Topological Aspects of Graphene Dirac Fermions and Bulk-Edge - PowerPoint PPT Presentation

20 COQUSY06, Sep28, 2006 ト 10 -3 -2 -1 1 2 3 -10 -20 -30 Topological Aspects of Graphene Dirac Fermions and Bulk-Edge Correspondence in a Magnetic Field Department of Applied Physics University of Tokyo Y. Hatsugai with T. Fukui ( Ibaragi U.) H. Aoki (U. Tokyo) Ref. Y. Hatsugai, T. Fukui and H. Aoki, to appear in Phys. Rev. B, cond-mat/0607669

Today’s Talk Graphene as a basic platform of Dirac Fermions Massless Dirac Fermions in Condensed Matter Physics Anomalous Quantum Hall Effect (QHE) in Graphene Topological Aspects of Graphene (Bulk) Topological Stability of the Dirac Fermions Topological Stability of the Anomalous QHE Adiabatic Principle and Topological Equivalence Quantum phase Transition by chemical potential shift Technical development for calculating Chern numbers (Lattice Gauge Theory) Topological Aspects of Graphene (Edge) Without Magnetic field Topological Origin of Zero Modes in Graphene (c.f. d-wave superconductors) With Magnetic field Edge States of Dirac Fermions Bulk --- Edge Correspondence (Analytical & Numerical) Edge States and Bloch States ( complex energy structure )

Massless Dirac Fermions in Condensed Matter Gapless Superconductor with point Nodes d-wave superconductivity Graphene as a 2D Carbon sheet Wallace (1946) Fig.Zhang et al. (2005)

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

Conventional QHE Landau Level and Integer QHE � k E ( B = 0) = � 2 2 mk 2 E µ F ǫ n = � ω ( n + 1 D ( E ) � 2) D ( E ) = δ ( E − ǫ n ) n µ F E [ � ω ] 1/2 3/2 5/2 σ xy [ e 2 /h ] σ xy ( µ F ) = e 2 n = 1 , 2 , 3 , · · · � n, 1 2 3 ǫ n − 1 < µ F < ǫ n µ F E [ � ω ] 1/2 3/2 5/2

QHE and Band Structures QHE of electrons and holes 2D organic metal (TMTSF) 2 PF 6 (Chaikin et al) SD W 2D FISDW T SC Lab. de Physique des Solides metal B p

QHE of Semiconductors Landau Level of Conduction band (Electrons) � k E ( B = 0) = ± � 2 2 mk 2 E g E ǫ n : zero point energy 1 / 2 µ F δ ( E − ǫ n ) E g / 2 + � ω ( n + 1  2 )   n ≥ 0  ǫ n = − E g / 2 + � ω ( n + 1 2 ) � n   n < 0  D ( E ) = µ F (Eg+1)/2 (Eg+3)/2 (Eg+5)/2 σ xy ( µ F ) = e 2 � -+1 +2 ( n > 0) µ F n σ xy [ e 2 /h ] � × n + 1 ( n ≤ 0) --3 --2 --1 ǫ n − 1 < µ F < ǫ n σ xy : e 2 E [ � ω ] h × integer

QHE of Graphene (Gapless Semiconductor) Landau Level of Doubled Dirac Fermions � k E ( B = 0) = ± � c | � k | E µ F ǫ n : No zero point energy shift δ ( E − ǫ n ) √ ǫ n = ± C Bn � � C = c (2) e � n McClure, 1956 D ( E ) = σ xy ( µ F ) = e 2 � (2 n + 1) 1 √ √ √ 3 2 2 5 1 3 5 7 9 σ xy [ e 2 /h ] n = 0 , 1 , 2 , 3 , · · · ǫ n − 1 < µ F < ǫ n Zheng-Ando 2002 σ xy : e 2 Gusynin-Sharapov, 2005 h × odd integer Peres-Guinea-Neto 2006 ...

Many Relevant Papers 89 Matches on Graphene in cond-mat in the past year Experiments and theories

Motivations Here How does the Anomalous QHE persist for higher Energy? Quantum Phase Transition? Is it specific to the honeycomb lattice? Edge State? Laughlin 81 Halperin 82 How do the edge states look like? Edge States & Topological Numbers Hatsugai 1993 How about the bulk-edge correspondence?

Hofstadter diagram for the honeycomb Tight-binding model on a honeycomb lattice � t ij e i θ ij H = � ij � � 2 πφ P = θ ij � ij �∈ P t t φ P = φ = p q t ( p, q ) = 1 Rammal 1985 E=0 Landau level : φ ∝ B outside Onsager’s semiclassical quantisation scheme

Bulk by the topological invariant σ xy Hall conductance by Chern number Counting vortices in the band j xy = e 2 � 1 � σ j A ℓ = � ψ ℓ | d ψ ℓ � C ℓ , C ℓ = dA ℓ , h 2 π i BZ ℓ =1 Thouless-Kohmoto-Nightingale-den Nijs 1982 ǫ ℓ ( k ) < µ F , ℓ = 1 , · · · , j with randomness Aoki-Ando 1986 Integration of the NonAbelian Berry Connection of the “Fermi Sea” σ xy = e 2 1 � Tr j dA FS Fermi Sea of j filled bands 2 π i h BZ A FS = Ψ † d Ψ , Ψ = ( ψ 1 , · · · , ψ j ) Hatsugai 2004 Tolological Invariant on Discretized Lattice Lattice in k space σ xy = e 2 1 Technical Advantage for large Chern Numbers � F 1234 2 π i h F 1234 = Im log U 12 U 23 U 34 U 41 Fukui-Hatsugai-Suzuki 2005 j Ψ † U mn = det Ψ n = ( ψ 1 ( k n ) , · · · , ψ j ( k n )) m Ψ n ,

Hall Conductace vs chemical potential Accurate Hall conductance over the whole spectrum D(E) -2 30 2 σ xy [ e 2 /h ] Electron Like Hole Like 20 in this region in this region 10 µ/t, t ≈ 1[eV] for graphene -3 -2 -1 1 2 3 -10 φ = 1 / 31 -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 Accurate Hall conductance over the whole spectrum D(E) -2 30 Dirac behavior 2 σ xy [ e 2 /h ] in this region 20 10 µ/t, t ≈ 1[eV] for graphene -3 -2 -1 1 2 3 -10 φ = 1 / 31 -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 Accurate Hall conductance over the whole spectrum Quantum phase transition D(E) at the van Hove Energies Singularity breaks Topological Stability -2 30 Dirac behavior 2 σ xy [ e 2 /h ] in this region 20 10 µ/t, t ≈ 1[eV] for graphene -3 -2 -1 1 2 3 -10 φ = 1 / 31 -20 � (1 + cos k x + cos k y ) 2 + (sin k x + sin k y ) 2 E ( k x , k y ) = ±

(Sheng et al, cond-mat/0602190) E

Is this anomalous behavior specific to the honeycomb lattice ? No! No! : It has topological Stability Vanishing DOS of the Dirac Fermions’ Anomalous behavior of the Hall conductance 30 20 10 Chiral Symmetry t’/t = 1 : Square Lattice -3 -2 -1 1 2 3 -10 (Bipartite Structure) t’/t =0 :Honeycomb Lattice -20 t’/t=-1 : Flux State -30 π To demonstrate t ’ introduce t t t t 2nd nearest neighbor t ’ hopping t ’ t t

Dirac Cones are Stable! The Dirac Cornes are not accidental It has topological stability − 3 < t ′ Doubled Dirac Cones t < 1 t’/t = 1 : Square Lattice t’/t =0 :Honeycomb Lattice t’/t=-1 : Flux State π

Density of States Vanishing DOS near the zero energy (a) (c) honeycomb: t � /t=0 � flux: t � /t=-1 D(E) 0 0 -4 -2 2 4 -4 -2 2 4 E t’/t = 1 : Square Lattice t’/t =0 :Honeycomb Lattice t’/t=-1 : Flux State π Stability of the Dirac Cornes!

Dirac Cornes Adiabatic Equivalence t’/t = 1 : Square Lattice t’/t =0 :Honeycomb Lattice t’/t=-1 : Flux State π

Topological Stability of the Dirac Cornes � � 0 ∆ H ( k x , k y ) = ∆ ∗ 0 ∆ = − t (1 + e ik y + e ik x (1 + re − ik y ) , r = t ′ /t E ( k x , k y ) = ±| ∆ | � (1 + cos k x + cos k y ) 2 + (sin k x + sin k y ) 2 = ± General zeros of Dirac Cones ∆ ( k x , k y ) ∆ ( k x , k y ) , k x : 0 → 2 π : loop C ( k y ) in C loop C ( k y ) moves : k y : 0 → 2 π

Topological Stability of the Dirac Cornes � � 0 ∆ H ( k x , k y ) = ∆ ∗ 0 ∆ = − t (1 + e ik y + e ik x (1 + re − ik y ) , r = t ′ /t E ( k x , k y ) = ±| ∆ | � (1 + cos k x + cos k y ) 2 + (sin k x + sin k y ) 2 = ± General zeros of Dirac Cones ∆ ( k x , k y ) ∆ ( k x , k y ) , k x : 0 → 2 π : loop C ( k y ) in C loop C ( k y ) moves : k y : 0 → 2 π The loop cut the origin Dirac Cones Topological Stability of the doubled Dirac Cones

Hofstadter Diagrams Adiabatic Equivalence & Duality Hall Conductance v.s. chemical potential t’/t = 1 : Square Lattice t’/t =0 :Honeycomb Lattice t’/t=-1 : Flux State π

van Hove singularity & Hall Conductance

by Adiabatic Principle σ xy Hofstadter Diagram and with t’/t σ xy σ xy t’/t = 1 : Square Lattice t’/t =0 :Honeycomb Lattice t’/t=-1 : Flux State π

Adiabatic Connections Near zero Field Honeycomb Lattice flux π Main Gaps Preserve Near E=0 Honeycomb System is Topologically Equivalent to flux System Near E=0 π Honeycomb Lattice Square Lattice Main Gaps Preserve Near Band Edges Honeycomb System is Topologically Equivalent to Square System Near the Band Edges

Honeycomb From Diophantine Equations σ xy As for the flux system and square system, π is determined by a Diophantine equation σ xy By the Adiabatic Equivalence, of the honeycomb σ xy is determined algebraically. Master Eq. Φ = P Q, J ≡ Pc J , (mod Q ) , | c J | < Q/ 2 TKNN1982 By the Adiabatic Equivalence Q = 1 2 = q + 1 φ = 1 Φ = P 2 + φ 2 q , q Dirac Fermion Type Quantization: P = q + 1 , Q = 2 q Algebraically J = q − 1 + 2( N + 1) = q + 2 N + 1 on Honeycomb Lattice c J = 2 N + 1 , N = 0 , 1 , 2 , · · ·

Edge states of Graphene Without magnetic field YH with S. Ryu (now KITP) With magnetic field Recent works

Graphene on a Cylinder Zigzag Edges (on Cylinder ) (b) � H total = H ( k y ) Fourier Transform in y direction k y Total System as a sum 1D unit cell of 1D system parametrized by ( j +1, j ) k y 1 2 ( j , j ) e 2 1 2 e 1 ( j , j -1) 2 1 1D system with parameter k y