Many-body excitations in undoped Graphene

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1 Department of Physics, Sharif University of Technology, Tehran , Iran M. Ebrahimkhas: Tehran, Iran E. Ghorbani: Isfahan, Iran A. Gruneis: Vienna, Austria Oct. 20, 2011

2 Phase diagram of correlations Z. Y. Meng, et al, Nature (2010) SAJ, EPJB (2009)

3 Spin liquid interpretation G. Baskarna, SAJ, Phys. Rev.Lett. (2002) A. Vaezi, X. -G. Wen, arxiv: (2010) RPA calculation of two-spinon states Slave particle: spinon states

4 Table of contents 1 One-particle physics of graphene: Dirac One-particle spectrum Chirality 2 3 Time resolved photoemission (TRPES) Neutron Scattering 4 Which model: Dirac or Heisenberg?

5 One-particle spectrum Chirality Dirac cones in graphene and graphite S.Y. Zhou, et al, Nature Phys. (2006) Hψ A = t (ψ B1 + ψ B2 + ψ B3 ) Hψ B = t (ψ A1 + ψ A2 + ψ A3 ) ( ) ( ψa 0 Φ ( H = ) ( ) k) ψa ψ B Φ( k) 0 ψ B ) Φ( k) = t (e i k. δ 1 i k. δ + e 2 i k. δ + e 3 A. Bostwick, et al, Nature Phys. (2007) k = K + p Φ( p) px + ip y Near K, K : H K = v F σ. p, H K = v F σ. p

6 Chirality One-particle spectrum Chirality C. W. J. Beenakker, Rev. Mod. Phys. (2009) Eigenvalues and eigen-vectors of H = v F σ. k: k, λ = 1 ( ) 1 2 λe iϕ, k Φ( k) = v F k e iϕ k, λ = ± M. I. Katsnelson, et al, Nature Phys. (2006) Scattering matrix elements: k, λ ˆV k, λ = Ṽ ( k k ) 1 + λλ e iϕ k iϕ k λ = λ No back-scattering! λ = λ No forward scattering! 2

7 Plasmons E. H. Hwang, et al, PRB (2007) Plasmons: Require doping Are singlet excitations Require long range Coulomb interactions

8 Importance of a gap in the PH continuum Gap in PHC: Room for spin collective modes! Similarity with Luttinger liquids χ 0 (q, ω) = 1 N k f k+q f k ω (ε k+q + ε k )

9 Importance of correlation in pi-π conjugated molecules E. C. da Silva, et al, J. Chem. Phys. (1994) E. Ghorbani, SAJ, F. Shahbazi, Iran. J. Phys. Res. (2011) Triplets are lowest excited states Molecular analogue of triplet excitons of semi-conductors RVB interpretation: Triplons: spinon + spinon Singlons: doublon + holon K. H. Mood, SAJ, E. Adibi, G. Baskaran, M. R. Abolhassani, arxiv: What is the fate of this triplet in graphene and/or graphite?

10 Weak coupling approach Key features: We consider undoped graphene No plasmons possible We consider short range interactions drastic changes in Dirac liquid What is the analogue of triplet excitons in graphene? What is the analogue of lowest triplet state of pi-π conjugated molecules in graphene?

11 Why short range interactions are important?, Eur. Phys. Jour. B (2009) Short range interactions: Are relevant in RG sense Additional non-dirac fixed points Induce singlet correlations in the ground state

12 Shor range interactions spin liquids? Z. Y. Meng, et al, Nature (2010) Moderate short range interactions: Produce spin liquid states! Triplons as excitations Charge fluctuations doublon-holon singlet pairs?

13 Strongly Correlated Hamiltonian Considre triplet operators in S z = +1 sector: c k v k+ q, v k c k+ q

14 Time evolution of triplet operators Which gives where η k, q = e iϕ k iϕ k+ q.

15 The eigen-value problem Two modes: T ± q (1 Ũχ± 0 ) = 0 χ± 0 ( q, ω q) = 1 Ũ where and T ± q = k ( c k v k+ q + v k c k+ q ) (1 ± η k, q ) χ ± 0 ( q, ω q) = 1 1 4N ω q ε k ε k k+ q + i0 1 (1 ± η ω q + ε k + ε k+ q i0 k, q )

16 Role of (1 ± η k, q ): Chirality helps! ω q = v F ( k + ) k + q 1±η k, q = 1±e iϕ k iϕ k+ q 1±e iπ χ T + mode required very high Ũ. χ 0 remains finite T mode can be excited by moderate Ũ.

17 Numerical results M. Ebrahimkhas, SAJ, G. Baskaran, arxiv (2009) E. Sarvestani, SAJ (unpublished) Doped Graphene: in (a), (b), µ = 0.4 ev, while for (c), (d), µ = 0.6 ev. From (a) to (d), U = 1.8, 2.0, 2.0, 2.2 in units of t. Undoped Graphene: in (a), (b), U = 1.8t, 2.0t, respectively. Linear DOS, similar to Dirac Fermions Linear dispersion: Comparable to Dirac Fermions

18 Ground state in QMC Geminal approach Singlet correlations Possible RVB description?

19 Excited states in QMC K. H. Mood, SAJ, E. Adibi, G. Baskaran, M. R. Abolhassani, arxiv: Genuine bound states Triplets are first excited states (spinon+spinon?) There is another singlet above it (doublon-holon?)

20 Exact diagonalization support Role of the longer range part of interaction V Trend of singlet state is not plamonic doublon + holon? U 3.5t Triplon=spinon + spinon? RVB ground state offers unified understanding of both states

21 Indirect evidence? Time resolved photoemission (TRPES) Neutron Scattering Where does the extra decay rate in HOPG come from? GW not enough Long range Coulomb interactions is not enough Short range correlations play role? Phonons cease to exist beyond 0.2 ev G. Moos, et al, PRL (2001) M. Ebrahimkhas, SAJ, PRB (2009)

22 Where in BZ to search? Time resolved photoemission (TRPES) Neutron Scattering SAJ, G. Baskaran, EPJB (2005)

23 Which model: Dirac or Heisenberg? Spin Excitations from strong coupling side

24 Dirac: Weak coupling approach Which model: Dirac or Heisenberg? Fermionic Hamiltonian: The standard model H F = t ( ) c iσ c jσ + c jσ c iσ + CoulombInteractions i,j,σ ṽ F k ψ kσ σ. kψ kσ Dirac at low-energies Explains many phenomena, including QHEs, Disorder effects, Transport, Functionalization, Adding Gauge fields, etc. Appealing, and easy to calculate and understand! Robust picture against many types of perturbing interactions

25 Heisenberg: Strong coupling approach Which model: Dirac or Heisenberg? Strongly Correlated Hamiltonians: H Hubbard = H t + U j ( n j 1 ) ( n j 1 ) 2 2 H S = J 1 Si. S j + J 2 Si. S j +... i,j i,j J 1 t 2 /U, J 2 t 4 /U allow for charge fluctuations H. Mosadeq, F. Shahbazi, SAJ, JPCM (2011) RVB wave functions become natural: Z. Noorbakhsh, F. Shahbazi, SAJ, G. Baskarna, JPSJ (2009) Gauge files are possible to add to mimic distortions Non-trivial topological phases A. Vaezi, et al (arxiv: ) Consistent spectrum of spin excitations, and much more H. Mosadeq, F. Shahbazi, SAJ, Asia Pacific Workshop 2008, Tokyo, Japan

26 Which model: Dirac or Heisenberg? Graphene = Dirac + Heisenberg? Thank you for your attention