Topological Aspects of the n=0 Landau Level in graphene Chiral - PowerPoint PPT Presentation

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009 ト Topological Aspects of the n=0 Landau Level in graphene Chiral Symmetry & Hall Plateau Transition Y. Hatsugai (Univ. Tsukuba) T. Kawarabayashi (Toho Univ.) H. Aoki (Univ. Tokyo) T.Kawarabayashi, Y. Hatsugai and H. Aoki, preprint

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009 ト Today’s Talk Quantum Hall effect of graphene: stability and topology Need to fill many Landau Levels with van Hove singularity Numerical technique : Non Abelian formulation & lattice gauge Chiral Symmetry of graphene : speciality of the n=0 L.L. Sublattice symmetry as chiral symmetry Physical outcomes of chiral symmetry Quantum Hall plateau transition of graphene Spatially correlated randomness for hopping as ripples Role of the correlation Quantum Hall critical states

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

QHE of Graphene (Gapless Semiconductor) Landau Level of Doubled Dirac Fermions � k E ( B = 0) = ± � c | � k | E µ F δ ( E − ǫ n ) ǫ n : No zero point energy shift √ 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 ...

Hall conductance as a topological invariant By Chern numbers of Bloch electrons 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 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 { van Hove singularity Rammal 1985 Need to sum over them

Bulk of the Filled Fermi sea & Dirac Sea σ xy Integration of the NonAbelian Berry Connection of the “Fermi Sea” & “Dirac Sea” Numerical 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 | d ψ 1 � · · · 1 | d ψ M � A F S ≡ Ψ † d Ψ = . . ...   . . . .   � ψ † � ψ † M | d ψ 1 � · · · M | d ψ 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 π i Hatsugai 2004

Bulk of the Filled Fermi sea & Dirac Sea σ xy Integration of the NonAbelian Berry Connection of the “Fermi Sea” & “Dirac Sea” Numerical 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 Technology 1 � ψ † � ψ †   1 | d ψ 1 � · · · 1 | d ψ M � A F S ≡ Ψ † d Ψ = . . ...   . . . .   � ψ † � ψ † M | d ψ 1 � · · · M | d ψ 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 π i Hatsugai 2004

Numerical Technique from the Lattice gauge theory Topological Invariant on Discretized Lattice N B Lattice in k space ( discretization for the integral ) Technical Advantage for large Chern Numbers k � Brillouin Zone σ xy = e 2 1 � gauge invariant F 1234 � 2 2 π i h � 1 F 1234 = Im log U 12 U 23 U 34 U 41 j Ψ † U mn = det Ψ n = ( ψ 1 ( k n ) , · · · , ψ j ( k n )) m Ψ n , Fermi Sea of j filled bands U µ ( k � ) U µ ( k � ) ≡ � n ( k � ) | n ( k � + ˆ µ ) � / N µ ( k � ) N µ ( k � ) = |� n ( k � ) | n ( k � + ˆ µ ) �| ˜ F 12 ( k � ) ˜ F 12 ( k � ) ≡ ln U 1 ( k � ) U 2 ( k � + ˆ 1) U 1 ( k � + ˆ 2) − 1 U 2 ( k � ) − 1 − π < ˜ (principal value) F 12 ( k � ) /i ≤ π Fukui-Hatsugai-Suzuki 2005

Numerical Technique from the Lattice gauge theory Topological Invariant on Discretized Lattice N B Lattice in k space ( discretization for the integral ) Technical Advantage for large Chern Numbers k � Brillouin Zone σ xy = e 2 1 � gauge invariant F 1234 � 2 2 π i h Technology 2 � 1 F 1234 = Im log U 12 U 23 U 34 U 41 Chern number extension of the KSV formula for polarization j Ψ † U mn = det Ψ n = ( ψ 1 ( k n ) , · · · , ψ j ( k n )) m Ψ n , Fermi Sea of j filled bands U µ ( k � ) U µ ( k � ) ≡ � n ( k � ) | n ( k � + ˆ µ ) � / N µ ( k � ) N µ ( k � ) = |� n ( k � ) | n ( k � + ˆ µ ) �| ˜ F 12 ( k � ) ˜ F 12 ( k � ) ≡ ln U 1 ( k � ) U 2 ( k � + ˆ 1) U 1 ( k � + ˆ 2) − 1 U 2 ( k � ) − 1 − π < ˜ (principal value) F 12 ( k � ) /i ≤ π Fukui-Hatsugai-Suzuki 2005

Hall Conductace vs chemical potential Accurate Hall conductance over whole spectrum D(E) single band model -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 Dirac Like Hatsugai-Fukui-Aoki ’06 φ = 1 / 31 -20 in this region

Chern numbers ( ) based on Realistic Band Calc. σ xy

Chern numbers ( ) based on Realistic Band Calc. σ xy σ xy M.Arai and Y.Hatsugai, Phys.Rev. B79, 075429 (2009) quantized everywhere

Chern numbers ( ) based on Realistic Band Calc. σ xy σ xy M.Arai and Y.Hatsugai, Phys.Rev. B79, 075429 (2009) quantized everywhere

Chern numbers ( ) based on Realistic Band Calc. σ xy Fermi surface σ xy E F σ xy M.Arai and Y.Hatsugai, Phys.Rev. B79, 075429 (2009) quantized everywhere

Chern numbers ( ) based on Realistic Band Calc. σ xy Fermi surface σ xy E F σ xy M.Arai and Y.Hatsugai, Phys.Rev. B79, 075429 (2009) quantized everywhere

Chern numbers ( ) based on Realistic Band Calc. σ xy Fermi surface σ xy E F Fermi surface σ xy σ xy E F M.Arai and Y.Hatsugai, Phys.Rev. B79, 075429 (2009) quantized everywhere

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009 ト Chiral Symmetry of Graphene Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping only between

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009 ト Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping only between

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between Protect “Massless” Dirac cones

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between Protect “Massless” Dirac cones Energy dispersion Density of States Perturbation preserving this chiral symmetry t ’ t t t ’ Example : t’ terms t ’ t

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between Protect n=0 Landau level at “E=0”

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between Protect n=0 Landau level at “E=0” with chiral symmetry

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between Protect n=0 Landau level at “E=0” without chiral symmetry with chiral symmetry & " ' ! ( Splitting of n=0 L.L. % ' by Chiral symmetry breaking % " H. Aoki, T.Fukui, Y. Hatsugai, % & Int. J. Mod. Phys. 21 1133 (2007) # $ ! " !

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between Protect zero mode edge states at Zigzag boundaries (Fujita)

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between Protect zero mode edge states at Zigzag boundaries (Fujita) with chiral symmetry

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between Protect zero mode edge states at Zigzag boundaries (Fujita) chiral symmetry breaking with chiral symmetry only at edges Splitting of S. Ryu & Y. Hatsugai, the zero mode edge state Bulk is gapless still. Physica E22, 679 (2004)

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between Protect zero mode edge states at Zigzag boundaries (Fujita) chiral symmetry breaking with chiral symmetry only at edges Splitting of S. Ryu & Y. Hatsugai, the zero mode edge state Bulk is gapless still. Physica E22, 679 (2004)

Chiral Symmetry of Graphene Chiral Symmetry Equivalence of Hopping between Protect zero mode edge states at Zigzag boundaries (Fujita) chiral symmetry breaking chiral symmetry breaking with chiral symmetry both bulk and edges only at edges Splitting of S. Ryu & Y. Hatsugai, Full gap opens the zero mode edge state Bulk is gapless still. Physica E22, 679 (2004)