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

Outline

I. Density-Matrix Renormalization-Group Method1.1 Problem1.2 Basic idea of DM truncation1.3 Algorithm : infinite-system & finite-system method1.4 Characteristics of DMRG method

II. Magnetic Properties of Nanographite Ribbons2.1 Introduction2.2 tight-binding model on nanographite ribbons2.3 electron-electron coupling2.4 Prospect of future studies

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

jiji

jijiji

ji

iiiji

ji

SSJH

SSJcctH

nnUcctH

,

,,,

.

,,,,.

)(

)(

h.c.

h.c.

Hubbard model :

t-J model :

Heisenberg model :

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

1.2 Basic idea of Density-Matrix truncation

1.2.1 Truncation of Hilbert space

Exact Diagonalization

L site system

・・・・・

1s 2s 3s 4s 5s Ls

basis : Ll ssss 21

ls

s sal

lw.f. :

'' llii sHsH Hamiltonian :

# of basis : nL (n : degree of freedom/site) ･･･ exponential growth with L

memory overflow occurs at quite small L

Reduction of Hilbert space by truncation

L sites

・・・・・・

block l : Ll sites block r : Lr sites

li ri: nLl - basis : nLr - basis

basis for whole system :

: nLl nLr = nL -basisrl iii

truncate !!

li~

: m -basis

rl iii ~whole system : : m nLr - basis

･･･ if m is small enough, Hii' is diagonalizable

Ll + Lr = L

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 ?

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 Hl

Hl

Hlr

Hr

H = Hl + Hlr + Hr

keep the m-lowest eigenstates of Hl as a basis set

algorithm of RSRG

・・・

HL

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

Hb

1~

LH

1~

LH

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)

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

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 : jiji

jigs ,

,

i j

j

jijiii ,',*)',(

Density Matrix for the left block

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

2

target

~ trunc･･･ Truncation error is minimized

m

i

iP1

)(1It can be shown that (where P(i) : i-th eigenvalue of DM)

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 (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

・・・・・・

H : n2m2×n2m2 matrix : diagonalizable

i l : nm -basis

・・・

form and diagonalize DM

new block newi : m -basis

substitute

･･･ right block is the reflection of the left block

1.3.2 finite-system algorithm

・・・・・・

・・・

form and diagonalize DM

fixed L

・・

・・・

use as a block with Ll+1 sites

Ll sites Lr sites1 1

draw a block with Lr sites

stock

After a few iterations of the sweep procedure one can obtain highly accurate results on a finite (L sites) system

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.

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).

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

L-sitesEquivalent to L/-chain system

mL/states are needed

# of states we must keep increases exponentially with the system width

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

graphite : sp2 carbons materialElectron state around Fermi energy Ef

= -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

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. : 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-C60 : 2D polymerized rhombohedral C60 phase (Makarove et al., Nature 413 718(2001).)

- Ferromagnetism with Tc ~ 500 (K)

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)

Magnetic field(kOe)

RhC60 (Makarova et al., Nature 413, 716 (2001).)

Hysteresis loop Saturation of magnetization

T-dependence of saturated magnetization

Tc ~ 500 (K)

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

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

)( ,,.

h.c.jiji

cctH

ji.

: sum only between nearest-neighboring sites

t ~ 3 (eV)

: sublattice A

: sublattice B

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 sheetHowever,

presence of edges in graphite ribbons makes essential modification on the band structure

Zigzag ribbon : (almost) flat band appears at E = Ef = 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)

Band structure of armchair ribbon

L=4 L=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

)13()23

1cos(22

)3()13

cos(22

)13(0)0(

nLtn

nt

nLtn

nt

nLka

Band structure of zigzag ribbon

L=4 L=6L=5

L = 30

(Wakabayashi, Ph.D Thesis(2000))

DOS has a sharp peak at Fermi energy E = Ef = 0

Flat band appears for 2/3 < k <

“edge state”

Harper’s eq. : Apply H to one-particle w.f. : 0,

ijji

ijc

0 cba

If E = 0, 0Ha

c

b

0)1(

0 rike)1(

0 rike ikre0 )2(

0 rike

Amplitude :

2/cos201 k

22/cos202 k

mkm 2/cos20

Wave function for E = 0 and wave number k on A-sublattice

(Wakabayashi, Ph.D Thesis(2000))

k = k = 2k = 7k = 8

perfect localization penetration

- 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

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 :

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).

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

0,,

iizi nnS

(Wakabayashi et al., J.Phys.Soc.Jpn. 65,1920(1996).)

(Okada and Oshiyama, PRL 87,146803 (2001).)

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 BAtotal NNS 21

(where NA(NB) is # of sites on A(B) sublattice)

In the case of graphite ribbons, NA=NB

0,, iizi nnS

We perform DMRG calculation on Hubbard model

- zigzag ribbon : N = 2, 3

- # of kept states m : up to typically 1000.

i

iijiji

nnUcctH,,,,

.

)( h.c.

charge gap :

spin gap :

local spin polarization :

Spin-spin correlation :

220220220 ,2,1,1 MMMMMMc EEE

220220 ,1,1 MMMMs EE

(M=NL: # of sites, E0(n↑,n↓) : lowest energy in the subspace (n↑,n↓) )

,,2

1ii

zi nnS

,,,,,,,,4

1jijijiji

zj

zi nnnnnnnnSS

N=2 Zigzag ribbon

Charge (spin) gap opens

for )( spinchargecc UUU

0spincharge cc UU

N=3 Zigzag ribbon

Charge gap opens

for 0charge cUU

Distribution of Szi for N = 2 in the lowest energy state of

Zigzag edge favors spin polarization

U=0

U=4

U=1

1i

zi

ztotal SS

U=0

U=1

U=4

Distribution of Szi for N = 3 in the lowest energy state of 1

i

zi

ztotal SS

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

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 : 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

effective spin

effective spin

Singlet stateJeff

Heisenberg model on zigzag ribbon

: Effective model for spin-degree of freedom

s(N=4) < s(N=2)

ji

ji SSJH,

Spin gap

Distribution of Szi for N = 4 in the lowest energy state of 1

i

zi

ztotal SS

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

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

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 = Ef = 0

(might be) relevant to paramagnetism in nanographite

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

)( spinchargecc UUU 0spincharge cc UU:

0,,

iizi nnS for all site