Resonating Valence Bond point of view in Graphene

Transcription

1 Resonating Valence Bond point of view in Graphene S. A. Jafari Isfahan Univ. of Technology, Isfahan 8456, Iran Nov. 29, Kolkata S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene /2

2 OUTLINE Introduction S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 2/2

3 Introduction to Graphene b K "Hard" condensed matter Room temperature QHE High mobility cariers Possibility of ballistic transport y x a a2 ky kx b2 Relativitic laboratory on the table top Robust electronic structure features A.H. Castro-Neto, et al, Rev. Mod. Phys. (29) Γ M S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 3/2

4 Fermionic Language (Bloch) ρ(ω) ω H = t c iσ c jσ + h.c. i,j σ v F k. σ ε λ ( k) = λv F k, λ = ± k λ Û kλ =Ũ( k ( ) k) +λλ e (θ k θ k ) /2 Dirac cone at K points Eigenstates have definite chirality (λ,λ = ±) No back scattering S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 4/2

5 Spin Language (Pauling) Spin-Fermion dictionary: Spin Singlet Valence Bond Basic object is b ij = (c i c j c i c j )/ 2, rather than c iσ Benzene: Resonance of Valence Bonds: Graphen: Possibility of longer range valence bonds: C + C Ψ Singlet ground state Triplet excitations Insulating behavior L. Pauling, The Nature of the Chemical Bond (963) S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 5/2

6 Correlations at Molecular Level in pi-π Systems Molecular Orbital Picture J. Jortner, et al, J. Chem. Phys (965) Ground State n=2, S= n=,s= n=, S= Triplet Excitation Singlet Excitation Experiment n=2, S= n=,s= n=, S= ~ 2 ev Spin Gap Remarkable ev triplet-singlet splitting Importance of correlations A general trend in these systems: Low energy triplet excitations Molecular analogue of triplet excitons S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 6/2

7 From Bloch to Pauling: Complementary pictures Phase diagram on the "correlation" axis: Fermi velocity v F /v F S.A. Jafari, Eur. Phys. J. B (29) Graphene is here Dirac Liquid Hubbard U QCP U c =3.29 Mott Insulator Some phenomena are better seen in Bloch s picture (QHE, transport, etc.) Some phenomena are better seen in Pauling s viewpoint (e.g. Pairing correlation, RVB-ness) S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 7/2

8 RPA Formulation What is the fate of low-lying triplet in Graphene? ω 2 2 Γ Μ Μ Κ Γ U = χ() ( q,ω)= N k Κ Γ K M χtriplet( q, RPA χ () ( q,ω) ω) = Uχ () ( q,ω) f +, k+ q f, k hω (ε + k+ q ε k )+i + ( ) cos(θ k θ k ) 2 S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 8/2

9 D-like Bound-state Formation Effectively D phase space for scattering at the edge of particle-hole continuum: U = χ() g( k, k+ q) k ω v F ( k+ q + k )+i + k k + q q Leading / ω v F q singularity dominates: ω Iχ () q ( q, ω) η ω vf q ω/v F Assymptotic behavior at PHC edge: ω v F q Rχ () ( q, ω) = a + b qη vf q ω q ω( q) = v F q cq 3 S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 9/2

10 Neutral collective mode dispersion near Γ point: ω( q) = v F q cq 3 Undoped Graphene Neutron Scattering intensity: Why the direct observation of S= excitaiton is difficult? Energy in units of t Γ Κ 5 6 Μ Γ qa G. Baskaran, S.A. Jafari, PRL (22), ibid (24) Γ Μ Γ Μ Γ S.A. Jafari, G. Baskaran, Eur. Phys. J. B (25) S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene /2

11 Doped Graphene I: Continuum M. Ebrahimkhas, S.A. Jafari, G. Baskaran, arxiv:9.8 Very small 2DEG-like portion appears for q k F For small q the spin- excitation branch acquires a gap S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene /2

12 Doped Graphene: II U=.8t, 2.t, 2.2 t (a,b,c) µ =.4,.5,.6 ev (a,b,c) ω gap = ( 6π UA h 4kc v F )v 2 F + 4µ 2ln( Λ µ ) Genuine inverse-log dependence on µ Gap Shorter Valence Bonds Superconducting Correlations Analogue of 4meV magnetic mode in HTSC U = 2.t, µ =.4 ev, t 2.8 ev. M. Ebrahimkhas, et al, arxiv:9.8 S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 2/2

13 Evidence: Time Resolved Photoemission Theory Experiment g=.25 g=.3 errorbar ω max = 2. ev Decay rate (/fs) (a) (b) ω (ev) TRPES: A missing decay channel at.4 2 ev?? G. Moos et al, PRL 87 (2) (c) + (d) M. Ebrahimkhas, S.A. Jafari, Phys. Rev. B (29) S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 3/2

14 DMFT study ρ(ω) U=. U=4. U=7. U= ω (a) U=. U=4. U= (c) U=. U=4. U= (b) U=. U=2. U= (d) U=. U=2. U=4. Dirac theory at ev (graphene) and 9 K (optical lattices) Possible to tune U/t in optical lattices Looking at graphene from a different angle ρ(ω) = 2π ω vf 2 The Fermi velocity vanishes at semi-metal to Mott insulator transition SMIT: The slope of cone decreases untill gets flat S.A. Jafari, Eur. Phys. J. B (29) S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 4/2

15 Phase Diagram: To View From Left or Right? Fermi velocity v F /v F S.A. Jafari, Eur. Phys. J. B (29) Graphene is here Dirac Liquid QCP U c =3.29 Mott Insulator Hubbard U Reference state "menu": Weak coupling phase Slater detarminants H = t i,j (c i c j)+h.c+ Interactions and/or gauge fields Effectively one-body (relativistic) Q.M. Suitable language for many phenomena Strong coupling regime Resonating valence bond WFs FQUE one-body Q.M. not suitable for insulating phase of graphene X. Du, et al, Nature (29) Pobbile hight temperature superconductivity and heavily doped graphene Pathak, et al, arxiv (28), Sahebsara et al, arxiv (29) RVB correlations manifest at molecular level K. Haqiqi, et al. (unpublished) Effective Hamiltonian: H = i,j JS i.s j +... S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 5/2

16 RVB: The ground state State E=J hðlþ Honeycomb lattice NNRVB.43(2) a ¼ :84 hðþ ¼ z ¼ hðlþ ¼ ; l > ð; 3Þ.587(2) a ¼ 4:7 a ¼ 2=9 z ¼ 5:9 al ¼ ; l > ð; 3; 5Þ.539(2) a ¼ 9:55 ða; a2þ ¼ ð=9; 2=3Þ z ¼ : hðlþ ¼ ; l > 2 Exponential.5437(2) ða; a2þ ¼ ð2=2; =4Þ al ¼ :32; l > 3 p E=J Ms ða; a2þ Honeycomb lattice (2).3(2) (/6, 2/9) (2).26(2) (2/23, 2/9) (2).25(2) (2/9, /4) (2).2(2) (/9, /4) 4.543(2).9(2) (/8, 2/7) a l = h(2l + )/h(2l ) H = J Si. S j, (i,j) ( ) i j i j Valence bond singlets i,j 2 Ψ = h(i j )...h(i n j n) (i,j )...(i n,j n) iα A,i β B Long-range RVB state, h(i i) = ri r j 2.5 gives the best variational estimate of the ground state energy.26 sublattice magnetization left strong quantum fluctuations Good reference state for undoped (half-filled) graphene Z. Noorbakhsh, F. Shahbazi, S.A. Jafari, G. Baskaran, J. Phys. Soc. Jpn. (29) S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 6/2

17 RVB: Excited States SPW: solid line SMA: points ω Iχ () ( q, ω) q η ω vf q Rχ () ( q, ω) = a + b qη vf q ω q ω( q) = v F q cq 3 Single mode approximation used (dots) Good agreement with spin-wave (solid line) gap-less near Γ point, linear dispersion H. Mosadeq, F. Shahbazi, S.A. Jafari (unpublished) RPA approximation Qualitative agreement with exact diagonalization gap-less at Γ, linear dispersion S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 7/2

18 Robustness of the RVB picture H = J i,j S i.s.j + J 2 i,j S i.s j RVB states of the honeycomb lattice are robust up to J2 /J.3 H. Mosadeq, F. Shahbazi, S.A. Jafari (unpublished) S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 8/2

19 Diffusion Monte Carlo Spin gap lower than the charge gap Spin gap rapidly vanishes RVB correlations build in K. Haqiqi, S.A. Jafari, et al. (in preparation) S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 9/2

20 Summary and RVBs describe the strong coupling limit of "graphene" For undoped graphene they are long ranged By doping VBs get shorter Development of SC correlations Triplets are lowest excitaitons (DMC, ED) Linear dispersion can be seen in both Pauling and Bloch point of view Thank You For Your Attention S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene 2/2