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 zigzag ribbon armchair ribbon 1st, Feb, 2002 at National Center for Theoretical Sciences

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 we must solve eigenvalue problem of a large Hamiltonian matrix without (or, at least, with controlled, unbiased) approximation Strong correlation between (quasi-) particles ･･･ many-body problem Hubbard model : t-J model : Heisenberg model :

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 We want to treat larger system with smaller memory/ CPU time controlled (unbiased) accuracy Variational Monte Carlo method - rather large system size - results depend on the trial function 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 : n L (n : degree of freedom/site) ･･･ exponential growth with L memory overflow occurs at quite small L

6. Reduction of Hilbert space by truncation L sites ・・・ block l : L l sitesblock r : L r sites : n L l - basis: n Lr - basis basis for whole system : : n L l n Lr = n L -basis truncate !! : m -basis whole system : : m n Lr - basis ･･･ if m is small enough, H ii' is diagonalizable L l + L r = L

7. 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 truncation = discarding the contribution of the basis to wave function of whole system = loss of information 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 properties of the system basic idea : highly-excited states of a local block do not contribute to the low-energy properties of whole system ・・・ diagonalize a block Hamiltonian H l HlHl H lr HrHr H = H l + H lr + H r keep the m-lowest eigenstates of H l as a basis set

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

10. reconsideration of RSRG (ex.) one-particle in a 1D box ・・・ Isolate a part of system keep the low-energy states of a block g.s.w.f. of whole system low-energy states of the block very small contribution at the connection x ψ 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 basic scheme : keep the eigenstates of ρ with m-largest eigenvalues as a basis set ・・・ Target state : Density Matrix for the left block

12. The basis set with DM scheme is optimal to keep the information of the target state 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 ･･･ Truncation error is minimized It can be shown that (where P(i) : i-th eigenvalue of DM)

13. 1.3 Algorithm of DMRG 1.3.1 Infinite-system algorithm (i) Form H of whole system from operators of four blocks (ii) Diagonalize H (n 2 m 2 ×n 2 m 2 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 ・・・ H : n 2 m 2 ×n 2 m 2 matrix : diagonalizable : nm -basis ・・・ form and diagonalize DM new block : m -basis substitute ･･･ right block is the reflection of the left block

14. 1.3.2 finite-system algorithm ・・・ form and diagonalize DM fixed L ・・ ・・・ use as a block with L l +1 sites L l sites L r sites11 draw a block with L r sites stock After a few iterations of the sweep procedure one can obtain highly accurate results on a finite (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 subspace with 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=1/2 XXZ chain of 200 sites Numerical data is in excellent agreement with exact results T.Hikihara and A. Furusaki, PRB58, R583 (1998).

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

18. 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 Nanotube : 1D Nanographite ribbon : 1D

19. graphite : sp 2 carbons material Electron state around Fermi energy E f =  -electron network on honeycomb lattice (# of  -electron) (# of carbon site) = 1 : half-filling 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 = E f = 0 (McClure, Phys. Rev. 104, 666 (1956). - weak temperature dependence - typical value at room temp. :  dia ~ 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-C 60 : 2D polymerized rhombohedral C 60 phase (Makarove et al., Nature 413 718(2001).) - Ferromagnetism with T c ~ 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)) (Shibayama et al., PRL 84, 1744 (2000)) Susceptibility measurement Crossover from diamagnetism (high T) to paramagnetism (low T)

22. Magnetic field(kOe) RhC 60 (Makarova et al., Nature 413, 716 (2001).) Hysteresis loopSaturation of magnetization T-dependence of saturated magnetization T c ~ 500 (K)

23. 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 Armchair ribbon Zigzag ribbon Edge bonds are terminated by hydrogen atoms

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

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

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

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

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

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

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

32. Lieb’s theorem : the ground state is spin-singlet Non-zero local spin-polarization is prohibited Detailed investigation on magnetic properties is desired. However, 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 N A (N B ) is # of sites on A(B) sublattice) In the case of graphite ribbons, N A =N B

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 : local spin polarization : Spin-spin correlation : (M=NL: # of sites, E 0 (n ↑,n ↓ ) : lowest energy in the subspace (n ↑,n ↓ ) )

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

35. N=3 Zigzag ribbon Charge gap opens for

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

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

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

40. Heisenberg model on zigzag ribbon : Effective model for spin-degree of freedom  s (N=4) <  s (N=2) Spin gap Distribution of S z i 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 (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 (ii) Open end of carbon nanotubes ･･･ open end of zigzag nanotube = zigzag edge

42. Flat band ferromagnetism 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 Azupyrene defect Four hexagons are replaced by two pentagons and two heptagons

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

44. Effect of electron-electron couplings Summary(continued) - 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