next? Large deformation of the Hamiltonian can cause the ground state to cross over other eigenstates. When such Level crossing happens in QHS, the adiabatic curvature diverges and the chern no is no longer defined. The transition between chern nos plateau take place at level crossing

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16 Z2 Invariants T symmetry identifies two important subspaces of Bloch Hmailtonian H(k) and the corresponding occupied band wave function u_i(k)>. Even subspace: T symmetry requires that H(k) belong to the even subspace at the point k=0 and as well as three M points.

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25 Insulators a material in which no electrons that are not bound with their respective place inside material happens in specific type of materials, and depends on thing such as no of electrons per atom and how they are arranged in solid

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27 Effect of the sample boundary In 1960, Kohn characterized the insulating state in terms of the sensitivity of electron inside the material to effect on the sample boundary. The presence of a bulk gap does not guarantee that electrons will always show insensitivity to boundary.

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29 Electrons feels force perp. to it motion and Applied field. Cause to move in circular orbit radius depending on the field.

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32 Berry curvature: Symmetry Consideration Time reversal (i.e. motion reversal ) T r T = r, T v T = v. Tε n k T =ε n k n k = n k Inversion Symmetry: I r I = r, I v I = v. Iε n k I =ε n k n k = n k

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34 KdS=2π n=2π 2 g Define vector potential (Berry Connection) s Hall Conductivity as Curvature

35 Topological invariance

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37 What happens if

38 Gauss-Bonnet

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41 Chern number and Magnetic monopole

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43 Observation: The magnetic BZ is topologically a Torus T2. Application of Stoke s thm to Eq. would give cond. 0 if A(k1,k2) is uniquely defined on the entire torus. A possible non 0 value of cond. Is a consequences of a non-trivial topology of A. In order to understand non-trivial topology of A, let us first discuss a guage transformation of a special kind. Introduce a transformation

44 Non-trivial topology arises when the phase of wavefn can not determined uniquely over entire MBZ The previous transformation implies that over all pha se factor can be chosen arbitrary. The phase can be determined by demanding that a a component of the state vector u(x0,y0) is real. This convention is not enough to fix the phase on the entire MBZ, since u(x0,y0) vanishes for some value of (k1,k2) Consider a simple case when u(x0,y0) vanishes only at one point (k10,k20) in MBZ.

45 Phase can be fixed by deman ding that a component of the State vector u(x0,y0) is real. However, this is not enough to fix the phase over the MBZ, when u vanishes at some pt. Divide Torus in 2 pieces H1 & H2 such that H1 contains (k10,k20). Adopt different convention in H1 so that another component u(x1,y1) real. The overall phase Is uniquely determined.

46 The Chern no is topological in the sense that it is invariant under small deformation of the Hamiltonian Small changes of the Hamiltonian result a small change of the Berry Curvature (adiabatic curvature),one might think small change in chern no, but chern no is invaria nt. Therefore, we observe plateau. But how chern no change from one plateau to the next? Large deformation of the Hamiltonian can cause the ground state to cross over other eigenstates. When such Level crossing happens in QHS, the adiabatic curvature diverges and the chern no is no longer defined. The transition between chern nos plateau take place at level crossing

47 What are topological band insulators? Topology characterizes the identity of objects up to deformation, e.g. genus of surfaces Figure courtesy C. Kane Similarly, band insulator can be classified up to the deformation of band structure. Modify smoothly preserving gap. H

48 Imagine an interface when a crystal slowly interpolates as a function of x between a QHS (n=1) and a trivial Insulator (n=0). Somewhere along the way the energy Gap has to go to zero, because otherwise it is impossib le for the topological invariant to change. There will be low energy electronic state bound to the region where Energy gap passes through zero.

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50 Can one realize a quantum hall like insulator WITHOUT a magnetic field? Yes: Kane and Mele; Bernevig & Zhang (2005), Spin-orbit interaction» spin-dependent magnetic field Spin-orbit interaction is Time Reversal symmetric: Spin-Hall Effect

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53 Two Dimensional Topological Insulator (Quantum Spin Hall Insulator) Time reversal symmetry =>two counter-propagating edge modes Requires spin-orbit interactions Protected by Time Reversal. Only Z2 (even-odd) distinction. (Kane-Mele)

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55 Special features of 2D T-I edge states Single Dirac node impossible in 1D with time reversal symmetry Stable to (non-magetic) disorder: (no Anderson localization though 1D) Single Dirac Node Experiments: transport on HgTe quantum well (Bernevig et al., Science 2006; Konig etal. Science 2007)

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59 If number edge states pair are even, then all right movers would hybridized with left movers with Exception their partner, hence no transport. On other hand odd pair after hybridization there Will be still edge state connecting the bands. Which of these two alternative occures is determined by the topological class of bulk band structure.

60 In strong topological insulator, the Fermi surface for the surface state encloses an odd number of degeneracy points.

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