1960-15 ICTP Conference Graphene Week 2008 25 - 29 August 2008 - PowerPoint PPT Presentation

1960-15 ICTP Conference Graphene Week 2008 25 - 29 August 2008 Bulk--edge correspondence in graphene with/without magnetic field Topological aspects of Dirac fermions in real materials Y. Hatsugai Institute of Physics, University of Tsukuba, Japan

σ xy = e 2 Graphene Week08 ICTP Trieste, Aug. 28, 2008 1 � 20 Tr dA h 2 πi 10 -3 -2 -1 1 2 3 -10 -20 -30 Bulk--edge correspondence in graphene with/without magnetic field Topological aspects of Dirac fermions in real materials Institute of Physics University of Tsukuba JAPAN Y. Hatsugai

Edge States in Condensed Matter Physics Bound states in quantum mechanics Levinson’s theorem, Friedel’s sum rule Surface states in Semiconductors Solitons in polyacetylene Su-Schriefer-Heeger ‘79 Edge states in quantum Hall effects Halperin ‘82 Hatsugai ‘93 Local moments in integer spin chains near the impurities Kennedy ‘90 Hagiwara-Katsumata-Affleck-Halperin-Renard ‘90 Zero bias conductance peaks of the d-wave superconductors Hu, ‘94 Zero energy localized states of graphene Fujita et al.‘96 Ryu-Hatsugai‘02 Quantum Spin Hall Edge states Kane-Mele‘05 Bernevig-Hughes-Zhang ‘06 more possibilities Edge states in 2D cold atoms in optical lattice Scarola-Das Sarma., PRL 98, 210403 ‘07 One-way edge modes in gyromagnetic photonic crystals Wang et al., PRL 100, 013905 ‘08

Why the Edge States are there?? Accidental ? NO ! Inevitable reasons Universal Structures behind: Bulk determines the edges : Bulk-Edge Correspondence Protected by Topological constraints and also additional Symmetry

Why the Edge States are there?? Bulk-Edge Correspondence Revisit the past and discuss graphene Quantum Hall effects Several New results They Can Be detected by STM experiments !

Quantum Hall Effects by edge states L R Edge states and Hall conductance σ xy Halperin ‘82 = E L Without boundaries Landau gauge in y � H 2 D = H 1 D ( k y ) R k y H 1 D ( k y ) E F : harmonic osc. centered at k y � x � ∼ � 2 B k y k y = 2 π ( n y + Φ ) n y = 0 , ± 1 , ± 2 , · · · L y Φ 0 With boundaries E 2 states are carried from L to R Φ = 0 Φ =Φ 0 Edge potential Gapless excitations E F Left edge Right edge Left edge Right edge n E F Laughlin’s undetermined : # of Landau Levels below Edge states are essential in the QHE !

Hall Conductance has a Topological meaning Discussion by the Bloch electrons ( Peierls substitution ) preserve U(1) gauge symmetry without cutoff ambiguity recover continuum theory by scaling limit ( weak field limit) φ = Ba 2 � c † i e iθ ij c j � H = x 2 πφ = θ ij � m,n+1 (m,n+1) (m+1,n+1) Φ 0 � ij � � ij �∈ P y � y � a m,n m+1,n x � m,n P : plaquette (m,n) (m+1,n) When E F is in the j-th gap Two topological quantities a = e 2 Sum of the First Chern numbers below E F � σ bulk C � Thouless-Kohmoto-Nightingale-den Nijs ‘82 xy h � : � � ( k ) <E F = e 2 Winding number of the edge state σ edge h I ( α j , C j ) xy in the complex energy surface Hatsugai ‘93a Bulk ---- Edge Correspondence Hatsugai ‘93b σ bulk = σ edge xy xy

Observation of Anomalous QHE in Graphene Anomalous QHE of gapless Dirac Fermions σ xy = e 2 h (2 n + 1) , n = 0 , ± 1 , ± 2 , · · · = 2 e 2 h ( n + 1 2) Zhang et al. Nature 2005 Novoselov et al. Nature 2005

Graphene under magnetic field Landau gauge � k y In continuum, 2D = (1D harmonic oscillators with parameter ) k y � Bloch electrons, 2D = (1D Harper problem with parameter ) k y k y Hofstadter diagram for the honeycomb Energy One particle Energy φ vs flux/hexagon (in flux quantum) Rammal 1985 φ

Topological meaning of the Hall Conductance σ xy TKNN formula: as a topological invariant Kubo formula Thouless-Kohmoto-Nightingale-den Nijs ‘82 σ xy = e 2 � C � h Sum over the bands below E F � : � � ( k ) <E F 1 � C � = F � � :First Chern number of the -th Band 2 πi T 2 :BZ intrinsically integer F � = dA � = � dψ � | dψ � � unless the energy gap collapses ∀ k, ǫ � ( k ) � = ǫ � ± 1 ( k ) A � = � ψ � | dψ � � regularity of the Berry connection H ( k ) | ψ � ( k ) � = ǫ ( k ) | ψ � ( k ) � k ∈ T 2 = { k = ( k x , k y ) | 0 ≤ k x , k y ≤ 2 π } BZ ∂ d = dk µ ∂k µ

Bulk Hall conductance of graphene Hall conductance by Chern number Counting vortices in the band j xy = e 2 1 � � σ j C � , C � = dA � , A � = � ψ � | dψ � � h 2 πi BZ � =1 Thouless-Kohmoto-Nightingale-den Nijs 1982 ǫ � ( k ) < µ F , � = 1 , · · · , j with randomness Aoki-Ando 1986 graphene Sum over the filled bands Need to sum many bands until E=0 Numerical difficulty for the weak field E=0 (experimental situation) { Need to fill negative energy Dirac sea E=0 { Need to sum over them

σ xy Bulk of the Filled Fermi sea & Dirac Sea Integration of the NonAbelian Berry Connection of the filled “Fermi Sea” & “Dirac Sea” Technical advantage for graphene H j ( k ) | ψ j ( k ) � = ǫ j ( k ) | ψ j ( k ) � Ψ =( | ψ 1 � , · · · , | ψ M � ) Collect M states below the Fermi level � ψ † � ψ †   1 | dψ 1 � · · · 1 | dψ M � . . ... A F S ≡ Ψ † d Ψ = . .   . .   � ψ † � ψ † M | dψ 1 � · · · M | dψ M � Matrix vector potential of the Fermi ( Dirac ) Sea Non Abelian extension for the Chern numbers σ xy = e 2 1 � T 2 Tr M dA F S h 2 πi Hatsugai 2004

Hall Conductace vs chemical potential Hall conductance of graphene over the whole spectrum v D(E) -2 -2 30 2 2 σ xy [ e 2 /h ] Electron Like Hole Like 20 in this region in this region 10 µ/t, µ/t, t t ≈ 1[eV] for graphene -3 -3 -2 -2 - - -1 1 1 1 2 2 3 3 YH, T. Fukui & H. Aoki, ‘06 Zheng-Ando 2002 -10 Gusynin-Sharapov, 2005 Peres-Guinea-Neto 2006 φ = 1 / 31 -20 � � � (1 + cos k x + cos k y ) 2 + (sin k x + sin k y ) 2 E ( k E ( k E ( k x , k y ) = ± k ) k ) ± ± (1 (1 +

Hall Conductace vs chemical potential Hall conductance of graphene over the whole spectrum g g p p D(E) ( ) ( ) -2 30 30 30 0 2 2 2 b b σ xy [ e 2 /h ] in n n n 20 20 20 2 xy 10 10 µ/t, t ≈ 1[eV] for graphene -3 -2 -1 - 1 1 1 1 2 3 YH, T. Fukui & H. Aoki, ‘06 Zheng-Ando 2002 -10 -10 Gusynin-Sharapov, 2005 Peres-Guinea-Neto 2006 φ = 1 / 31 -20 -20 � (1 + cos k x + cos k y ) 2 + (sin k x + sin k y ) 2 E ( k x , k y ) = ±

Hall Conductace vs chemical potential Hall conductance of graphene over the whole spectrum g g p p Quantum phase transition D(E) ( ) ( ) at the van Hove Energies Singularity breaks Topological Stability -2 30 30 30 0 2 2 2 b b σ xy [ e 2 /h ] in n n n 20 20 20 2 xy 10 10 µ/t, t ≈ 1[eV] for graphene -3 -2 -1 - 1 1 1 1 2 3 YH, T. Fukui & H. Aoki, ‘06 Zheng-Ando 2002 -10 -10 Gusynin-Sharapov, 2005 Peres-Guinea-Neto 2006 φ = 1 / 31 -20 -20 � (1 + cos k x + cos k y ) 2 + (sin k x + sin k y ) 2 E ( k x , k y ) = ±

3 types of Edge states in Graphene I. QH edge states in graphene YH, T. Fukui & H. Aoki, ‘06 See also, L. Brey, H. A. Fertig, ‘06 Edge transport : D. A. Abanin, P. A. Lee, L. S. Levitov ‘07 II. Zero modes edge state without magnetic field Fujita et al., ‘96 S. Ryu & YH, ‘02 III. Zero modes with magnetic field M. Arikawa, H. Aoki & YH arXiv:0805.3240 & 0806.2429

� Laughlin’s Argument & Edge States L y B Adiabatic Charge Transfer I y I y V x Edge States determine the Hall Conductance

� � � � Analytic Continuation of the Bloch State to the complex energy (Riemann surface) B C j = I j − I j − 1 E Chern # = winding # Difference between the neighboring gaps Unifi d Unifi fied n C n C plex ex ergy rgy su su face Bulk-Edge Correspondence 0 j � of the topological numbers j =1 , ··· � bulk = σ xy edge σ xy Complex Energy surface YH, T. Fukui & H. Aoki, ‘06 of Harper eq. R + genus g=q -- 1: number of the gaps φ = p/q � � � � - R

Another type of Edge states in Graphene Quantum Hall edge states Topologically protected edge states Symmetry protected edge states: zero modes ( topological origin & stability ) without magnetic field Fujita et al. ‘96 : discovery S. Ryu & YH ‘02 : topological reason with magnetic field M. Arikawa, H. Aoki & YH arXiv:0805.3240 & 0806.2429 New feature : topological compensation It can be observed by STM experiments under magnetic field

Without magnetic field

Graphene on a Cylinder Localized Boundary State in Carbon Sheet (1) now called as Graphene Tight-binding Model Calculation “ Peculiar Localized State at Zigzag Graphite Edge “ M. Fujita, K. Wakabayashi, K. Nakada and K. Kusakabe, JPSJ 65, 1920 (1996) cnt-fujita – p.

Localized Boundary State in Carbon Sheet (2) Local Spin Density Functional Appr. Calculation “Magnetic Ordering in Hexagonally Bonded Sheets with First-Row Elements”, Okada, Oshiyama, Phys. Rev. Lett. 87, 146803 (2001) cnt-oshiyama – p.