Lecture notes on topological insulators

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1 Lecture notes on topological insulators Ming-Che Chang Department of Physics, National Taiwan Normal University, Taipei, Taiwan (Dated: November 1, 18) Contents I. D Topological insulator 1 A. General theory 1 1. Edge state in D topological insulator. Lattice with inversion symmetry 3. Z integer ν as a topological invariant 3 B. Bernevig-Hughes-Zhang model 3 1. Time reversal and space inversion 5. Z topological number 5 3. Orbital basis versus spin basis 5 4. QWZ model versus BHZ model 6 References 6 I. D TOPOLOGICAL INSULATOR pairs, e iχ 1 e iχ 1 w() = e iχ e iχ.... (1.) Then one can show that its determinant at a TRIM is, det w( i ) = N wn( i ), (1.3) n=1 where w n () = e iχ n. For an antisymmetric N N matrix M, one can define its pfaffian (H.E. Haber s note), A. General theory pf M = P ( 1) P M i1j 1 M ij M in j N, (1.4) Analogous to the connection between 1D charge pump and D quantum Hall effect (see??), there is also a close connection between 1D spin pump and D topological insulator (TI). The ey ingredient that leads to the Z values of P θ is the TRS of the 1D lattice at t =, T/. We can identify the parameter t with the Bloch momentum y of a D lattice system (see Fig. 1), then y =, π play similar roles to t =, T/. Mathematically, one expects a similar Z number from the following expression, ( 1) ν = w n ( 1 ) w n ( ) w n ( 3 ) w n ( 4 ) n w n ( 1 ) w n ( ) w n ( 3 ) w n ( 4 ) = ±1, (1.1) where i are the TRIM shown in Fig. 1. Introducing the sewing matrix for N filled Kramer T/ t π 3 = ( π,) y 1 = (, ) 4 = ( π, π ) x = (, π ) FIG. 1 The domain of (, t) variables in the 1D Fu-Kane model. The first quadrant of the Brillouin zone in a D lattice model. The four corners are the TRIM. where P is a permutation from (1,,..., N) to (i 1, i,, i N ), such that i 1 < j 1, i < j,, j N < j N, and i 1 < i < < i N. ( 1) P is the sign of permutation p. For example, a pf = a. (1.5) a For the determinant in Eq. (1.3), we have pf w( i ) = N w n ( i ). (1.6) n=1 Pfaffian pf M can be considered as the square root of determinant det M, because of the general relation, det M = (pf M). (1.7) If we define the a cumulative index for filled bands at each TRIM as, δ i n filled then Eq. (1.1) can be re-written as ( 1) ν = w n ( i ) w n ( i ) = pf w( i) det w(i ), (1.8) 4 pf w( i ) 4 det w(i ) = δ i. (1.9) i=1 i=1

2 (c) The parity eigenvalue ζ ni = ±1 is the same for the two Bloch states (with α = ±) in a Kramer pair. Note that there is a slight difference between Bloch state ψ nα and cell-periodic state u nα, t T/ y π FIG. Comparison of the edge states in 1D spin pump, D topological insulator, and (c) D trivial insulator. is a schematic plot of the edge state in Fig.??. and (c) show the energy levels in edge BZ. 1. Edge state in D topological insulator If the 1D spin pump or the D lattice in Fig. 1 has an edge at certain cutoff x, then (or x ) is no longer a good quantum number. In Fig.??, we have shown the evolution of the edge states in a spin pump. By analogy, a D TI would have similar edge states inside the energy gap, crossing each other at TRIM (Fig. ). In comparison, even though the edge states in a trivial insulator also need to cross each other at TRIM (Fig. (c)), the ways they lin together are different. This is due to the fact that when δ 1 δ and δ 3 δ 4 have opposite signs, ν = 1, and we have a TI accompanied by switch of Kramer-pair partner. When δ 1 δ and δ 3 δ 4 have same sign, ν =, and we have a trivial insulator with no switch of Kramer-pair partner. If one plots a horizontal line (chemical potential) inside the energy gap, then it would cut the edge states in odd number of times, but cut those in (c) even number of times. The former cannot be avoided by shifting or distorting the energy levels of edge states, while the latter can be avoided. Thus, the edge states in TI are robust, while those in trivial insulator are not.. Lattice with inversion symmetry Even though ν is nown to have two possible values, and 1, it is not a trivial tas to get explicit values of χ ni at the first place. Fortunately, if the lattice has space inversion symmetry (SIS), then w n ( i )/ w n( i ) is simply equal to the parity ζ n ( i ) of the Bloch state ψ ni± (a Kramer pair with index α = ±). This fact is proved below, following Nomura, 13. First, under space inversion, ψ nα (r) Πψ nα (r) = ψ nα ( r) = ψ n α (r), (1.1) where Π is the SI operator. cell-periodic state, y π The same is true for the u nα (r) Πu nα (r) = u n α (r). (1.11) If the lattice has SIS, then ψ nα (but not u nα ) are parity eigenstates at = i, Πψ niα(r) = ζ ni ψ niα(r). (1.1) ψ n+gα = ψ nα, (1.13) but u n+gα = e ig r u nα. (1.14) Therefore, only the Bloch state can have ψ n G α = ψ n G α. Second, define a different type of sewing matrix as follows, for a particular Kramer pair with index n (suppressed), v αβ () = u α ΠΘ u β. (1.15) (1) The matrix v αβ is anti-symmetric. Pf: Using the relation ψ 1 Θ ψ = Θ ψ 1 Θ ψ = ψ Θψ 1, and Π = Π, one has v αβ = u α ΠΘ u β (1.16) = u α Θ (Π u β ) (1.17) = u β ΠΘ u α (1.18) = v βα. (1.19) () The matrix v αβ is unitary. Pf: Using the relation ψ 1 ψ = ψ ψ 1, one has ( vv ) = v αγ αβ vγβ (1.) β = u α ΠΘ u β u γ ΠΘ u β (1.1) β = u α u γ (1.) = δ αγ. (1.3) Similarly, one can also show that ( v v ) αγ = δ αγ. Therefore, it is unitary. The matrix w αβ is also unitary, and is antisymmetric at i. Therefore, sewing matrices w αβ ( i ) and v αβ ( i ) differ by a multiplicative constant (the parity!) at TRIM: w αβ ( i ) = u iα Θ u iβ (1.4) = ψ iα Θ ψ iβ (1.5) Under a gauge transformation, = ψ iα Π Θ ψ iβ, Π = 1 (1.6) = ζ i ψ iα ΠΘ ψ iβ (1.7) = ζ i u iα ΠΘ u iβ (1.8) = ζ i v αβ ( i ). (1.9) u + u + = e iϕ u +, (1.3) u u = u, (1.31) the off-diagonal matrix element of v() transforms as, v() v () = e iϕ v(). (1.3)

3 3 EBZ EBZ CdTe HgTe CdTe CdTe HgTe CdTe FIG. 3 Time reversal conjugate pairs and effective Brillouin zone. Folding the EBZ into a cylinder, with open edges. Therefore, one can adjust its phase such that v( i ) = 1. Thus, As a result, It follows that, w n ( i ) = ζ n ( i )v n ( i ) = ζ n ( i ). (1.33) w n ( i ) w n ( i ) = ζ n( i ). (1.34) δ i = n filled ζ n ( i ), (1.35) which is the cumulative parity of filled Bloch states (pic only one ζ n ( i ) for each Kramer pair) at a TRIM, and ( 1) ν = 4 δ i. (1.36) i=1 3. Z integer ν as a topological invariant To understand the Z topology, we follow Moore and Balent s argument for D TI (Moore and Balents, 7). Because of time-reversal symmetry, the degenerate Bloch states for and in a Brillouin zone are time-reversal conjugate (see Fig. 3). As their Berry curvatures cancel with each other, the first Chern number for a filled band vanishes. Since the domain of independent Bloch states cover only half of the BZ (called effective Brillouin zone, or EBZ), one may wonder if the integral of the Berry curvature over the EBZ could be quantized. Unfortunately, since the EBZ does not form a closed surface (see Fig. 3), no quantization is guaranteed. To fix this, one can put two fictitious caps to close the EBZ. This closed surface should have an integer C 1, but its value depends on the caps of choice. However, Moore and Balents proved that C 1 mod is independent of the caps of choice. Therefore, C 1 mod is an intrinsic property of the EBZ itself. We thus have two topological classes: being the usual insulator, and 1 being the topological insulator. d<d c d>d c FIG. 4 HgTe is sandwitched between CdTe, forming a quantum well. The positions of the discrete energy levels in the QW depend on the width of the QW. Fu and Kane have shown that the Z integer can also be written as (Fu and Kane, 6), ν = 1 ( ) d Ω z d A mod. (1.37) π EBZ EBZ The second term is the Berry connection integrated around the boundary of the EBZ. This is different from the Chern number in a quantum Hall system, which only has the first term, and is an integral over a closed surface (the whole BZ). The topological number here is for a manifold with edge. Eq. (1.37) loos lie the generalized Gauss-Bonnet formula for a D surface M with edge, in which the Berry curvature is replaced by the Gaussian curvature G, and the Berry connection is replace by the geodesic curvature g of the boundary, χ = 1 ( ) d r G dr g. (1.38) π M M For example, for a torus, χ =, for a dis-lie surface (which has a boundary), χ = 1, and for a sphere, χ =. So far we have learned that the Z topological invariant ν can be calculated via, 1. The sign of pf w/ det w at TRIM (or, the parities of Bloch states at TRIM, if there is inversion symmetry).. The Gauss-Bonnet-lie integral formula above. In addition, ν can also be characterized by 3. The winding number between two patches of gauge choices over the Brillouin zone (Fu and Kane, 6). 4. The vorticity of the pfaffian of a sewing matrix m (see Prob. ). 5. The axion angle of electromagnetic response (see Chap. 9). Except for the parities in 1., none of these is easy to evaluate. For a crystal without inversion symmetry, one can deform it to one that has SIS and determine its ν by parities. This is a valid shortcut if the energy gap remains open during the process of deformation. B. Bernevig-Hughes-Zhang model The first experimentally confirmed (D) TI is made from semiconductor quantum well. Bul HgTe has inverted band structure (due to spin-orbit coupling) near

4 4 E() CB t sp 4-band model E g s p x ±ip y HH LH FIG. 5 Typical energy bands near the energy gap of a semiconductor, in which each line is two-fold degenerate. The conduction band is originated from the s-orbital. The valence band from the p-orbitals are composed of heavy-hole (HH) band, light-hole (LH) band, and spin-orbit split-off (SO) band. SO ε, ε s p tss, t pp, tsp R R + a µ FIG. 6 The hopping amplitude of an electron hopping from a s-orbital to the p x ± ip y orbitals of a nearest-neighbor atom is designated as t sp. Each atom has s-orbital and p- orbital, with energies ε s and ε p. An electron can hop between s-orbitals, between p-orbitals, or between an s-orbital and a p-orbital. the Fermi energy. Unfortunately, it s a metal, not an insulator. Nevertheless, one can sandwitch it between CdTe (an ordinary insulator), forming a quantum well (QW) and opening an energy gap near the Fermi energy (see Fig. 4). When the HgTe layer is thic, the discrete QW energy levels remain inverted, similar to the bul states. However, if the HgTe layer is thinner than a critical width d c, then the electron (E1) and hole (H1) levels in the QW would switch positions. Therefore, one can chec if the topology of the QW states (signified by the emergence of helical edge states) depends on the width of the QW (König et al., 7). Typical semiconductor band structure near = is shown in Fig. 5. In a QW, the LH bands split off the HH bands, so in the simplified Bernevig-Hughes-Zhang (BHZ) model, only conduction band and one HH valence band are considered (each are two-fold degenerate). In order to investigate the parities of Bloch states at TRIM, we follow the lattice version of the BHZ model proposed by Fu and Kane, 7 (also, see Nomura, 16) For an atom at site R, four states are considered, s, s, p x + ip y, p x ip y. (1.39) Without ambiguity, one can write p x + ip y and p x ip y simply as p and p. They are the states with quantum numbers m j = 3/ and m j = 3/. We consider only the electron hopping between nearest neighbors. The relevant parameters are on-site energies ε s, ε p, and hopping amplitudes t ss, t pp, and t sp (and its complex conjugate t ps ), see Fig. 6. In a D square lattice, the vectors R + a µ (µ = ±x, ±y) point to the four nearest neighbors of R. The tight-binding Hamiltonian is therefore given as, H = H + H 1, H = (ε s Rsσ Rsσ + ε p Rpσ Rpσ ),(1.4) Rσ=± and (given t ps = t sp ) H 1 = Rσ µ=±x,±y (t ss R + a µ sσ Rsσ (1.41) t pp R + a µ pσ Rpσ + e iθµσ t sp R + a µ sσ Rpσ + e iθµσ t sp Rpσ R + a µ sσ ), where θ µ = (ˆx, a µ ). That is, θ x =, θ y = π/, θ x = π, θ y = 3π/. Such a system has both TRS and SIS (details later). Because of the lattice translation symmetry, the Hamiltonian can be diagonalized using the momentum basis. We therefore introduce the Fourier transformation (N is the total number of lattice sites), Rsσ = 1 e i R sσ, (1.4) N Rpσ = 1 e i R pσ, (1.43) N and get (the lattice constant is set as one) = H() (1.44) εs t ss (cos x + cos y ) t sp (σ z sin y i sin x ). t sp (σ z sin y + i sin x ) ε p + t pp (cos x + cos y )

5 5 Or, εs + ε p H() = (t ss t pp )(cos x + cos y ) (t ss + t pp )(cos x + cos y ) τ z 1 + t sp sin x τ y 1 + t sp sin y τ x σ z. (1.45) The Pauli matrices τ x,y,z are for the orbital degree of freedom, and σ x,y,z are for the spin degree of freedom. 1. Time reversal and space inversion Time reversal flips spin, but not orbital, therefore it acts only on the spin degree of freedom. The TR operator is thus, iσy K Θ = = 1 iσ iσ y K y K. (1.46) The s-orbital is even under space inversion, while the p-orbital is odd under SI. That is, Therefore, Π = Π sσ = sσ, (1.47) Π pσ = pσ. (1.48) 1 = τ 1 z 1. (1.49) One can chec that the Hamiltonian H() is indeed invariant under these two transformations,. Z topological number ΘH()Θ 1 = H( ), (1.5) ΠH()Π 1 = H( ). (1.51) Since the BHZ model has SI symmetry, one can calculate the Z topological number from the parities of the Bloch states at TRIM (see Fig. 1). At a TRIM, the first line of Eq contributes a constant energy shift and can be ignored, and the third line is zero. Therefore, H( 1 ) = (t ss + t pp ) τ z 1, (1.5) H( ) = τ z 1, (1.53) H( 3 ) = τ z 1, (1.54) H( 4 ) = + (t ss + t pp ) τ z 1. (1.55) Since they are proportional to the parity operator Π = τ z 1, an energy eigenstate at i is also a parity eigenstate. Let s assume ε s > ε p in the following discussion. If ε s ε p > 4(t ss + t pp ) >, then the energies at 1 are ε 1+ = + (t ss + t pp ), (1.56) ε 1 = (t ss + t pp ). (1.57) The degenerate eigenstates ψ 1+ has even parity, while ψ 1 have odd parity. Only the state ψ 1 is filled, so δ( 1 ) = 1. Similarly, one can also get δ( ) = δ( 3 ) = δ( 4 ) = 1. Therefore, ( 1) ν = 1, or ν =. (1.58) This is a trivial insulator. On the other hand, if ε s ε p < 4(t ss + t pp ), then ε 1 > ε 1+. Due to the band inversion, now the states ψ 1+ are filled instead. Therefore, δ( 1 ) = 1. The other three parities are not changed. As a result, This is a topological insulator. 3. Orbital basis versus spin basis ( 1) ν = 1, or ν = 1. (1.59) Note that in Eq. 1.44, every bloc of the Hamiltonian matrix is diagonal. In this case, it is possible to bloc-diagonal the 4 4 matrix by re-arranging the order of the basis, s, s, p, p (1.6) s, p, s, p. (1.61) For convenience, we call the first choice the orbital basis, and the second the spin basis. Under the spin basis, the Hamiltonian becomes (first switch the nd and the 3rd rows, then switch the nd and the 3rd columns of the matrix) where H() = ( h() h ( ) ), (1.6) h() εs t = ss (cos x + cos y ) t sp ( i sin x + sin y ) t sp (i sin x + sin y ) ε p + t pp (cos x + cos y ) = ε () + t sp sin x τ y + t sp sin y τ x + (t ss + t pp )(cos x + cos y ) τ z. (1.63) Because of the bloc diagonalization, the up and down spins are explicitly decoupled. Note that the TR and SI operators also are altered under the new basis. They now become Θ = iσ y K 1, Π = 1 τ z. (1.64)

6 6 ε ( ) FIG. 7 In real space, there is a helical edge state along the boundary of a D TI. In momentum space, the energy dispersion curves of the edge states cross each other at a TRIM. 4. QWZ model versus BHZ model When written in the bloc-diagonal form, the Hamiltonian h() is similar to the QWZ model Hamiltonian for the QAHE (see??). That is, the BHZ model is composed of two independent QWZ subsystems, h() and h ( ). One can write h() = ε () + d() τ, (1.65) then the Hall conductivity of the subsystem is, σ xy = e 1 d 1 h 4π d 3 d d d. (1.66) x y BZ One can chec that if ε s ε p > 4(t ss + t pp ), then the signs of d z () are all positive at the TRIM (analogous to Fig.??). If ε s ε p < 4(t ss + t pp ), then d z ( 1 ) becomes negative, while the other three signs remain the same (see Fig.??). According to the analysis of the QAHE in??, the first case has σ xy =, while the second case has σ xy = e /h. When the subsystem is in the QAH phase, according to the discussion in??, it has chiral edge states. Since this subsystem consists only of spin-up electrons (see Eq. 1.61), the edge-state electrons are spin-up. On the other hand, the conjugate subsystem h ( ) has σ xy = e /h. Its edge electrons transport along the opposite direction and the spins are down (see Fig. 7). In momentum space, the energy dispersion of the edge state is linear in the small -limit. One has positive slope (positive velocity), and the other has negative slope (negative velocity). Because of the Kramer degeneracy, these two dispersion curves have to cross each other at a TRIM (see Fig. 7). This point degeneracy can be lifted only if the TRS is broen. The topological phase of the BHZ model is called a D TI phase, aa a quantum spin Hall (QSH) phase. Its edge state, with one spin moving along one direction, and the opposite spin moving along the opposite direction, is called a helical edge state. It is robust in the sense that, even if there is a non-magnetic impurity V imp (r) blocing the way, the electron will not be bac scattered since that requires a spin flip. In the Born approximation, the transition amplitude for bacscattering is zero since (ψ e is an edge state) ψ e V imp (r) θψ e =. (1.67) See Prob. of Chap??. If there is a magnetic impurity that breas TRS, then an electron could be bacscattered, accompanied by a µ spin flip. Also, in the presence of electron interaction, there is a possibility that the edge is spontaneously magnetized. Should this happen, then the edge state is no longer protected by the TRS. Exercise 1. Start from the tight-binding Hamiltonian in Eqs. (1.4) and (1.41), switch to the momentum basis, and verify Eq. (1.44).. In addition to w, s, and v, a fourth type of sewing matrix is defined as m αβ () = u α Θ u β. (1.68) Consider the case with only one Kramer pair (α, β = ±): Show that the matrix m is unitary and antisymmetric. Show that m( ) = w()m ()w T (). (1.69) (c) With the help of the identity pf(bab T ) = (det B)(pf A), show that log[det w()] = log[pf m( )] log[pf m ()] = log[pf m( )] + log[pf m()]. (1.7) (d) Let x and y be or π. Write pf m( x, y ) as m( x ); pf w( x, y ) as w( x ). Show that ( y = y ), P θ = 1 π d x x log m( x ) + i m(π) log πi π π m(). (1.71) Note that w( x ) = m( x ) = ±1. (e) Finally, show that P θ P θ ( y = π) P θ ( y = ) = 1 d log[pf m()] mod, (1.7) πi EBZ where EBZ = [ π, π] [, π] is the upper half of the BZ. That is, the Z topological invariant is the total vorticity of the zeros of pf[m()] in the effective BZ. For more details, see Fu and Kane, 6 and Sec. 4.5 of Fruchart and Carpentier, 13. References Fruchart, M., and D. Carpentier, 13, Comptes Rendus Physique 14, 779. Fu, L., and C. L. Kane, 6, Phys. Rev. B 74, Fu, L., and C. L. Kane, 7, Phys. Rev. B 76, 453. König, M., S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenamp, X.-L. Qi, and S.-C. Zhang, 7, Science 318(5851), 766. Moore, J. E., and L. Balents, 7, Phys. Rev. B 75, Nomura, K., 13, Fundamental theory of topological insulator, unpublished, written in Japanese. Nomura, K., 16, Topological insulator and superconductor (Maruzen Publishing Co.), in Japanese.