Introduction to topological insulators. Jennifer Cano

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1 Introduction to topological insulators Jennifer Cano Adapted from Charlie Kane s Windsor Lectures: Review article: Hasan & Kane Rev. Mod. Phys. 2010

2 What is an insulator? No low-energy excitations Electrical resistor

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4 <latexit sha1_base64="/ecato+vgjp179didgcwt14amtu=">aaaca3icbvdjsgnbeo2jw4zbqde9nabbu5wjirkiqs/ejgawsglo6vssjj0l3tvigaje/buvhhtx6k9482/sladnffdwek+kqnpejivgx/m2uguls8sr6dxm2vrg5pa9vvpryaw4lhkoq1xzmaypaiijqam1sahzpqlvr3858qv3olqig1scrnd0wtcqhcezgqll7/wo4s5pz2n+nfdoidxg0h82rn3ospaddxloghseufosjvouwvzxox3y2icauwra110nwmbcfaouyzhpxboixvusc3vda+adbibjh4b00cht2gmvqqdpwp09ktbf64hvmu6fyu/peipxp68ey6fqteqqxqgbnyzqxjjiseeb0lzqwfeodgfccxmr5t2mgectw8ae4m6+pe8q+zzr5nybk2zxyhphmuyta3jexhjgiuskleizcpjinskreboerbfr3fqytkas6cwu+qpr8we34jzo</latexit> <latexit sha1_base64="/ecato+vgjp179didgcwt14amtu=">aaaca3icbvdjsgnbeo2jw4zbqde9nabbu5wjirkiqs/ejgawsglo6vssjj0l3tvigaje/buvhhtx6k9482/sladnffdwek+kqnpejivgx/m2uguls8sr6dxm2vrg5pa9vvpryaw4lhkoq1xzmaypaiijqam1sahzpqlvr3858qv3olqig1scrnd0wtcqhcezgqll7/wo4s5pz2n+nfdoidxg0h82rn3ospaddxloghseufosjvouwvzxox3y2icauwra110nwmbcfaouyzhpxboixvusc3vda+adbibjh4b00cht2gmvqqdpwp09ktbf64hvmu6fyu/peipxp68ey6fqteqqxqgbnyzqxjjiseeb0lzqwfeodgfccxmr5t2mgectw8ae4m6+pe8q+zzr5nybk2zxyhphmuyta3jexhjgiuskleizcpjinskreboerbfr3fqytkas6cwu+qpr8we34jzo</latexit> <latexit sha1_base64="/ecato+vgjp179didgcwt14amtu=">aaaca3icbvdjsgnbeo2jw4zbqde9nabbu5wjirkiqs/ejgawsglo6vssjj0l3tvigaje/buvhhtx6k9482/sladnffdwek+kqnpejivgx/m2uguls8sr6dxm2vrg5pa9vvpryaw4lhkoq1xzmaypaiijqam1sahzpqlvr3858qv3olqig1scrnd0wtcqhcezgqll7/wo4s5pz2n+nfdoidxg0h82rn3ospaddxloghseufosjvouwvzxox3y2icauwra110nwmbcfaouyzhpxboixvusc3vda+adbibjh4b00cht2gmvqqdpwp09ktbf64hvmu6fyu/peipxp68ey6fqteqqxqgbnyzqxjjiseeb0lzqwfeodgfccxmr5t2mgectw8ae4m6+pe8q+zzr5nybk2zxyhphmuyta3jexhjgiuskleizcpjinskreboerbfr3fqytkas6cwu+qpr8we34jzo</latexit> <latexit sha1_base64="/ecato+vgjp179didgcwt14amtu=">aaaca3icbvdjsgnbeo2jw4zbqde9nabbu5wjirkiqs/ejgawsglo6vssjj0l3tvigaje/buvhhtx6k9482/sladnffdwek+kqnpejivgx/m2uguls8sr6dxm2vrg5pa9vvpryaw4lhkoq1xzmaypaiijqam1sahzpqlvr3858qv3olqig1scrnd0wtcqhcezgqll7/wo4s5pz2n+nfdoidxg0h82rn3ospaddxloghseufosjvouwvzxox3y2icauwra110nwmbcfaouyzhpxboixvusc3vda+adbibjh4b00cht2gmvqqdpwp09ktbf64hvmu6fyu/peipxp68ey6fqteqqxqgbnyzqxjjiseeb0lzqwfeodgfccxmr5t2mgectw8ae4m6+pe8q+zzr5nybk2zxyhphmuyta3jexhjgiuskleizcpjinskreboerbfr3fqytkas6cwu+qpr8we34jzo</latexit> Quantum Hall devices are not insulating!! h/e 2 = 25.8k Nobel Prize 1985 Klaus von Klitzing Data: Weis and von Klitzing, Phil. Trans. R. Soc. A 369, 3954 (2011)

5 Topological invariants do not change under smooth deformations Example 1: genus Example 2: Hall conductivity <latexit sha1_base64="vz9slf2mzitk4wivg6zdrbc8ot0=">aaab/xicbvdlssnafj3uv62v+ni5gsycq5ouqtdc0y0rqwaf0mywmu7aototmdmryyj+ihsxirj1p9z5n07bllt1wixdofdy7z1bzkjsjvntfrywl5zxiqultfwnzs17e6epokri0sari2q7qiowkkhdu81io5ye8ycrvjc8hputeyivjcsttmpicdqxnkqyasp59l5x0t5hfvaqjua5vcz31eobb5edijmbncdutsogr923v7q9ccecci0zuqrjorh2miq1xyymst1ekrjhieqtjqeccak8bhl9cb4apqfdsjosgk7u3xmz4kqlpdcdhombmvxg4n9ej9hhmzdreseacdxdfcym6gioo4a9kgnwldueyunnrrapkerym8bkjgr39uv50qxwxkfi3pyuaxd5hewwdw7aexdbkaibk1ahdydbi3ggr+dnerjerhfry9paspkzxfah1ucpxwgufw==</latexit> xy 2 = N e /h

6 Surface spectrum explains Hall conductivity E Bulk-edge correspondence EF x Landau levels bend up at edges Image: Hasan & Kane RMP Hall conductivity from bulk eigenstates: TKNN invariant (Thouless, Kohmoto, Nightingale, den Nijs PRL 1982)

Energy eigenstates in a crystal are Bloch wavefunctions Translation symmetry: Bloch s theorem: T (R) i = e ik R i i = e ik r u(k)i Brillouin zone contains distinct k ky π Eigenvalues of

7 Energy eigenstates in a crystal are Bloch wavefunctions Translation symmetry: Bloch s theorem: T (R) i = e ik R i i = e ik r u(k)i Brillouin zone contains distinct k ky π Eigenvalues of Hamiltonian form band structure H(k) u n (k)i = E n (k) u n (k)i -π -π π kx = ΔE Γ π kx Two band structures are topologically equivalent if they can be deformed into each other without closing energy gap

8 Berry phase Berry connection: A = ihu(k) r k u(k)i I Berry phase: C = C A dk Wave function phase ambiguity Berry phase defined mod 2π: u(k)i!e i (k) u(k)i A! A + r k (k) I C! C + C r k (k) dk 2 n Berry curvature: F = r k A gauge invariant!

9 Zak phase: Berry phase around Brillouin zone k=π/2 Zak PRL 62, 2747 (1989) Z 2 k=0 k=2π k=π = 0 k u(k)idk P = e I 2 i@ k! x k=3π/2 A(k)dk Polarization of infinite crystal defined mod e: P=e P=0 Differences in polarization are well-defined: Z P t P (t)dt = e 2 ZZ dtdk r A Modern theory of polarization King-Smith & Vanderbilt PRB (1993) Resta Ferroelectrics 136, 51 (1992)

10 Berry phase in 1d: SSH model (Su, Schrieffer, Heeger PRL 1979) H = X i (t + )c A,i c B,i +(t )c A,i+1 c B,i +h.c. A, i A, i+1 E(k) > 0 B, i < 0 A, i B, i A, i+1 4 -π π k H(k) =d(k) dy dy d x (k) =(t + )+(t d y (k) =(t )sin(k) ) cos(k) dx dx d z (k) =0 Symmetry enforces dz(k)=0 < 0 Berry phase π > 0 Berry phase 0 Interpret difference in Berry phase as difference in polarization!!

11 Domain wall states (Su, Schrieffer, Heeger PRL 1979) > 0 < 0 Low-energy continuum theory: H =2 x tk y (k, t) Domain wall:! (x) k! i@ x δ(x) x Zero-energy state localized at domain wall: (Jackiw & Rebbi PRD 1976) (x) =e R x dx 0 2 (x 0 ) t 1 0 ψ(x) 2 x

12 t=0 Thouless charge pump (Thouless PRB 1983) t=t/2 t=t P = e/2 P = e/2 Integer charge pumped in one full cycle is a topological invariant: Z T Z T Z! 2 P = t P (t)dt = 1 2 dt dkr A 0 0 Chern number e = ne Chern number is an integer topological invariant characterizing Bloch wavefunctions of two variables k t

13 Chern number = winding Berry phase 2 e P = Z T t (t)dt = (T ) (0) = 2 n 2 n =1 (t) 0 0 t T t=0 t=t/2 t=t Practical way to compute Chern number

14 Integer quantum Hall: Laughlin flux argument 2. Faraday s law: generate electric field E t 2 R I = 3. Hall conductance determines current Z T 4. Charge pumping: Q = t 0 I(t)dt = 1. Thread flux quantum xy h e Laughlin PRB 1981 (t = 0) = 0 (t = T )=h/e 5. Energy gap ) Q = ne Thouless pump! xy = n e2 h Comparison to TKNN invariant: n = 1 2 Z d 2 kr A kx ky Chern number with t ky

15 Chern number surface state Bulk diagnosis Topological invariant: TKNN or Chern number n = 1 2 Z d 2 kr A kx t or ky Bulk-edge correspondence If x-axis is time: 1d Thouless pump If x-axis is k: Laughlin argument Surface diagnosis

16 Bulk-edge correspondence Surface states are required but dispersion is determined by microscopics Conduction band Conduction band Conduction band EF EF EF Valence band Valence band Valence band n+ = # bands with positive slope that cross EF n- = # bands with negative slope that cross EF Chern number = n+ n-

17 Time-reversal symmetry 1 H(k) = H( k) 2 = I Degenerate Kramer s pairs at Time-reversal-invariant-momenta (TRIM) k=0, k=π Time-reversal forbids Chern number! conduction bands Surface state: Kramers pair conduction bands Two surface states, protected by time-reversal Conventional insulator valence bands valence bands valence bands valence bands Topological insulator Z2 topological invariant ν: even vs odd bands cross EF over half the BZ Fermion parity pump

18 Bulk Z2 invariant Kane & Mele PRL 95, (2005) BZ ky (0,π) (π,π) (0,0) (π,0) kx ( 1) = Y a 2TRIM Pf[w( a )] p Det[w( a )] = ±1 w mn (k) =hu k,m u k,n i Gauge invariant but must choose continuous gauge In practice, difficult to compute

19 Practical calculation of Z2 invariant: winding Berry phase π π -π -π ky 0 -π -π ky 0 Yu, Qi, Bernevig, Fang, Dai PRB 84, (2011) Soluyanov and Vanderbilt Phys. Rev. B 83, (2011) Berry phase imitates surface spectrum (proof: Fidkowski, Jackson, Klich PRL 2011) Z2Pack software package: Gresch, et al, PRB 95, (2017)

20 Easier computation of Z2 invariant with symmetry Spin conservation: quantum spin Hall effect Inversion symmetry Conventional insulator Z2 invariant given by product of inversion eigenvalues of occupied bands: Quantum spin Hall ( 1) = Y a 2TRIM Y i 2i ( a ) Fu & Kane PRB 76, (2007) Each spin has Chern number: n = -n ν = n mod 2 Kane & Mele PRL 95, (2005), Bernevig & Zhang PRL 96, (2006)

Staggered flux Haldane PRL 1988 Haldane z z 3.

21 Quantum spin Hall effect in graphene with spin orbit coupling z = sublattice z = valley s z =spin Fig: Castro Neto, et al, RMP (2009) Three ways to open energy gap: 2 1. Sublattice potential (hbn) hbn z 2. Staggered flux Haldane PRL 1988 Haldane z z 3. Spin-orbit coupling Kane and Mele PRL 1995 SOC z z s z Breaks inversion symmetry Breaks time-reversal Quantum Hall effect Preserves all symmetries Quantum spin Hall effect

22 Quantum spin Hall in HgTe quantum wells Theory: Bernevig, Hughes, Zhang Science 2006 BHZ model CdTe HgTe CdTe d d<dc normal band order d>dc inverted (topological) band order dc = 6.4nm CdTe HgTe Band inversion at k=0

Science 2007 G=0, d<dc Conventional insulator G=2e2/h, d>dc

23 Experimental realization HgTe quantum wells Experiment: Konig, Wiedmann, Brune, Roth, Buhmann, Molenkamp, Qi, Zhang Science 2007 G=0, d<dc Conventional insulator G=2e2/h, d>dc Topological insulator Quantized conductance independent of sample width

24 Topological insulators in 3d Strong index ky Four Z2 indices: ( 0 ; 1, 2, 3 ) } Weak indices ky Refs: Fu, Kane, Mele PRL 2007 Moore, Balents PRB 2007 Roy PRB 2009 Ex: (1;000) kx Surface Brillouin zone kx Ex: (0;100) Bulk construction (not unique) k=0 plane is 2D TI; k=π is conventional Stacked 2D TIs

25 Topological insulators in 3d Bi1-xSbx Band gap ~.03eV Theory: Fu & Kane PRL 2007 Expt: Hsieh, et al, Nature 2008 Bi2Se3 Band gap ~.3eV Theory: Zhang, et al, Nat. Phys Expt: Xia et al, Nat. Phys Figures from Hasan & Kane RMP

26 What comes after time-reversal symmetry? 10-fold way classification Ryu, Schnyder, Furusaki, Ludwig, New J. Phys. (2010) Classified by time-reversal and charge-conjugation symmetries no symmetry Integer quantum Hall fermions w/ time-reversal 2d and 3d topological insulators (weak index not captured)

Symmetry, Topology and Phases of Matter

Symmetry, Topology and Phases of Matter E E k=λ a k=λ b k=λ a k=λ b Topological Phases of Matter Many examples of topological band phenomena States adiabatically connected to independent electrons: - Quantum