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INTERNATIONAL RESEACH SCHOOL AND WORKSHOP ON ELECTRONIC CRYSTALS ECRYS2011 August 15 -27, 2011 Cargèse, France

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Dirac electrons in solid Hidetoshi Fukuyama Tokyo Univ. of Science

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Acknowledgement Bi Yuki Fuseya (Osaka Univ.) Masao Ogata (Tokyo Univ.) α-ET 2 I 3 Akito Kobayashi(Nagoya Univ.) Yoshikazu Suzumura (Nagoya Univ.)

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Dirac electrons in solids contents 1)“elementary particles” in solids <= band structure, locally in k-space 2)Band structure similar to Dirac electrons Examples: bismuth, graphite-graphene molecular solids αET 2 I 3, FePn, Ca 3 PbO 4x4 (spin-orbit interaction), 2x2 (Weyl eq.) 3)Particular features of Dirac electrons small band gap => inter-band effects of magnetic field effects Hall effect, magnetic susceptibility

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Dirac equations for electrons in vacuum Equivalently, In special cases of m=0, Weyl equation for neutrino 4x4 matrix 2x2 matrix

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“Elementary particles in solids” band structures, locally in k-space SiInSb electrons holes Semiconductors, Carrier doping electron doping ->n type hole doping -> p type Dispersion relation ＝＞ effective masses and g-factors “elementary particles” Luttinger-Kohn representation (k ・ p approximation)

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LK vs. Bloch representation Bloch representation: energy eigen-states Ψ nk (r)= e ikr u nk (r) : u nk (r+a)=u nk (r) Luttinger –Kohn representation [ Phys. Rev. 97, 869 (1955) ] Χ nk (r)= e ikr u nk0 (r) k 0 = some special point of interest If ε n (k) has extremum at k 0 Spin-orbit interaction “k ・ p method” Hamiltonian is essentially a matrix

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LK vs. Bloch * LK forms complete set and are related to Bloch by unitary transformation * k-dependences are completely different, * in Bloch, both e ikr and u nk (r), the latter being very complicated, while in LK only in e ikr as for free electrons. * just replace k=> k+eA/c in Hamiltonian matrix once in the presence of magnetic field

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Dirac types of energy dispersion(1) *Graphite [ P. R. Wallace (1947),J.W. McClure(1957)] semimetal （ n e =n h ≠ ０） *graphene: special case of graphite （ n e =n h = ０） Geim H = v( k x σ x + k y σ y ) Weyl eq. for neutrino Isotropic velocity McClure(1957)

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Dirac types of energy dispersion(2) *Bi, Bi-Sb [M. H. Cohen and E. I. Blount (1960), P.A. Wolf(1964)] ： semimetals strong spin-orbit interaction This term is negligible *α-ET 2 I 3 ： molecular solids S. Katayama et al.[2006] A. Kobayashi et al.(2006) H = k ・ V ρ σ ρ σ 0 = 1, σ α α= x,y,z Tilted Weyl eq. Tilted Dirac eq. Anisotropic velocity Anisotropic masses and g-factors

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*FePn Hosono(2008) Ishibashi-Terakura(2008) DFT in AF states HF : JPSJ Online—News and Comments [May 12, 2008] * Ca 3 PbO : Kariyado-Ogata(2011)JPSJ Dirac types of energy dispersion(3)

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Dirac electrons in solids Bulk *Bi *graphite-graphene *ET 2 I 3 *FePn *Ca 3 PbO cf. topological insulators at surfaces Effective Hamiltonian

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Characteristics of energy bands of Dirac electrons *narrow band gap, if any *linear dependence on k (except very near k 0 ) Gapless (Weyl 2x2) negligible s-o => effects of spins additive Finite gap(mass)(4x4) s-o => spin effects are essential

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Essence of Luttinger-Kohn representation

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Luttinger-Kohn representation

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Particular features of Dirac electrons Narrow band gaps =>Inter-band coupling “ Inter-band effects” Different features form effective mass approximation in transport and thermodynamic properties. Especially, in magnetic field Hall effects, orbital magnetic susceptibility

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10 th ICPS (1970) - corresponds to the Peierls phase in the tight-binding approx. ε n (k) => ε n (k+eA/c)

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Landau-Peierls Formula χ LP = 0 if DOS at Fermi energy =0 p ・ A : p has matrix elements between Bloch bands

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Orbital Magnetism in Bi Landau-Peierls formula (in textbooks) is totally invalid !! Expt. Indicate importance of inter-band effects of magnetic field. Landau-Peierls Formula χ LP = 0 if DOS at Fermi energy =0

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HF-Kubo: JPSJ 28 (1970) 570 Diamagetism of Bi P.A. Wolff J. Phys. Chem. Solids (1964) Dirac electrons in solids! Strong spin-orbit interaction

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Exact Formula of Orbital Susceptibility in General Cases In Bloch representation

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With Gregory Wannier ＠ Eugene, Oregon (1973)

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Weak field Hall conductivity, σ xy One-band approximation based on Boltzmann transport equation, General formula based on Kubo formula : HF-Ebisawa-Wada PTP 42 (1969) 494. Inter-band effects have been taken into account => Existence of contributions with not only f’(ε) but also f(ε) HF for graphene (2007) Weyl eq. A. Kobayashi et al., for α-ET 2 I 3 (2008) Tilted Weyl eq. Y. Fuseya et al., for Bi (2009) Tilted Dirac eq.

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Bi Wolf(1964) Assumption = isotropy of velocity “Isotropic Wolf” Δ=E G /2 = original Dirac

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In weak magnetic field R=0, but not 1/R=0 Fuseya-Ogata-HF, PRL102,066601(2009)

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Isotropic Wolf model (original Dirac) Under magnetic field, k=> π=k+eA/c * Reduction of cyclotron mass = enhancement of g-factor => Landau splitting = Zeeman splitting both can be 100 times those of free electrons * Energy levels are characterized by j=n+1/2 +σ/2 orbital and spin angular momenta contribute equally to magnetization * Spin currents can be generated by light absorption Fuseya –Ogata-HF, JPSJ Under strong magnetic field

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Molecular Solids ET 2 X layered structure ET layers Anions layers S S S S S S S S ET molecule (ET=BEDTTTF) ET 2 X - => ET +1/2 ET layers conducting X- closed shell

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Degree of dimerization (effectively ¼-filled for weak, ½ for strong) and degree of anisotropy of triangular lattice, t’/t Hotta,JPSJ(2003), Seo,Hotta,HF:Chemical Review 104 (2004) 5005. ET 2 X Systems ET=BEDT-TTF S S S S S S S S α Spin Liquid Dirac cones

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α-ET2I3 JPSJ 69(2000)Tajima-Kajita T-indep. R under high pressure Kajita (1991,1993) p =19Kbar μ eff deduced by weak field Hall coefficient has very strong T-dep. n eff is also, since σ=neμ μ eff α-ET 2 I 3 by charge order

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Hall coefficient in weak magnetic field depends on samples, some change signs at low temperature.

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Tight-binding approximation

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fastest slowest Energy dispersion Massless Dirac fermion in α-(BEDT-TTF) 2 I 3 Katayama et al. (2006) Tilted Dirac cone Confirmed by DFT : Kino et al. (2006) Ishibashi (2006) NMR ： Takahashi et al. (2006) Kanoda et al. （ 2007 ） Shimizu et al.(2008) Interlayer Magnetoresistance Osada et al.(2008) Tajima et al.(2008) Morinari et al. (2008) Tilted Weyl Hamiltonian Kobayashi et at. (2007) Hall effect: Tajima et al. (2008) Kobayashi et al. (2008)

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The conventional relation R H ∝ 1/n is invalid. ------ typically, R H =0 at μ=0 ( n eff =0 for semicoductors) sharp μ-dependence in narrow enegy range of the order of Γ. 1/Γ: elastic scattering time extremely sensitive probe! Orbital susceptibility conductivity Hall conductivity X=μ/Γ Transport properties: Hall effect Kobayashi et al., JPSJ 77(08)064718 σ μν =σ 0 K μν μ:chemical potential 2d model Without tilting=graphene

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Effect of Tilting Kobayashi-Suzumura-HF,JPSJ 77, 064718(2008) Based on exact gauge-invariant formula X=ε/Γ

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speculations on T-dep. with μ=0 for T/Γ>1 σ xx = K xx σ xx (T) =- ∫ dεf’(ε ） σ(ε ）～ Γ/T weak T dep. of σ => Γ ~ T, Then σ xy = ~ 1/T 2 R ~ 1/T 2 K xy σ=neμ n~ T 2 μ ~1/T 2 α= 0 Stronger T-dep In expts ?

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Possible sign change of Hall coefficient; A. Kobayashi et al., JPSJ 77(2008) 064718. Asymmetry of DOS relative to the crossing energy, ε 0. Chemical potential crosses ε 0 as T->0 if I 3 - ions are deficient of the order of 10 -6 (hole-doped) Hall coefficient can change sign, in accordance with expt. by Tajima et al. as below. Prediction, diamagnetism will be maximum, when Hall coefficient changes sign. Bulk 3d effects Cf. specific heat

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Under strong perpendicular magnetic field p=18kbar α-(BEDT-TTF) 2 I 3 N. Tajima et al. (2006) T0T0 T1T1 *For tilted-cones, inter-valley scattering plays important roles. *Mean-filed phase transition(T 0 ) to pseudo-spin XY ferromagnetic state. *Possible BKT transition at lower temperature. A.Kobayashi et al, JPSJ78(2009)114711 T 0 T1T1

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Landau quantization Massless Dirac fermions under magnetic field At H=10T T0T0 With tilting M. O. Goerbig et al. (2008) T. Morinari et al. (2008) Electron correlation can play important roles! Effective Coulomb interaction Zeeman energy

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Kosterlitz-Thouless Transition in Strong Magnetic Field Long-range Coulomb interaction :spin ↑ 、 ↓ ： pseudo-spin （ valley) R,L Tilted Weyl Hamiltonian v: cone velocity pseudo-spin （ valley) Katayama et al. (2006) Zeeman term w: tilting velocity Kobayashi et at. (2007)

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Wave function of N=0 states (Landau gauge) Å X-direction: localized Y-direction: plane wave Magetic length magnetic unit cell : a flux quantum Φ 0 |Φ| 2 Wannier functions (ortho-normal) can be defined on magnetic lattice Fukuyama (1977, in Japanese) To treat interaction effects, “Wannier function” for N=0 states

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Effective Hamiltonian on the magnetic lattice Landau quantization (N=0) ＋ Zeeman energy ＋ long-range Coulomb interaction Effective Hamiltonian SU(4) symmetric independent of tilting Breaking SU(4) symmetry Induced by Tilting! V term ： intra-valley scatteringW term ： inter-valley scattering for α-(BEDT-TTF) 2 I 3 H=10T :tilting parameter

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Ground state of the effective Hamiltonian In the absence of tilting Spin-polarized state the phase transition can occur at finite T in the mean-field approximation. W-term :Pseudo-spins are bound to XY-plane. V-term ： symmetric in the spin and pseudo-spin space In the presence of tilting Pseudo-spin ferromagnetic state Only E z -term breaks the symmetry If the interaction is larger than E z,

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Mean field theory (finite T) :Pseudo-spin operator ： interactions between pseudo-spins Taking fluctuations of pseudo-spins in XY-plane, Spin-polarized state Pseudo-spin XY ferro Effective “spin model” on the magnetic lattice Tc ~ 0.5 I

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Kosterlitz-Thouless transition Expanding the free energy from long-wavelength limit, The fluctuations are described by the XY model Berenzinskii-Kosterlitz-Thouless transition (J. M. Kosterlitz, J. Phys. C7 (1974) 1046. ) (in the present case) vortex and anti-vortex excitations Tc~ 0.5 I nearest-neighbor interaction nearly isotropic if I 00 =I

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Under strong perpendicular magnetic field p=18kbar α-(BEDT-TTF) 2 I 3 N. Tajima et al. (2006) T0T0 T1T1 *For tilted-cones, inter-valley scattering plays important roles. *Mean-filed phase transition(T 0 ) to pseudo-spin XY ferromagnetic state. *Possible BKT transition at lower temperature. A.Kobayashi et al, JPSJ78(2009)114711 T 0 T1T1

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Graphenes Checkelsky-Ong,PRB 79(2009)115434 BKT transition T=0.3K at 30T K. Nomura, S. Ryu, and D-H Lee, cond-mat/0906.0159 Without tilting (W=0) : electron-lattice coupling

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Massless Dirac electrons in α-ET 2 X *Described by Tilted Weyl equation *Unusual responses to weak magnetic field Hall coefficient Inter-band effects of magnetic field (vector potential, A) are crucial. *Under strong magnetic field possible Berezinskii-Kosterlitz-Thouless transition * Further many-body effects ?

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Massless Dirac electrons in α-ET 2 X *Described by Tilted Weyl equation *Unusual responses to weak magnetic field Hall coefficient Inter-band effects of magnetic field (vector potential, A) are crucial. *Under strong magnetic field possible Berezinskii-Kosterlitz-Thouless transition * Further many-body effects ?

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Ca 3 PbO Synthesis not yet. Similarity to and differences from Bi Kariyado-Ogata to appear in JPSJ

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Dirac electrons in solids Summary * Examples: bismuth, graphite-graphene molecular solids αET 2 I 3, FePn, Ca 3 PbO 4x4 (spin-orbit interaction), 2x2 (Weyl eq.) * Particular features are “small band gap” => inter-band effects of magnetic field effects Hall effect, magnetic susceptibility ~~ Targets Effects of boundary( surfaces, interfaces)

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Supplement FePn Superconductivity

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Year 2008 : New High-T c “Fever” derived from Hosono’s Discovery Pb Nb NbC NbN Nb 3 Ge MgB 2 Hg Year T c (K) Onnes 1913Physics 1911 LaBaCuO LaSrCuO YBaCuO BiCaSrCuO HgCaBaCuO HgCaBaCuO(High-Pressure) 1986 BednorzMuller 1987Physics 2001 Akimitsu LaFePO LaFeAsO LaFeAsO(High-Pressure) SmFeAsO Hosono 1 st International Symposium June 27-28, Tokyo 1 st Proceedings Vol. 77 (2008) Supplement C November 28 1 st Focused Funding Program Transformative Research-Project on Iron Pnictides Call for proposal: July-August Start: October (till March 2012) TlCaBaCuO 20082008 Prepared by JST

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World-wide Competition and Collaboration triggered by TRIP Oct 2008 – Mar 2012 Leader: Hide Fukuyama 24 Research Subjects 0.3-0.8 M$/ 3.5 Yrs Collaboration Leader: Hideo Hosono Mar 2010 – Mar 2013 Outcome New priority program ‘High-temp. superconductivity in iron pnictides’ (SPP 1458) From 2010; 6 Yrs (3Yrs + 3Yrs) Collaboration Collaboration JST-EU Strategic Int. Cooperative Program on (3-Yrs period) ‘Superconductivity’ (3-Yrs period) Under ex ante evaluation Under ex ante evaluation International Workshop on the Search for New SCs Co-sponsored by JST-DOE-NSF-AFOSR May 12-16, 2009, Shonan Collaboration Frontiers in Crystalline Matter Reported by National Academy of Sciences Oct 2009 P108-109 Box 3.1 Iron-Based Pnictide Materials: Important New Class of Materials Discovered Outside the United States Prepared by JST

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A15-MgB 2 -Cuprates-FePn *A15 : BCS, structural change *MgB 2 : BCS, strong ele-phonon, 2bands *Cuprates: strong correlation in a single band, Doped Mott, t-J model *FePn: strong correlation in multi bands structural change

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Journal of the Physical Society of Japan Vol. 77 (2008) Supplement C Proceedings of the International Symposium on Fe-Pnictide Superconductors Published in JPSJ online November 27, 2008 Preface Outline ＊ Layered Iron Pnictide Superconductors: Discovery and Current Status＊ Layered Iron Pnictide Superconductors: Discovery and Current Status Hideo Hosono ＊ A New Road to Higher Temperature Superconductivity＊ A New Road to Higher Temperature Superconductivity S. Uchida ＊ Doping Dependence of Superconductivity and Lattice Constants in Hole Doped La 1-x Sr x FeAsO＊ Doping Dependence of Superconductivity and Lattice Constants in Hole Doped La 1-x Sr x FeAsO Gang Mu, Lei Fang, Huan Yang, Xiyu Zhu, Peng Cheng, and Hai-Hu Wen ＊ Se and Te Doping Study of the FeSe Superconductors＊ Se and Te Doping Study of the FeSe Superconductors K. W. Yeh, H. C. Hsu, T. W. Huang, P. M. Wu, Y. L. Huang, T. K. Chen, J. Y. Luo, and M. K. Wu Total ~50 papers

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In 2011, Special Issue : Solid State Communications, to appear.

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S. Nandi et al.: Phys. Rev. Lett. 104 (2010) 057006 1111 R. Parker et al.: Phys. Rev. Lett. 104 (2010) 057007 111 FePn Phase diagram Tet Ort T S >T N for x>0 T-W Huang et al.: Phys. Rev. B82 (2010) 104502 Tet Ort J. Zhao et al.: Nature Mater. 7 (2008) 953 122 No T N 11 Courtesy: Ono

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1111 Tet Ort J. Zhao et al.: Nature Mater. 7 (2008) 953 Courtesy: Ono Basic difference from cuprates Parent compound Cuprates : Mott insulator (odd) 1 band FePn : semimetal (even) multi-band Importance of magnetism : spin-fluctuations Roles of many bands : Mazin, Kuroki Effects of crystal structure: Lee plot (Pn height-Kuroki) film MKWu Electronic inhomogeneity Phase separation

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Minimum Courtesy: Yoshizawa Ba122Co

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Analysis for softening in C 66 of Ba(Fe 1-x Co x ) 2 As 2 Co ( % )Θ ( K )Δ ( K ) 3.7 %75.55.4 6 %17.28.3 10 %- 3015.6 M.Yoshizawa et al., arXiv:1008.1479v3 (Aug 2010) Increasing of Co doping in Ba(Fe 1-x Co x ) 2 As 2 reduces Θ and enhances Δ. C 66 of Ba(Fe 1-x Co x ) 2 As 2 Constant Θ changes its sigh from + to – over quantum critical point.

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Temperature dependence in elastic constants of Ba(Fe 0.9 Co 0.1 ) 2 As 2 C 66 reveals huge softening of 28% from room temperature down to T sc =23K. No sigh of softening in (C 11 –C 12 ) / 2 and C 44. Electric quadrupole of O u is relevant Courtesy: Goto little change by H

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1d bands Labbe-Friedel:band Jahn Teller Gorkov:dimerization along chains 3d bands <= band calc. by Mattheiss Bhatt-McMillan, Bhatt: 2 close-lying saddle points based on dx2-y2 band Matheiss dz2 Tc Klein ele-phonon A15

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FePn: Coulomb interaction +el-ph interaction due to multi-orbit(multi-band)

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END

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