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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
zigzag ribbon
armchair ribbon
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
2.2
2.3
2.4
Introduction
tight-binding model on nanographite ribbons
electron-electron coupling
Prospect of future studies
I. Density-Matrix Renormalization-Group Method
1.1 Problem
investigation of the properties of
strongly correlated systems on lattice sites
Hubbard model :
H  t 
(c  c


H  t 
(c


i . j 
t-J model
:
i . j 
Heisenberg model :



i,

i ,
j ,
c j ,
 h.c.)  U  ni , ni ,
i
 
 h.c.)  J  Si  S j
i , j 
 
H  J  Si  S j
i , j 
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
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
- minus sign problem
- flexible
- slow convergence at low-T
Variational Monte Carlo method
- rather large system size
- results depend on the trial function
We want to treat larger system
with smaller memory/ CPU time
controlled (unbiased) accuracy
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
s1
s2
s3
s4
s5
sL
・・・・・
L site system
basis :
w.f. :
s  
l
s1  s2  sL
   asl  sl 
sl 
Hamiltonian :
Hii'  sl  H sl '
# 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
basis for whole system :
L sites
・・・
block l : Ll sites
il : nLl - basis
i  il  ir : nLl nLr = nL -basis
・・・
block r : Lr sites
Ll + Lr = L
ir : nLr - basis
truncate !!
~
il
: m -basis
~
whole system : i  i l  ir : m nLr - basis
･･･ if m is small enough, Hii' is diagonalizable
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 ?
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
H = Hl + Hlr + Hr
Hlr
・・・
・・・
Hl
Hr
diagonalize a block Hamiltonian Hl
keep the m-lowest eigenstates of Hl as a basis set
algorithm of RSRG
Hb
HL
・・・
L sites (m-basis)
HL+1=HL+Hb
L+1 site (nm-basis)
n-basis
(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 HL  1 (m×m matrix)
into the new basis
~
(v) Go to (ii) by Substituting H
L  1 for HL
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?
reconsideration of RSRG
・・・
・・・
Isolate a part of system
・・・
(ex.) one-particle in a 1D box
keep the low-energy states of a block
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
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 : gs   i, j i j
i, j
・・・
・・・
i
j
Density Matrix for the left block
 (i, i' )   *i, ji ' , j
j
basic scheme : keep the eigenstates of ρ
with m-largest eigenvalues as a basis set
The basis set with DM scheme is
optimal to keep the information of the target state
~



･･･Truncation error
trunc  target
2
is minimized
m
It can be shown that
  1   P(i ) (where P(i) : i-th eigenvalue of DM)
i 1
Calculations become more accurate as m gets larger
･･･ m : controlling parameter of DMRG
the truncation error
rapidly decreases with m
(In many cases,)
very high-precision can be
achieved with feasible m
1.3 Algorithm of DMRG
1.3.1 Infinite-system algorithm
H : n2m2×n2m2 matrix : diagonalizable
・・・
i
・・・
 l
: nm -basis
･･･right block is the reflection
of the left block
substitute
form and diagonalize DM
・・・
new block
inew
: m -basis
(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
1.3.2 finite-system algorithm
fixed L
・・・
Ll sites
・・・
1
1
Lr sites
form and diagonalize DM
・・・
stock
use as a block
with Ll+1 sites
・・・
draw a block
with Lr sites
・・
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
T.Hikihara and A. Furusaki,
PRB58, R583 (1998).
Numerical data is in excellent agreement with exact results
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-sites
Equivalent to L/x-chain system
mL/x 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
Nanotube : 1D
Nanographite
ribbon : 1D
Graphite
Nanoparticle : 0D
graphite : sp2 carbons material
Electron state around Fermi energy Ef
= p-electron network on honeycomb lattice
(# of p-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. : 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)
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))
RhC60 (Makarova et al., Nature 413, 716 (2001).)
Magnetic field(kOe)
Hysteresis loop
T-dependence of
saturated magnetization
Tc ~ 500 (K)
Saturation of magnetization
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
Zigzag ribbon
Armchair ribbon
Edge bonds are terminated
by hydrogen atoms
Definition of the site index
i=1
N = finite, L →∞ :
zigzag ribbon
2
3
L = finite, N →∞ :
armchair ribbon
N
j = 1 2 3 4 5 6
･･････
: sublattice A
: sublattice B
L
Tight-binding model
H  t 

(
c
 i, c j ,  h.c.)
i . j   

i. j 
: sum only between nearest-neighboring sites
t ~ 3 (eV)
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
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 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
D a ( k  0) 
0
n


2 2t cos(
p )  t
3n  1


n 1


2 2t cos(
p )  t
3n  2


( L  3n  1)
( L  3n)
( L  3n  1)
Energy gap of tight-binding model
can be obtained exactly
Band structure of zigzag ribbon
L=4
L=5
L=6
(Wakabayashi,
Ph.D Thesis(2000))
L = 30
Flat band appears
for 2p/3 < k < p
DOS has a sharp peak
at Fermi energy E = Ef = 0
“edge state”
Harper’s eq. : Apply H to one-particle w.f. :
   ijcij 0
i, j
b
a
c
If E = 0, H   0
a  b  c  0
Wave function for E = 0 and wave number k on A-sublattice
0eik ( r 1) 0eikr 0eik ( r 1) 0eik ( r  2)
Amplitude :
0
1  0  2 cosk / 2
2  0  2 cos k / 2 
2
m  0  2 cosk / 2
m
(Wakabayashi,
Ph.D Thesis(2000))
k=p
k = 8p/9
k = 7p/9
perfect localization
k = 2p/3
penetration
- 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
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
Mean-field analysis
(Wakabayashi et al.,
J.Phys.Soc.Jpn. 65,1920(1996).)
Infinitesimal interaction U
of Hubbard type causes
spontaneous spin-polarization
Siz  ni ,  ni ,  0
around zigzag edge sites
(Okada and Oshiyama, PRL 87,146803 (2001).)
DFT calculation
Appearance of
spontaneous spin-polarization
at zigzag edge
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 Stotal  12 N A  NB
(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 Siz  ni ,  ni ,  0 is prohibited
Detailed investigation on magnetic properties is desired.
We perform DMRG calculation on Hubbard model
H  t 

(
c
 i, c j ,  h.c.)  U  ni,ni,
i . j   
i
- zigzag ribbon : N = 2, 3
- # of kept states m : up to typically 1000.
charge gap : Dc  E0 M 2  1, M 2   E0 M 2 1, M 2   2E0 M 2 , M 2 
spin gap :
Ds  E0 M 2  1, M 2 1  E0 M 2 , M 2 
(M=NL: # of sites, E0(n↑,n↓) : lowest energy in the subspace (n↑,n↓) )
local spin polarization :
Spin-spin correlation :
S
z
i

1

ni ,  ni ,
2
Siz S jz 


1
n n  ni , n j ,  ni , n j ,  ni , n j ,
4 i , j ,

N=2 Zigzag ribbon
Charge (spin) gap opens
for U  Uccharge (Ucspin )
Uccharge  Ucspin  0
N=3 Zigzag ribbon
Charge gap opens
for U  U ccharge  0
z
  Siz  1
Distribution of Szi for N = 2 in the lowest energy state of Stotal
i
U=0
U=1
U=4
Zigzag edge favors spin polarization
z
  Siz  1
Distribution of Szi for N = 3 in the lowest energy state of Stotal
i
U=0
U=1
U=4
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 spin
Singlet state
Jeff
- bulk sites form spin-singlet state
- 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
Heisenberg model on zigzag ribbon
 
H  J  Si  S j
: Effective model for spin-degree of freedom
i , j 
Spin gap
Ds(N=4) < Ds(N=2)
z
  Siz  1
Distribution of Szi for N = 4 in the lowest energy state of Stotal
i
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
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
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
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(continued)
Effect of electron-electron couplings
- zigzag ribbon
charge (spin) gap appears for U  U ccharge (Ucspin ) : Uccharge  Ucspin  0
ground state is spin-singlet : Siz  ni ,  ni ,  0 for all site
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

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