Lecture notes on topological insulators

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1 Lecture notes on topological insulators Ming-Ce Cang Department of Pysics, ational Taiwan ormal University, Taipei, Taiwan Dated: October 5, 08 Contents I. Carge polarization, anomalous quantum Hall effect A. Modern teory of carge polarization B. Rice-Mele model C. Qi-Wu-Zang model 4 D. Edge state in te Qi-Wu-Zang model 6 References 7 I. CHARGE POLARIZATIO, AOMALOUS QUATUM HALL EFFECT In addition to te quantum Hall effect, Berry pase is essential to te calculation of carge polarization. It could also be a ey ingredient in anomalous Hall effect. We illustrate tese connections in tis capter. A. Modern teory of carge polarization Te electric polarization P of an infinite crystal, or a crystal wit periodic boundary condition PBC is generically not well defined. Te reason is tat, in a periodic solid, te electric polarization depends on one s coice of te unit cell. Te teory of electric polarization in conventional textboos applies only to solids consisting of well localized carges, suc as ionic or molecular solids Clausius-Mossotti teory. It fails, for example, in a covalent solid wit bond carges suc tat no natural unit cell can be defined see Fig.. A crucial observation made by R. Resta is tat, even toug te value of P may be ambiguous, its cange is well defined Resta, 99. It was soon pointed out by King-Smit and Vanderbilt tat P as a deep connection wit te Berry pase of te Bloc state King-Smit and Vanderbilt, 993. Unit cell + - P P FIG. For an infinite lattice, or a lattice wit periodic boundary condition, te polarization is ill-defined. It depends on te coice of te unit cells. - + To illustrate te modern teory of polarization, let s consider a one-dimensional D lattice wit periodic boundary condition PBC. Tere are lattice points located at R l = la l Z, and a is te lattice constant. Te Fourier transformation of te Bloc state ψ n = e ix u n is called te Wannier state, nr l = π = π e ir l ψ n,. wic is localized near a lattice point R l. states form an ortonormal basis, Te Bloc ψ n ψ n = δ nn δ.. Liewise, te Wannier states are also ortonormal to eac oter, n R nr = δ nn δ RR..3 In an insulator, te electric polarization can be related to te carge center of te Wannier function, P = q filled n n0 x n0, q = e.4 = q u n e i x e ix u n.5 i n = q u n e i x i u n..6 n Decompose te x-integral as a sum of unit-cell integrals, and write x = ma+x x [0, a], it can be sown tat see omura, 03, p.85 u n e i x i u n = δ, u n i u n cell.7 = δ, u n i u n,.8 were cell integrates over only one unit cell, and cell = is used in te second equation. Terefore, P = q u n i u n.9 n = qa n π π d π A n..0

2 π 0 π FIG. Te domain of Bloc momentum and parameter is similar to a of a two dimensional lattice. Under a gauge transformation, u n = eiχ n u n, were χ n is single-valued mod π, te polarization contributed from band-n is, P n = P n qa χ nπ χ n π.. π Single-valuedness of χ n gives χ n+π = χ n + πma m Z, terefore, P n = P n qma.. Te polarization could cange by qma under a gauge transformation. Tat is, only te fractional part of P n and P is pysical. ote tat te integral in Eq..0 is noting but te Berry pase divided by π, P = qa n γ n π..3 Te Berry pase of D Bloc states is first studied by Za, 989 and is called Za pase. In addition to being gauge dependent, te value of Za pase is also coordinate dependent see Prob., but te relative pases between energy bands are fixed. For a D lattice wit space inversion symmetry, if te origin is at a symmetric point, ten te Za pase can only be 0 or π, and P n can only be 0 or qa/. Tis can be understood as follows: It was sown earlier tat if te lattice as space inversion symmetry, ten see Eq.?? Terefore, after te inversion, P n P n = qa A n = A n..4 π π d π A n = P n..5 Since P n is allowed to ave te freedom of canging by qma, te constraint given by te inversion symmetry is P n = P n mod qa.6 P n = 0 or q a mod qa..7 Put it in anoter way, for a lattice wit space-inversion symmetry, te carge center n0 x n0 in a unit cell can only be at 0 or a/ on top of a lattice point or in te middle between two lattice points. On te oter and, if tere is no inversion symmetry, ten γ n and P n can be any value. An experimental confirmation of te Za pase wit cold atoms can be found in Atala et al., 03. Even toug P n is gauge dependent, te cange of polarization P n under an adiabatic and continuous deformation is gauge invariant and well-defined. From ere on we consider only one filled band, and use to label te degree of ion displacement. It varies from 0 to as te ions sift adiabatically from an initial position to a final position. Te difference of polarizations between tese two states is, π d P P = qa π π [A, A, ]..8 Te parameter defines a second dimension in addition to. We can coose te parallel transport gauge, suc tat tere is no pase accumulation along te -direction, ten P can be written as a counter-clocwise line integral around te rectangle in Fig., P = qa d A,,.9 π = qa d F z,.0 π were F z = A A, and A = i u u. If = 0, ten te rectangle is similar to a, were opposite edges abor te same states. Terefore, te integral of te Berry curvature is an integer π, and P = qac.. Tat is, under a periodic parametric variation, te carge transport is quantized. Tis fact can be utilized to build a quantized carge pump Touless et al., 98. However, be aware tat even toug te quantization is precise in te adiabatic limit, it is no longer if te pumping is fast. Te analysis above is based on one-particle states in D. But te same sceme can be extended to real solids wit electronic interactions in tree dimensions. An essential alteration is to replace te one-particle states by te Kon-Sam orbitals in te density functional teory King-Smit and Vanderbilt, 993. B. Rice-Mele model To illustrate te parametric carge pumping, we consider a dimerized D lattice Rice and Mele, 98, H = i= t0 + i δ c i c i+ +.c. i c i c i,. were t 0 is te opping amplitude between nearest neigbors, ±δ modulate te opping strengt, ± are te onsite potentials for staggered sublattices, and c + = c

3 3 PBC. Te famous Su-Srieffer-Heeger model Su et al., 979 for polyacetylene is a special case of te Rice-Mele model wit = 0 see Fig. 3. Since te lattice is dimerized, it is convenient to write c i=j as c j, c i=j as d j, and double te size of te unit cell to a. Te Hamiltonian becomes, H = + j= j= t 0 δ c j d j +.c. + c j c j t 0 + δ c j+ d j +.c. d j d j..3 Expand c j and d j, c j = e ij a c,.4 d j = e ija d,.5 were is te total number of unit cells, ten H = = c, d t 0 cos a t 0 cos a a iδ sin c a + iδ sin d.6 c, d H c d..7 δ > 0 δ < 0 FIG. 3 Two different locations of double bonds in polyacetylene. a t cos a δ sin a P0, 0 P δ0,0 δ b One can write H = σ, in wic = t 0 cos a a, δ sin,..8 Te band energies are ε ± = ± + t 0 cos a + δ sin a /..9 Te two energy bands touc at a, δ, = ±π, 0, 0. We now demonstrate te carge pumping under te periodic variation, δt, t = δ 0 cos t, 0 sin t..30 ote tat te Hamiltonian H as te same structure as te one for te spin-/ system in Eq.??. plays te role of te magnetic field, and te two sublattices play te roles of te spin up/down degrees of freedom. Terefore, te Berry pase due to te variation of parameters can be determined by te solid angle extended by a closed pat in te -space. First, consider te case wit = 0, ten = t 0 cos a a, δ sin, 0..3 FIG. 4 a Two semi-circle pats due to te increasing of a from 0 to π. Pat- is for δ > 0; pat- is for δ < 0. b A closed pat on te δ plane. If δ > 0, ten wen a increases from 0 to π, follows te pat- in Fig. 4a. On te oter and, if δ < 0, ten wen a increases from 0 to π, follows te pat- in Fig. 4a. Te two pats togeter extends a solid angle Ω = π. Terefore, te difference of polarizations, P δ 0, 0 P δ 0, 0 = qa γ π, γ = Ω = q a..3 Tis is a sensible result since a cain wit modulation δ can be obtained from sifting a cain wit modulation δ by a/ Xiao et al., 00. We now consider te cyclic variation, = t 0 cos a, δ 0 sin a cos t, 0 sin t..33 Wen 0, te size of te loop traversed in Fig. 4a srins. It srins to 0 wen t = ± 0 and P δ, ± 0 P δ, ± 0 = 0. Maximum difference is reaced wen is zero.

4 4 Start from t = 0, ten see Fig. 4b under appropriate gauge coice, one could expect to ave a b P δ 0, 0 = q a ext, at t = π/, Also, at t = π, P 0, 0 = c m>0 <m<0 + + d P δ 0, 0 = q a 4,.36 and so no. Tis is consistent wit te result in Eq..3. After a full cycle in Fig. 4b, te polarization canges by 0 or qa. In te latter case, a quantized carge as been transported. Recent confirmations of te carge pumping related to te Rice-Mele model are reported in Lose et al., 05, and aajima et al., 06. C. Qi-Wu-Zang model In tis section, we introduce a simple D model proposed by Qi, Wu and Zang to illustrate te connection between Berry pase and quantized Hall conductivity Qi et al., 006. Te fermions are living on a D square lattice, and eac lattice point is allowed to ave two degrees of freedom quasi-spin. Te quasi-spin typically refers to te valence band wit p-orbitals and te conduction band wit s-orbitals. Te QWZ Hamiltonian is given as, H = H 0 + H m + H y,.37 H 0 = ε 0 + cos x a cos t y a 0, 0 cos x a cos y a 0 H m = m, 0 0 sin H y = x a i sin y a. sin x a + i sin y a 0 H m is a mass term tat breas parity symmetry see p. of Hug s tesis, and H y is te ybridization between s- and p-bands. It is also possible to interpret te quasi-spins as true spins, ten H m is a magnetization term tat breas timereversal symmetry, and H y is a spin-orbit coupling. In sort, were H = ε 0 + σ,.38 = sin x a, sin y a, m + t cos j a. j= <m< m< 4 FIG. 5 Distribution of vectors in te. Only te signs of z are sown. Te eigen-energies are, ε ± = ε 0 ±..40 For simplification, we will set t, a, =. difficult to see tat tere are energy gaps at It is not 0 = 0 ε ± 0 = ε 0 ± m,.4 0 = π, 0, 0, π ε ± 0 = ε 0 ± m +,.4 0 = π, π ε ± 0 = ε 0 ± m Te energy gap closes at m = 0,, and 4. We will see tat te topology of te Bloc states canges at tese critical points, wen te energy gap closes. Te distribution of te vectors in te canges wen an energy gap closes, see Fig. 5. Depending on te value of m, tere are four different regimes: m > 0 : z > 0 over te wole. < m < 0 : z < 0 near = < m < : z > 0 near = π, π and its equivalent points. 4 m < 4 : H z < 0 over te wole. Te in Eq..38 is te magnetic field for te quasi-spin. In case, te magnetic field only sweeps over te nortern emispere wen scans over te. According to te analysis in??, one needs only to coose one gauge A over te wole. Terefore, te topology is expected to be trivial and te Hall conductivity σ H = 0. In cases and 3, z canges sign, and two gauges, A and A S are required to avoid te singularity. Terefore te topology is non-trivial and σ H 0. Case 4 is similar to Case, but sweeps over te soutern emispere only. Te topology is again trivial and σ H = 0. Te simple picture presented above can be verified by actual calculation of σ H. First, we sow tat omura,

5 5 d S dω d y d x T S FIG. 6 Mapping a small area d in te D to a small area d S on te surface of. Its solid angle dω is equal to te area of d S projected on a unit spere S. 03 F ± z = 3 x y..44 Pf: Te Berry connections in -space are, A ± l = i, ±, ± l.45 = α i, ±, ± l α.46 = α l a ± α,.47 were a ± are te Berry connections in -space. Terefore, te Berry curvatures in -space are, F z ± = A± y A± x.48 x y = β a ± β α a ± α.49 x y y x = α β a ± β a± α.50 x y α β = α x β y ε αβγ b ± γ.5 = 3 x y,.5 in wic b ± γ = γ / 3 are te Berry curvatures in - space. End of proof. Suppose te lower band is completely filled, and te upper band is empty, ten σ H = e π = e 4π In te integrand, ĥ d F z.53 d 3 x y..54 x d x y d y is actually te area d S on te -surface in Fig. 6. After being divided by, it becomes te solid angle dω extended by FIG. 7 Te line segment x [ π, π] in a would map to a closed loop in -space. As one sweeps te line segment from y = π to y = 0, te loop in -space sweeps out a toruslie structure, as sown in te figures. Furter advancement of y from 0 to π would map out anoter alf of te torus not sown for te sae of clarity. Te winding number w depends on weter te origin is enclosed in te torus-lie structure or not. In a and d for m < 4 and m > 0, te origin is outside of te surface, so w=0. In b and c for m = 3 and m =, te winding numbers are and + respectively not easily seen, toug. [Tese figures are adopted from Asbót et al., 03, p.37.] tat area. Since te is a closed surface a D torus, or T, under a continuous mapping, it would map to a closed surface in -space see Fig. 7. Te integral in Eq..54 gives te total solid angle extended by tat -surface. For a closed surface, it must be an integer multiple of 4π, tus σ H = w e, w Z..55 Te integer w, wic is equal to te first Cern number C, is te number of times te -surface wraps over a unit spere S. It caracterizes te topology of te mapping and te Bloc states and is called te winding number or te wrapping number. We empasize tat w is an integer only if te base space is a close surface in tis case, T, wic requires te valence band to be completely filled tat is, an insulator. Te quantized Hall conductance in te QWZ model is a result of te magnetization m, not an external magnetic field as in te case of te quantum Hall effect. It is called alternatively as te quantum anomalous Hall effect QAHE. Teir difference is tat, in te QHE, te electron orbitals are quantized due to te external magnetic field; in te QAHE tere is no orbital quantization,

6 6 y m<0 m>0 x m>0 m<0 m>0 Te x-directions extend to infinity on bot ends, and PBC is imposed along te y-direction. We now solve te differential equation, Hpψx, y = εψx, y..59 Use te metod of separation of variables and write a b ψx, y = φ xφ y..60 FIG. 8 a A topological pase occupies te left side of te space. b A finite sample wit two boundaries. Te edge states move in a definite direction along an edge. but only spin re-orientation due to m. A recent confirmation of te QAHE can be found in Cang et al., 03. One could apply an external magnetic field to te QAH insulator aa Cern insulator. Ten tere will be orbital quantization and Landau levels in energy spectrum. Interested readers can consult p. 03 of Bernevig and Huges, 03 for more details. D. Edge state in te Qi-Wu-Zang model Topological materials insulators ave an important property: teir surface states are stable against perturbations. Tey can be destroyed only if te energy gap of te bul bands closes so tat te topology of te electronic states is trivialized. In general, te interface between two materials wit different topologies would ave robust interface states. A euristic explanation is as follows: to go from one material to anoter, te spatial-dependent energy gap in te sense of te Tomas-Fermi approximation needs to close near te interface, oterwise it is simply impossible for te topology to cange. Tis gapless region is were te surface states reside. Tae te D QWZ model as an example. Divide te space to two parts were { > 0 for x > 0, mx < 0 for x < 0,.56 so tat tere is a D boundary along te y-axis see Fig. 8a. For simplicity, consider only te small limit, H = ε 0 + m x i y x + i y m + O..57 Te exact profile of mx does not matter, as long as it is monotonic and smoot compared to te electron wavelengt. To solve for te surface states, one needs to re-quantize te Hamiltonian using te substitution r i, suc tat mx Hp = ε 0 + i x + y i mx x y..58 Since te y-direction is invariant under translation, a trivial solution is φ y = e iyy, a plane wave. Terefore, te equation for φ x is, mx i x + y i x φ y mx x = ε e y φ x..6 We can tae a guess at a solution tat is localized near te boundary, φ x = e x 0 dx mx a..6 b It can be verified as an eigenstate wit eigenvalue ε e y = y if a, b =, i. Furtermore, it decays to zero at x 0 and x 0, and as a pea at x = 0. On te oter and, if { > 0 for x < 0, mx.63 < 0 for x > 0, ten φ x = e x 0 dx mx..64 i is a localized eigenstate wit ε e y = y. Terefore, in a sample wit finite widt see Fig. 8b, te electrons on te rigt edge move wit velocity ε e y = / ; te ones on te left move wit velocity /. Tey are called ciral edge states. Te two edges can be treated as independent only if te strip is wide enoug compared to te decay lengt of te edge state so tat te edge states on two sides do not couple wit eac oter. In te small y limit, te energy dispersion ε e y of te edge states are linear in y. Tis is not so for larger y s, were numerical calculation is required. Exercise. Te value of Za pase depends on te coice of te origin. Under a sift of te origin, Tis leads to Sow tat ψ x ψ x = ψ x d..65 u x u x = u x de id..66 γ γ = γ + π d a,.67

7 7 were a is te lattice constant.. Consider te Rice-Mele model wit = 0, i.e., te Su-Scrieffer-Heeger SSH model. a Instead of Eq..5, use te following expansion, c j = e ija c,.68 d j = e ija d..69 Find out te Hamiltonian in momentum-space, H = σ. b Find out te winding number of around te origin wen runs troug te D Brillouin zone. Consider te cases wit δ > 0 and δ < 0 respectively. Tey represent two pases wit different topologies. ote: Te winding number can be written as, w = π d ẑ d d..70 But you do not need to rely on tis expression, just draw te trajectory of to find out w. c Te SSH Hamiltonian can be written as H = 0 Q Q 0..7 Find out Q and sow tat te following expression also gives te winding number: w = i d d ln Q..7 π d 3. Assume a SSH cain as two domains: δx = δ 0 < 0 for x 0, δx = δ 0 for x 0, and δx varies smootly and monotonically from δ 0 to δ 0 in between. We will find out a localized state tat is trapped in tis domain wall. a Te low-energy states are located near a = ±π. Write as π/a + q, and expand H to linear order in q. Tis is te Hamiltonian in te continuum limit. b Replace q wit /i/x to requantized te H in a, and find out te eigenstate ψ 0 x wit zero energy. Tis problem is first studied in Jaciw and Rebbi, In Prob. 3 of Cap, we derived te effective Hamiltonian of an electron moving in a non-uniform magnetic field Br, t = B 0 ˆmr, t, [ ] m p A n V n + ε n ψ n = i ψ n t,.73 were V n and A n can be find tere. a Sow tat an electron wit spin up/down feels an effective electromagnetic field, Ẽ α = ˆm ˆm r α ˆm t,.74 B γ = 4 ɛ αβγ ˆm ˆm ˆm..75 r α r β As a result, an electron wit velocity v is subject to a force Ẽ + v B. b A magnetic syrmion is a topological spin texture in magnetic materials. Because of te excange interaction, te spin texture as an associated magnetic field tat can be identified wit te Br, t-field above. Sow tat a syrmion moving rigidly no cange of sape wit velocity v s generates an effective electric field, Ẽ = v s B,.76 were B is te effective magnetic field of a static syrmion. References Asbót, J. K., L. Oroszlány, and A. Pályi, 03, Topological insulators, unpublised. Atala, M., M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloc, 03, at. Pys. 9, 795. Bernevig, B. A., and T. L. Huges, 03, Topological Insulators and Topological Superconductors Princeton University Press. Cang, C.-Z., J. Zang, X. Feng, J. Sen, Z. Zang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, et al., 03, Science 34069, 67. Jaciw, R., and C. Rebbi, 976, Pys. Rev. D 3, King-Smit, R. D., and D. Vanderbilt, 993, Pys. Rev. B 47, 65. Lose, M., C. Scweizer, O. Zilberberg, M. Aidelsburger, and I. Bloc, 05, at. Pys., 350. aajima, S., T. Tomita, S. Taie, T. Icinose, H. Ozawa, L. Wang, M. Troyer, and Y. Taaasi, 06, at. Pys., 96. omura, K., 03, Fundamental teory of topological insulator, unpublised, written in Japanese. Qi, X.-L., Y.-S. Wu, and S.-C. Zang, 006, Pys. Rev. B 74, Resta, R., 99, Ferroelectrics 36, 5. Rice, M. J., and E. J. Mele, 98, Pys. Rev. Lett. 49, 455. Su, W. P., J. R. Scrieffer, and A. J. Heeger, 979, Pys. Rev. Lett. 4, 698. Touless, D., M. Komoto, M. igtingale, and M. Den ijs, 98, Pys. Rev. Lett. 496, 405. Xiao, D., M.-C. Cang, and Q. iu, 00, Rev. Mod. Pys. 8, 959. Za, J., 989, Pys. Rev. Lett. 6, 747.