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Density-Matrix Renormalization-Group Study on  Magnetic Properties of Nanographite Ribbons

Density-Matrix Renormalization-Group Study on  Magnetic Properties of Nanographite Ribbons. T ． Hikihara and X ． Hu （引原 俊哉、胡暁） National Institute for Materials Science. 1st, Feb, 2002 at National Center for Theoretical Sciences. armchair ribbon. zigzag ribbon.

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Density-Matrix Renormalization-Group Study on  Magnetic Properties of Nanographite Ribbons

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  1. Density-Matrix Renormalization-Group Study on  Magnetic Properties of Nanographite Ribbons T．Hikihara and X．Hu （引原 俊哉、胡暁） National Institute for Materials Science 1st, Feb, 2002 at National Center for Theoretical Sciences armchair ribbon zigzag ribbon

  2. Outline I. Density-Matrix Renormalization-Group Method 1.1 Problem 1.2 Basic idea of DM truncation 1.3 Algorithm : infinite-system & finite-system method 1.4 Characteristics of DMRG method II. Magnetic Properties of Nanographite Ribbons 2.1 Introduction 2.2 tight-binding model on nanographite ribbons 2.3 electron-electron coupling 2.4 Prospect of future studies

  3. I. Density-Matrix Renormalization-Group Method 1.1 Problem investigation of the properties of strongly correlated systems on lattice sites Hubbard model : t-J model : Heisenberg model : Strong correlation between (quasi-) particles･･･ many-body problem we must solve eigenvalue problem of a large Hamiltonian matrix without (or, at least, with controlled, unbiased) approximation

  4. Numerical approach Exact Diagonalization (Lanczos, Householder etc.) - extremely high accuracy - applicable for arbitrary systems - severe restriction on system size (ex. Hubbard model : up to 14 sites) Quantum Monte Carlo method - rather large system size - flexible - minus sign problem - slow convergence at low-T Variational Monte Carlo method - rather large system size - results depend on the trial function We want to treatlarger system with smaller memory/ CPU time controlled (unbiased) accuracy extend the ED method by using truncated basis

  5. 1.2 Basic idea of Density-Matrix truncation 1.2.1 Truncation of Hilbert space Exact Diagonalization ・・・・・ L site system basis : w.f. : Hamiltonian : # of basis :nL(n : degree of freedom/site)･･･ exponential growth with L memory overflow occurs at quite small L

  6. Reduction of Hilbert space by truncation basis for whole system : : nLlnLr = nL -basis L sites ・・・ ・・・ block l : Ll sites block r : Lr sites Ll + Lr = L : nLl - basis : nLr - basis truncate !! :m-basis : mnLr- basis whole system : ･･･ ifmis small enough,Hii' is diagonalizable

  7. truncation = discarding the contribution of the basis to wave function of whole system = loss of information truncation procedure consists of (i) selecting an orthonormal set to expand the Hilbert space for the block (ii) discarding all but the m important basis ･･･ We can improve the procedure (i) to reduce the loss Question :Which basis set is optimal to keep the information ?

  8. 1.2.2 (Wilson's) Real Space RG Real Space RG (RSRG) :method to investigate low-energy propertiesof the system basic idea :highly-excited states of a local block do not contribute to the low-energy properties of whole system H = Hl + Hlr + Hr Hlr ・・・ ・・・ Hl Hr diagonalize a block HamiltonianHl keep them-lowest eigenstates of Hl as a basis set

  9. algorithm of RSRG Hb HL ・・・ L sites (m-basis) n-basis HL+1=HL+Hb (i) Isolate block L from the whole system (ii) Add a new site to block L and Form new block Hamiltonian HL+1 from HL and Hb (iii) Diagonalize the block HL+1 (nm×nm matrix) to obtain m-lowest eigenstates (iv) “Renormalize" HL+1 to (m×m matrix) into the new basis (v) Go to (ii) by Substituting for HL L+1 site (nm-basis) The RSRG scheme works well forKondo impurity problem random bond spin system etc. but RSRG becomes very poor for other strongly correlated systems Why?

  10. reconsideration of RSRG ・・・ ・・・ Isolate a part of system ・・・ keep the low-energy states of a block (ex.) one-particle in a 1D box ψ g.s.w.f. of whole system x low-energy states of the block very small contribution at the connection We must take account of the coupling between the blocks

  11. 1.2.3 Density-Matrix RG : S.R.White,PRL 69,2863(1992) ;PRB 48,10345(1993). utilize the density matrix for truncation procedure Target state : ・・・ ・・・ Density Matrix for the left block basic scheme : keep the eigenstates of ρ with m-largest eigenvaluesas a basis set

  12. The basis set with DM scheme is optimal to keep the information of the target state ･･･Truncation error is minimized It can be shown that (where P(i) : i-th eigenvalue of DM) Calculations become more accurate as m gets larger ･･･ m : controlling parameter of DMRG (In many cases,) the truncation error rapidly decreases with m very high-precision can be achieved with feasible m

  13. 1.3 Algorithm of DMRG 1.3.1 Infinite-system algorithm H: n2m2×n2m2 matrix : diagonalizable ・・・ ・・・ ･･･right block is the reflection of the left block : nm -basis substitute form and diagonalize DM ・・・ : m -basis new block (i) Form H of whole system from operators of four blocks (ii) Diagonalize H (n2m2×n2m2 matrix)to obtain (iii) Form the density matrix ρfor left two blocks (iv) Diagonalize ρ(nm×nm matrix) to obtain m-largest eigenvalues and eigenstates (v) Transform operators of left two blocks into the new m-basis (vi) Go to (i), replacing old blocks by new ones

  14. 1.3.2 finite-system algorithm fixed L ・・・ ・・・ 1 1 Lr sites Ll sites form and diagonalize DM ・・・ draw a block with Lr sites use as a block with Ll+1 sites stock ・・・ ・・ After a few iterations of thesweep procedure one can obtainhighly accurate resultson afinite (L sites) system

  15. Characteristics of DMRG DMRG = Exact Diagnalization in truncated basis optimized to represent a target state using DM scheme - Highly accurate especially for a lowest-energy state in a subspacewith given quantum number(s) 1D system - (In principle,) we can calculate expectation values of arbitrary operators in the target state (ex.) lowest energy for each subspace → charge (spin) gap, particle density at each site, two-point correlation function, three-point correlation ･･･ - less accurate for excited states → finite-T DMRG, dynamical DMRG 2D (or higher-D) system or 1D system with periodic b.c.

  16. Two-spin correlation function in the ground state of S=1/2XXZ chain of 200 sites T.Hikihara and A. Furusaki, PRB58, R583 (1998). Numerical data is in excellent agreement with exact results

  17. DMRG for 2D system - Single-chain system An accuracy with m states kept - double-chain system We need m2 states to obtain the same accuracy - 2D system Equivalent to L/x-chain system L-sites mL/xstates are needed # of states we must keep increases exponentially with the system width

  18. Nanotube : 1D Nanographite ribbon : 1D II. Magnetic Properties of Nanographite Ribbons 2.1 Introduction Nanographite : graphite system with length/width of nanometer scale - quantization of wave vector in dimension(s) - # of edge sites ~ # of bulk sites graphene sheet : 2D Graphite Nanoparticle : 0D

  19. graphite : sp2 carbons material Electron state around Fermi energy Ef =p-electron network on honeycomb lattice (# of p-electron) = 1 : half-filling (# of carbon site) Topology (boundary condition, edge shape etc.) is crucial in determining electric properties of nanographite systems (ex.) Nanotube : can be a metal or semi-conductor depending on chirality Nanographite ribbon : edge shape

  20. Experimental results on magnetic properties of nanographite Graphite sheet : large diamagnetic response - due to the Landau level at E = Ef = 0 (McClure, Phys. Rev. 104, 666 (1956). - weak temperature dependence - typical value at room temp. : cdia~ 21.0×10-6 (emu/g) Activated carbon fibers : 3D disorder network of nanographites (Shibayama et al., PRL 84, 1744 (2000); J. Phys. Soc. Jpn. 69, 754 (2000).) - Curie like behavior at low temperature ･･･ due to the appearance of localized spins in nanographite particles Rh-C60 : 2D polymerized rhombohedral C60 phase (Makarove et al., Nature 413 718(2001).) - Ferromagnetism with Tc ~ 500 (K)

  21. Activated Carbon Fiber Disordered network of nanographite particles Each nanographite particle - consists of a stacking of 3 or 4 graphene sheets - average in-plane size ~ 30 (A) (Kaneko, Kotai Butsuri 27, 403 (1992)) Susceptibility measurement Crossover from diamagnetism (high T) to paramagnetism (low T) (Shibayama et al., PRL 84, 1744 (2000))

  22. Magnetic field(kOe) RhC60(Makarova et al., Nature 413, 716 (2001).) Hysteresis loop Saturation of magnetization T-dependence of saturated magnetization Tc ~ 500 (K)

  23. Zigzag ribbon Armchair ribbon 2.2 tight-binding model on nanographite ribbons Nanographite ribbon : graphene sheet cut with nano-meter width Two typical shape of edge depending on cutting direction Edge bonds are terminated by hydrogen atoms

  24. : sublattice A : sublattice B Tight-binding model : sum only between nearest-neighboring sites t ~ 3 (eV) Definition of the site index N = finite, L →∞ : zigzag ribbon L = finite, N →∞ : armchair ribbon i = 1 2 3 N j = 1 2 3 4 5 6 ･･････ L

  25. Armchair ribbon :energy gap at k = 0 : Da = 0 (L = 3n-1) ~ 1/L (L = 3n, 3n+1) Zigzag ribbon :(almost) flat band appears at E = Ef = 0 !! “edge states” : electrons strongly localize at zigzag edges Band structure of graphite ribbons p-band structure of graphite ribbons can be (roughly) obtained by projecting the p-band of graphene sheet into length direction of ribbon p-band structure of graphene sheet However, presence of edges in graphite ribbons makes essential modification on the band structure

  26. Band structure of armchair ribbon L=4 L=5 L=6 (Wakabayashi, Ph.D Thesis(2000)) L = 30 At k = 0, armchair ribbon is mapped to 2-leg ladder with L-rungs Energy gap of tight-binding model can be obtained exactly

  27. Flat band appears for 2p/3 < k < p DOS has a sharp peak at Fermi energy E = Ef = 0 Band structure of zigzag ribbon L=4 L=5 L=6 (Wakabayashi, Ph.D Thesis(2000)) L = 30

  28. Amplitude : “edge state” Harper’s eq. : Apply H to one-particle w.f. : If E = 0, Wave function for E = 0 and wave number k on A-sublattice

  29. - These localized states form an almost flat band for 2p/3 < k < p - Edge states exhibit large Pauli paramagnetism (might be) relevant to Curie-like behavior of ACF at low-T (Wakabayashi, Ph.D Thesis(2000)) k = p k = 8p/9 k = 7p/9 k = 2p/3 perfect localization penetration

  30. 2.3 electron-electron couplings Localized “edge” states at zigzag edge of graphite ribbon sharp peak DOS at E = Ef = 0 might be unstable against electron-phonon and/or electron-electron couplings Electron-phonon coupling : Fujita et al., J.Phys.Soc.Jpn. 66,1864 (1997). Miyamoto et al., PRB 59, 9858 (1999). Lattice distortion is unlikely with realistic strength of electron-phonon couplings We consider the effect of electron-electron coupling

  31. Mean-field analysis (Wakabayashi et al., J.Phys.Soc.Jpn. 65,1920(1996).) Infinitesimal interaction U of Hubbard type causes spontaneous spin-polarization around zigzag edge sites (Okada and Oshiyama, PRL 87,146803 (2001).) DFT calculation Appearance of spontaneous spin-polarization at zigzag edge

  32. However, Lieb’s theorem : For the Hubbard model on a bipartite lattice, (i) if coupling U is repulsive (U > 0) and (ii) if the system is at half-filling then, (1) the ground state has no degeneracy (2) the total spin of the g.s. is (where NA(NB) is # of sites on A(B) sublattice) In the case of graphite ribbons, NA=NB the ground state is spin-singlet Non-zero local spin-polarization is prohibited Detailed investigation on magnetic properties is desired.

  33. We perform DMRG calculation on Hubbard model - zigzag ribbon : N = 2, 3 - # of kept states m : up to typically 1000. charge gap : spin gap : (M=NL: # of sites, E0(n↑,n↓) : lowest energy in the subspace (n↑,n↓) ) local spin polarization : Spin-spin correlation :

  34. N=2 Zigzag ribbon Charge (spin) gap opens for

  35. N=3 Zigzag ribbon Charge gap opens for

  36. Distribution of Szi for N = 2 in the lowest energy state of U=0 U=1 U=4 Zigzag edge favors spin polarization

  37. Distribution of Szi for N = 3 in the lowest energy state of U=0 U=1 U=4

  38. spin-spin correlation function AF correlation grows as U increases Spin-polarization induced in zigzag edge sites correlates ferrimagnetically resulting in the formation of effective spins on both edges

  39. Schematic picture of ground state of zigzag ribbon effective spin Singlet state - bulk sites form spin-singlet state Jeff - Effective spins appear in zigzag edges effective spin AF effective coupling between effective spins : Jeff ･･･ ground state is a spin-singlet (consistent with Lieb’s theorem) Jeff becomes smaller as the width N becomes larger ･･･spin gap becomes smaller small magnetic field can induce magnetization

  40. Heisenberg model on zigzag ribbon : Effective model for spin-degree of freedom Spin gap Ds(N=4) < Ds(N=2) Distribution of Szi for N = 4 in the lowest energy state of

  41. 2.4 Prospect of Future Studies Realization of nanographite system with edge (i) graphite ribbon Epitaxial growth of carbon system on substrate with step edges graphite ribbons with controlled shape (ii) Open end of carbon nanotubes ･･･ open end of zigzag nanotube = zigzag edge (iii) Carbon island in BNC system Honeycomb structure consisting of B, N, and C atoms Hexagonal BN sheet has a large energy gap ･･･ BN region can work as a separator between C regions (Okada and Oshiyama, PRL 87,146803 (2001).) BN - C boundary ~ open edge of C system

  42. Flat band ferromagnetism Azupyrene defect Four hexagons are replaced by two pentagons and two heptagons Azupyrene defect in armchair ribbon (Kusakabe et al., Mol.Cryst.Liq.Cryst. 305, 445 (1997)) Perfect flat band appears at E = 0 Ferromagnetism might appear for infinitesimal U

  43. Tight-bonding model : - armchair ribbon : energy gap at k = 0 appears depending of width Da = 0 (L = 3n-1) ~ 1/L (L = 3n, 3n+1) - zigzag ribbon : localized “edge state” appears for 2p/3 < k < p ･･･resulting in sharp peak of DOS at E = Ef = 0 (might be) relevant to paramagnetism in nanographite Summary Nanographite ribbon -1D graphene sheet cut with nano-meter width - p-electron system at half-filling - presence of edges is crucial for electronic/magnetic properties

  44. Summary(continued) Effect of electron-electron couplings - zigzag ribbon charge (spin) gap appears for ground state is spin-singlet : upon applying a magnetic field, - magnetization appears around zigzag edge site - spin-polarizations ferrimagnetically correlated each other forms a effective spin - effective coupling between effective spins in zigzag edges gets weaker as the width N increases : for all site

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