Journal of the Physical Society of Japan
Vol. 77, No. 5, May, 2008, 054703#2008 The Physical Society of Japan

Magnetism as a Mass Term of the Edge States in Graphene
Ken-ichi S ASAKI1;2  and Riichiro SAITO1;2
2

1
Department of Physics, Tohoku University, Sendai 980-8578
CREST, Japan Science and Technology Agency, Sendai 980-8578(Received January 28, 2008; accepted February 26, 2008; published April 25, 2008)

The magnetism by the edge states in graphene is investigated theoretically. An instability of thepseudo-spin order of the edge states induces ferrimagnetic order in the presence of the Coulombinteraction. Although the next nearest-neighbor hopping can stabilize the pseudo-spin order, a strong
Coulomb interaction makes the pseudo-spin unpolarized and real spin polarized. The magnetism of theedge states makes two peaks of the density of states in the conduction and valence energy bands near the
Fermi point. Using a continuous model of the Weyl equation, we show that the edge-induced gauge ﬁeld
and the spin dependent mass terms are keys to make the magnetism of the edge states. A relationshipbetween the magnetism of the edge states and the parity anomaly is discussed.
KEYWORDS: graphene, edge state, magnetism, Coulomb interaction, Weyl equation, gauge field, mass term,
parity anomaly
DOI: 10.1143/JPSJ.77.054703

1.

Introduction

The electronic properties of graphene1–3) have attracted
much attention mainly because of its relativistic character oflow energy electronic excitation. The energy band structureof graphene consists of two Dirac cones at the K and K0
points in the k-space. The electron dynamics around each
Dirac point is approximated by the Weyl equation which
describes a massless particle. When we consider solutionsof the Weyl equation for a ﬁnite (or semi-ﬁnite) graphene
cluster (ribbon) with the zigzag edges, the spatially-localizededge states4) exist around the Fermi energy.5) The existenceof the edge states depends on the shape of the edge for agraphene cluster. For example, the zigzag edge yields theedge states while the armchair edge does not. The energy
dispersion for the edge states which appears only between
the two Dirac points smoothly connects to the energy
dispersion of the delocalized states.6) A large local density ofstates (LDOS) by the edge states may induce magnetism4)
and superconductivity7) near the zigzag edge. The existenceof a mass for the Dirac particle is an important issue because
it gives rise to an energy gap at the Dirac points and relates
to the ordered states. We examine a mechanism that the
Coulomb interaction makes a mass and a magnetism.
Fujita et al. discussed the eﬀect of the Coulomb interaction on the edge state, in which the electronic spins are
localized at the edge to form a ferromagnetic state at
one zigzag edge and another ferromagnetic state with the
opposite spin at another zigzag edge.4) The occurrence of themagnetism is investigated by ﬁrst principles calculations,
too.8–10) If the ferromagnetic state appeared at one edge, wewould expect that two peaks for up and down spin states
appear in scanning tunneling spectroscopy (STS). However
this situation seems to be inconsistent with the STS measurements in which they observed only one LDOS peak near
the zigzag edge of graphite.11–15) Klusek et al.11) found apeak of LDOS in the energy range of 20 – 250 meV above
the Fermi level at the edges of circular pits on graphite
surface. Kobayashi et al.12,13) and Niimi et al.14,15) independently observed a peak in the LDOS below the Fermi
E-mail: sasaken@ﬂex.phys.tohoku.ac.jp

energy by 20 – 30 meV. Since the peak appears only conduction (Klusek et al.) or valence energy band (Kobayashi
et al. and Niimi et al.), it suggests that the edge states do notmake a magnetism. Thus it is an interesting problem forunderstanding the occurrence of the ferromagnetic order atthe edge in the presence of the Coulomb interaction. When
we see the calculated results of Fujita et al., the polarized
spin appears for the edge states at a much small on-site
Coulomb interaction U value compared with the nearest
neighbor interaction 0 (see Fig. 5 in ref. 4). Although we
reproduce their results numerically, the results are very
surprising. A possible reason why the spin ordering occurs
for such a small U is due to a special fact that the wavefunction of the edge states has an amplitude only one of thetwo sublattices (A and B) and thus the nearest neighborinteraction is suppressed. When we introduce the next
nearest neighbor interaction, n , the spin polarization around
U=0  0 disappears and spin ordering appears from ﬁnite
values of U depending on n , which we will show in thispaper.
The wavefunction for two sublattice structure is referred
to as the ‘‘pseudo-spin’’. An edge state can be described by apseudo-spin polarized state.6) A pseudo-spin structure givesa rich variety of interesting physical phenomena not only theedge states but also the extended states. For example, theabsence of backward scattering mechanism is relevant to thispseudo-spin nature,16,17) in which a 2 rotation of a pseudospin wavefunction around the K-point in the two-dimensional Brillouin zone does not give the original wavefunction but gives minus sign to the wavefunction. Thus, thepseudo-spin is quite similar to the real spin in the real space.
In this paper, we show that the pseudo-spin also plays animportant role for the magnetism (or real spin) of the edge
states, which is shown by a numerical analysis of the latticemodel and by a analytical study of the Weyl equation.
This paper is organized as follows. In §2 we explain themodel Hamiltonian and introduce symmetric and antisymmetric variables for the pseudo-spin. In §3 we show
numerical results for the ground state of the model. In §4
we use a continuous model to examine the mechanism of themagnetism of the edge states. Discussion and summary are
given in §5.

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J. Phys. Soc. Jpn., Vol. 77, No. 5

K. S ASAKI and R. SAITO
Table I.

2.

Parities with respect to spin and pseudo-spin.

Pseudo-Spin Representation of Hamiltonian

The Hamiltonianis given by H0" þ H0# þ HU whereP
H0s  0 hi; ji cys;i cs; j (s ¼ "; #) is the nearest-neighbortight-binding Hamiltonian (0  3 eV is the hopping integral), and HU is the Hubbard on-site interaction. HU iswritten asXn" ðrÞn# ðrÞ;ð1Þ
HU ¼ Urwhere U is the on-site energy and n" ðrÞ (n# ðrÞ) is the density
operator of up (down) spin electron at site r. Since thehexagonal lattice consists of two sublattice, A and B, HU isgiven as a summation over unit cells as
X X
HU ¼ Un"; p ðru Þn#; p ðru Þ;ð2Þ
ru p¼A;B

where n"; p ðru Þ (n#; p ðru Þ) is the density operator of up (down)
spin electron at p-sublattice (p ¼ A; B), and ru denotes the
position of a unit cell. For a unit cell, we introduce a density
and a magnetization at ru as np ðru Þ ¼ n"; p ðru Þ þ n#; p ðru Þ andmp ðru Þ ¼ n"; p ðru Þ  n#; p ðru Þ, respectively. Hereafter ru foreach variable is omitted for simplicity. From np and mp , wedeﬁne density and magnetization for a unit cell asn ¼ nA þ nB ;m ¼ mA þ mB :

ð3Þn and m are symmetric with respect to the sublattice. Here we
introduce pseudospin order and antiferromagnetic order for aunit cell,pn ¼ nA  nB ;pm ¼ mA  mB ;

ð4Þwhich are anti-symmetric with respect to the sublattice. pn(pm ) represents charge (spin) polarization within the hexagonal unit cell. HU can be rewritten in terms of n; m; pn , andpm as
U X 2HU ¼
n þ p2n  m2  p2m :ð5Þ
8 ruThis representation of HU shows that not only non-vanishingmagnetization (hmi 6¼ 0) but also antiferromagnetic (hmi ¼ 0
and hpm i 6¼ 0) or ferrimagnetic (hmi 6¼ 0 and hpm i 6¼ 0) spin
conﬁguration are favored to decrease HU where hOi denotes
the expectation value of operator O for the ground state.
By applying
the mean-ﬁeld approximation to eq. (2),P
HU ¼ U ru ; p hn"; p in#; p þ n"; p hn#; p i  hn"; p ihn#; p i,theHamiltonians for up and down spin electrons in graphene
are given as follows:U X hnA  mA in";A0H"  H0" þ;n";B0hnB  mB i
2 ru




ð6Þ
U X hnA þ mA in#;A0
H#  H0# þ:n#;B0hnB þ mB i
2 r

pn

pm

n

m

z

Spin (" $ #)

þ



þ



þ

Coupling
Bem

Pseudo-spin (A $ B)





þ

þ



Bq

where z ¼ diagð1; 1Þ and I ¼ diagð1; 1Þ. The pseudospin variables are proportional to z and aﬀect magnetization of the edge states as we will show in the following
sections.
In Table I, we show the parities of pn , pm , n, m, and z forchanging the direction of spin and pseudo-spin. The interaction terms in eq. (7) are invariant with respect to pseudospin parity: pn ! pn , pm ! pm , n ! n, m ! m, andz ! z , and we have H" $ H# for spin parity: pn ! pn ,pm ! pm , n ! n, m ! m, and z ! z . Since HU iseven parity with respect to spin and pseudo-spin, it isexpected that a ground state is realized by spontaneous
symmetry breaking if H0" þ H0# ( H0 ) is symmetric, too.
H0 is symmetric with respect to spin but asymmetric withrespect to pseudo-spin due to the presence of the zigzag
boundary. This can be explained as follows. A magnetic
ﬁeld, Bem ðrÞ, breaks the spin degeneracy of the ground stateby the Zeeman term, Bem ðrÞ  mðrÞ, which is odd parity
with respect to spin. Similarly, we can deﬁne a pseudo-magnetic ﬁeld, Bq ðrÞ, that couples to z and breaks the degeneracy of the pseudo-spin parity. Bq ðrÞ is deﬁned by the rotation of a deformation-induced gauge ﬁeld Aq ðrÞ as Bq ðrÞ ¼
r Aq ðrÞ,18) which is similar to that Bem ðrÞ is given by therotation of an electro-magnetic gauge ﬁeld Aem ðrÞ.
In the previous paper, we derived Aq ðrÞ and Bqz ðrÞ for
zigzag edges.5) Aq ðrÞ is given by a small change of the
hopping integral from 0 between an A-atom and one of
the nearest B-atoms [a ðrÞ where a ¼ 1; 2; 3, see Fig. 1] as
1
vF Aqx ðrÞ ¼ 1 ðrÞ  2 ðrÞ þ 3 ðrÞ ;
2
pﬃﬃﬃ
ð8Þ


3
vF Aqy ðrÞ ¼
2 ðrÞ  3 ðrÞ :2

zigzag

Klein
A

B

δγ 2
δγ 1
δγ 3

u

The Hamiltonians of eq. (6) can also be rewritten usingeqs. (3) and (4) as


n";A
UX
H" ¼ H0" þ½ðhpn i  hpm iÞz þ ðhni  hmiÞI
;
4 ru
n";B


n#;A
UX
H# ¼ H0# þ½ðhpn i þ hpm iÞz þ ðhni þ hmiÞI
;
4 ru
n#;B
ð7Þ

Fig. 1. Local modulation of the hopping integral is deﬁned by a(a ¼ 1; 2; 3), which is related to Aq ðrÞ through eq. (8). When we cut thebonds on the dashed (dotted) line by setting 1 ¼ 0 and 2 ¼ 3 ¼ 0(1 ¼ 0 and 2 ¼ 3 ¼ 0 ), we obtain the zigzag (Klein) edge.

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J. Phys. Soc. Jpn., Vol. 77, No. 5

Zigzag
A B

(b)

Klein
B A

(c) Klein
A

(b)

(a)

ZigzagA.
.

0.3
Local density of states (a.u.)(a)K. S ASAKI and R. SAITO

Density / Unit-cell

0.6
0.4

/

B

A

0.2
0
-0.2

Φ
Φ /

0

0

4

-1

Φ /

Fig. 2. Conﬁguration of Aq ðrÞ and Bqz ðrÞ for (a) the zigzag edge and (b)the Klein edge. When we cut the graphene sheet at the dashed line, theedge and Aq ðrÞ ¼ ðAqx ðyÞ; 0Þ appears. The direction of Aq ðrÞ for the Kleinedge is opposite to that of the zigzag edge. (c) A graphene with the zigzag
edge and the Klein edge.

When we cut the bonds on the dashed line at y1 in Fig. 1,
we get a zigzag edge, while we get a Klein edge19,20) for the
dotted line at y0 . In the case of the zigzag edge, since1 ðyÞ 6¼ 0 and 2 ðrÞ ¼ 3 ðrÞ ¼ 0, we get vF Aqx ðyÞ ¼
1 ðyÞ (> 0) and vF Aqy ¼ 0, and thus, Bqz ðrÞ ¼ @y Aqx ðyÞ asshown in Fig. 2(a). On the other hand, in the case of
the Klein edge, since 1 ðrÞ ¼ 0 and 2 ðyÞ ¼ 3 ðyÞ 6¼ 0,
we get vF Aqx ðyÞ ¼ 2 ðyÞ (< 0) and vF Aqy ¼ 0, and thus,
Bqz ðrÞ ¼ @y Aqx ðyÞ as shown in Fig. 2(b). The directions of
Aq ðrÞ and Bq ðrÞ for the Klein edge are opposite to thosefor the zigzag edge [Figs. 2(a) and 2(b)]. Kusakabe andMaruyama discussed the edge state magnetism for agraphene cluster with a zigzag edge at one edge and a Kleinedge at another edge,9) as shown in Fig. 2(c). In all cases, we
can explain the edge structure within the same frame.
Since Bqz ðrÞ ¼ @y Aqx ðyÞ appears at the zigzag edge,5) H0
can induce a pseudo-spin order, hpn i 6¼ 0, near the edge. In
the next section, we will show numerically that H0 of a
zigzag nanotube breaks the pseudo-spin parity of pn , which
is important to obtain magnetism for the total Hamiltonian,H" þ H# .
3.

2

0.1

Numerical Results and Analysis

In this section, we show numerical results for the groundstate of the mean-ﬁeld Hamiltonian H" þ H# in eq. (6).
hn=4i, hpn i, hpm i, and hmi are plotted for ð50; 0Þ zigzag nanotube with length L  4 nm, and LDOS curves are calculatedfor ð100; 0Þ zigzag nanotube with length L  20 nm. Hereð50; 0Þ (or ð100; 0Þ) is the chiral index of a zigzag nanotube.21) We set the origin of the Fermi energy EF ¼ 0 as
Hs ¼ U=2.
In Fig. 3(a), we plot hn=4i and hpn i in the case of U ¼ 0.
Since U ¼ 0, hmi ¼ hpm i ¼ 0. The solid (dashed) curves are
the results for the Fermi energy EF ¼ 0:01 eV (þ0:01 eV).
hn=4i and hpn i are modulated near the edges and their difference from the constant values (hn=4i ¼ 0:5 and hpn i ¼ 0)
is due to the presence of the edge states.4) The wavefunctionof the edge states is localized near the edges so that hn=4i
is diﬀerent from 0.5 (i.e., half ﬁlling n ¼ 2) only near theedges. Moreover, the wavefunction of the edge states ispolarized about the pseudo-spin and hpn i is nonzero, too.
hn=4i and hpn i are unstable against a small change of EF due

01

Fig. 3. (a) Self-consistent solution of hn=4i and hpn i for U ¼ 0 and
EF ¼ 0:01 eV. Due to the presence of the edge states consisting a ﬂatenergy band at EF ¼ 0, hn=4i and hpn i are unstable against the smallchange of EF . hOðru Þi depends only on the distance from an edge due torotational symmetry around the axis of the tube. (b) Corresponding LDOS
curve at L  0:5 nm. The degenerate spin up and down edge states make a
sharp peak at EF ¼ 0 in the case of U ¼ 0.

(b)

(a)
0.6
0.4

γ

0.2

.

Local density of states (a.u.)Φ

Density / Unit-cell

Φ

0.2

/

0.2
0
-0.2
0

2

4

γ

0.1

0

-1

0

1

Fig. 4. (Color online) (a) Self-consistent solution of hn=4i, hmi, hpn i, andhpm i for U ¼ 0 and EF ¼ 0:01 eV. The spin around the edge ispolarized and jhpm ij  jhmij shows that spin is a ferrimagnetic conﬁguration. (b) The LDOS curve at L  0:5 nm shows that spin up edge states(denoted by up-arrow) and spin down edge states (denoted by downarrow) make two sharp peaks (solid curve) near EF ¼ 0 when U 6¼ 0.

to the ﬂat energy band of the edge states. In Fig. 3(b), weplot the LDOS at L  0:5 nm in the case of EF ¼ 0. The spinup edge states and spin down edge states are degenerate in
the case of U ¼ 0 so that they make a sharp LDOS peakat EF ¼ 0. To show the (smooth) LDOS curve, we put aconstant width (0.05 eV) for each state.
In Fig. 4(a), we plot self-consistent solution of hn=4i, hmi,
hpn i, and hpm i in the case of U ¼ 0 . The result shows that,
hn=4i  0:5 and hpn i  0 hold at each hexagonal unit cell,and hmi and hpm i become nonzero near the zigzag edges.
Since HU can stabilize hni and hpn i according to eq. (5), thepseudo-spin polarization (hpn i 6¼ 0) which exists for U ¼ 0
disappears for the ground state due to a ﬁnite value of U.
The corresponding LDOS curve is shown in Fig. 4(b).
Because of U, the spin up (down) edge states are shiftedabove (below) the Fermi energy so that the ferrimagnetic
order (hmi 6¼ 0 and hpm i 6¼ 0) appears and that two LDOS
peaks appear around EF ¼ 0. The LDOS curve in the case of
U ¼ 0 is also shown in Fig. 4(b) for comparison.
The numerical results can be explained qualitatively using
eq. (7) as follows. When hpn i ¼ 0 and hpm i > 0, the energy

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J. Phys. Soc. Jpn., Vol. 77, No. 5

K. S ASAKI and R. SAITO

(b)

(a)

γ

3

γ

2

γ
γ

1

0

0

1

γ

. γ
. γ

2

2

1

0

0

1

γ

. γ

γ

2

3

. γ

0.1

-1

0

1

γ
. γ
. γ

(d) 0.2

. γ
. γ

γ

γ

0

3

(c)
3

Local density of states (a.u.)

0.2

Local density of states (a.u.)of a spin up electron is shifted below the Fermi energy
for a pseudo-spin up state (z ¼ 1). On the other hand, theenergy of a spin down electron is shifted above the Fermi
energy for the same state (a pseudo-spin up state). Thus,
when EF ¼ 0, the ground state has a ﬁnite positive valueof hpm i and hmi, which lowers the energy of a spin up
electron due to the last term in eq. (7) and stabilizes theground state conﬁguration further. In fact, the second termsof the right-hand side of eq. (7) give rise to an energy gap inthe energy spectrum. As we will show using a continuous
model for H0s in §4, the appearance of the gap will become
more clear since the term proportional to z acts as a massterm of Dirac fermion. It is noted that hpn i ¼ 0 is consistent
with the presence of a gap, and non-vanishing hpm i gives
diﬀerent signs of the mass terms for spin up and down
electrons.
Next, we consider the next-nearest neighbor (nnn) hopping, n , which is an intrinsic perturbation to the edge states.
In the previous paper, we showed that the nnn interaction,
Hnnn , gives a ﬁnite energy bandwidth to the edge states,
W ¼ n .22) n  0:10 is obtained by ﬁrst-principles calculation using the local density approximation.23) A ﬁnite
energy band width of the edge states suppresses the above
mentioned hpn i’s instability with respect to a small changeof EF (see Fig. 3).
To see the relationship between hpn i and hpm i in detail,we ﬁrst deﬁne the net pseudo-spin P
order Pn and the averaged
antiferromagnetic
order
P
:
P

m
n
u jhpn ðru Þij=n and Pm 
P
hp
ðr
Þi=n
where
the
summation
is taken over allmuuhexagonal unit cells. Since not only the edge states but also
extended states can contribute to Pi (i ¼ n; m), we consider
the diﬀerence between Pi for a tube with the zigzag edges(Ptube) and that for a corresponding periodic torus systemi
torus(Ptorus) which does not have edge. Pedge
 Ptubecanimm  Pmbe used to show the magnetism for the edge states.
In Fig. 5(a), we plot Pedge(solid curve) and Pedge(dashedmncurve) as a function of U=0 for n ¼ 0 (black), 0:10(red), and 0:20 (blue). When n ¼ 0, no pseudo-spin orderPedge
 0 for any positive value of U, while the antiferronmagnetic order Pedge
increases until U=0  2:1. Pedgemmdecreases when U=0 > 2:1. However Fujita et al. showedthat the magnetism due to the extended states increases.4) In
case of a ﬁnite value of n , the antiferromagnetic order issuppressed Pedge
 0 up to a ﬁnite value of U and themmagnetism (in the case of n ¼ 0) appears discontinuously
above the critical value of U. On the other hand, the pseudospin order appears Pedge
6¼ 0 below the critical value of U.
nThus, when EF ¼ 0, we see that n controls the occurrenceof the pseudo-spin order (Pedgen ) and the antiferromagnetic
order (Pedgem ) exclusively. In Fig. 5(b), we plot the LDOS
curves when U ¼ 0 for n ¼ 0:10 (solid curve) and n ¼
0:20 (dashed curve). When n ¼ 0:10 , magnetic order isrealized so that there are two peaks in the LDOS curve.
When n ¼ 0:20 , the pseudo-spin order Pedge
is realized andnthere is one peak below the Fermi energy in the LDOS
curve. In the case of U ¼ 0, the peak position appears at
E ¼ n .22) The pseudo-spin order
P shifts the peak position
above E ¼ n due to ðU=4Þ ru hpn iz in eq. (7). Thisis a possible reason why Kobayashi et al.12,13) and Niimi
et al.14,15) observed a peak in the LDOS below the Fermi
energy by 20 – 30 meV not by n  0:3 eV.23) Since the most

γ

0.1

0

-1

0

1

Fig. 5. (Color online) (a) Self-consistent solution of Pedge(solid curve)mand Pedge(dashed curve) as a function of U=0 for n ¼ 0 (black), 0:10n(red), and 0:20 (blue) in the case of EF ¼ 0. n controls the appearance
edgeof the pseudo-spin order (Pedgen ) and of the ferrimagnetic order (Pm ).
(b) The LDOS curves for n ¼ 0:10 (solid curve) and n ¼ 0:20(dashed curve) when U ¼ 0 and EF ¼ 0. (c) Pedge(solid curve) and Pedgemn(dashed curve) for n ¼ 0:20 with EF ¼ 0:10 . This corresponds tothat EF is located at the center of the edge energy band. The small stepon Pedge(at U=0 ¼ 1) is due to a ﬁnite diameter of a tube. (d) Themcorresponding LDOS curve when U ¼ 0 in (c). The peak structure issensitive to the position of EF .

localized edge states have pn ¼ 1 at a unit cell of the edgesite and the energy is given by n , we see that Hnnn ¼
n jpn j. Thus, the pn dependent energy density at a unit cellof the edge site, HU þ Hnnn ¼ ðU=8Þp2n  n jpn j þ   , may
become a negative value when the ground state shows apseudo-spin order, hpn i 6¼ 0. In fact, in the case of n ¼
0:20 , the magnetization disappears even for U  0 .
In Fig. 5(c), we plot Pedge(solid curve) and Pedge(dashedmncurve) as a function of U=0 for n ¼ 0:20 with EF ¼
0:10 . The results show that, the critical U value decreases
as compared with the case of EF ¼ 0 shown by the blue
curves in Fig. 5(a). The corresponding LDOS curve for U ¼
0 is plotted in Fig. 5(d). We see that there are one sharp
peak and a broaden peak. Although Pedge
Pedge
when
m
n
U ¼ 0 , this two peaks structure is not so clear as the two
peaks in the case of EF ¼ 0 with n ¼ 0:10 .
4.

Continuous Model

To understand the edge magnetism in detail, we solveeq. (7) analytically by means of a continuous model. It will
be shown that the magnetic order is explained by a gauge
ﬁeld for the edge states and spin-dependent mass term.

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J. Phys. Soc. Jpn., Vol. 77, No. 5

K. S ASAKI and R. SAITOThe low energy states around EF ¼ 0 consist of electrons
near the K-point and K0 -point. Since the K-point and K0 point are related to each other by time-reversal symmetry,it is suﬃcient to consider only the K-point when Bem ¼ 0.
Then, the low energy Hamiltonian is given by replacing H0s
in eq. (7) with HK asU
Uðhpn i  hpm iÞz  hmiI;4
4
UU
H# ¼ HK þ ðhpn i þ hpm iÞz þ hmiI;4
4

H" ¼ HK þð9Þ

where HK is given byHK  vF   ðp^ þ Aq ðrÞÞ;ð10Þ

and hni which is present in eq. (7) is dropped in eq. (9)because it can be absorbed in the energy. In eq. (9),Hs (s ¼ "; #) operates on a two-component wavefunction,
K
t
K
K
K
K
s ¼ ð A;s ; B;s Þ, where A;s and B;s are the pseudo-spin
up and down states, respectively. In eq. (10), vF is the Fermi
velocity, p^  ð p^ x ; p^ y Þ is the momentum operator, and  ðx ; y Þ where i (i ¼ x; y; z) are the Pauli matrices. Aq ðrÞ ¼
ðAqx ðrÞ; Aqy ðrÞÞ is a deformation-induced gauge ﬁeld that
represents a lattice deformation in the hexagonal unit cell[see eq. (8)].18,24) The corresponding deformation-induced
magnetic ﬁeld, Bqz ðrÞ  @x Aqy ðrÞ  @y Aqx ðrÞ, couples to thepseudo-spin, z . This is shown by HK squared,ð11Þ
H2K ¼ v2F ðp^ þ Aq ðrÞÞ2 þ h Bqz ðrÞz ;where z selects the direction opposite to Bqz ðrÞ in order todecrease the energy.
In the following, we obtain the wavefunction and theenergy eigenvalue E for the following Hamiltonian:


K
ð12Þ
HK þ ms z K
px ;s ðrÞ ¼ E px ;s ðrÞ;where the mass term is given byms Uðhpn i
4

hpm iÞ:where s ¼ "; #, and x (y) is parallel (perpendicular) to thezigzag edge.5) The unknown functions, GðyÞ and gs ðyÞ, andE can be determined by putting eq. (14) into eq. (12). We
obtainhdgs ðyÞ ¼ D coshð2gs ðyÞ þ fs Þ;dydGðyÞ ¼ D sinhð2gs ðyÞ þ fs Þ;
dy

ð15Þ
ð16Þ

ð17Þ
ð18Þ

Here, we consider a localized wavefunction and put GðyÞ ¼
jyj= into eq. (16) where  is localization length of the edge
state and the zigzag edge is located at y ¼ 0. Then we get8 >>  sinh1 h
>(y < 0)
<
D
 
ð19Þ
2gs ðyÞ þ fs ¼
>
h
>
1
>
(y
>
0).
þ
sinh
:
D
Next, we integrate eq. (15) with respect to y from g to g .
By considering g ! 0, only singular functions of Aqx ðyÞ andeq. (19) around y ¼ 0 survive, and we get  Z gh
1
 sinh¼
Aqx ðyÞ dy:
ð20Þ
D
g
Using eqs. (19) and (20), we see from eq. (15) that
px
D¼
Z g

q
cosh
Ax ðyÞ dy

ð21Þ

g

holds except very close to the edge. From eqs. (20) and (21),
we see that  in GðyÞ is given by
!
Z g
h
q
¼ px tanhAx ðyÞ dy :
ð22Þ

g
This result is surprising in the sense that  in the presenceof U is identical to  for U ¼ 0.5) The mass term wouldaﬀect , but it is not the case. The reason for this will be
discussed elsewhere. Finally, we get the energy eigenvalue
from eqs. (17) and (21),E2 ¼ m2s þ

ð13Þ

The mass term depends on real spin, that is, the negative(positive) sign in front of hpm i is for s ¼ " ð#Þ. Equation (12) describes Dirac fermion having a mass, ms , where
the dimension of ms is energy here. In obtaining eq. (12), we
neglect ðU=4Þhmi of eq. (9). The neglected term does notcouple to the pseudo-spin and only shifts the energy positionof each state so that it is not important for our discussion.
Further, we neglect the r dependence of ms in order tosimplify the argument.
Since there is translational symmetry along the edge, theeigenfunction of eq. (12) can be expressed by!
þgs ð yÞepx
x eGð yÞ g ð yÞ ;ð14Þpx ;s ðrÞ ¼ N exp ihe spx þ Ax ðyÞ þ h

where variables D and fs are deﬁned respectively as
ﬃ
1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
E2  m2s ;DvF
mstanhð fs Þ   :E

ðvF px Þ2
:
Z g
q
2
cosh
Ax ðyÞ dy

ð23Þ

g

The energy dispersion relation for the edge states of eq. (23)
is similar to the relativistic energy dispersion relation for theextended state, E2 ¼ m2s þ ðvF px Þ2 .
In eq. (18), we see that the sign of fs depends both on thesigns of ms and E. To obtain a ground state, we ﬁrst consider
the valence states E < 0. Then, we have fs ¼ signðms Þjfs j.
Using eq. (20), we can rewrite eq. (19) as
8Z
1 g q 01>>
þ
Ax ðy Þ dy0  signðms Þjfs j (y < 0)
>
< 2
2
g
ð24Þ
gs ðyÞ ¼
Z g
>
1
1
>
>
Aqx ðy0 Þ dy0  signðms Þjfs j (y > 0).
:
2 g
2
By
puttingeq. (23)
intoR eq. (18),
we
have
R

jfs j  j g g Aqx ðyÞ dyj when j g g Aqx ðyÞ dyj
0. Since
R g q0 for the zigzag edge [see Fig. 2(a)], weg Ax ðyÞ dy
get from eq. (24)8Z
> g q 0<
Ax ðy Þ dy0 (y < 0)g" ðyÞ 
ð25Þ
g
>
:
0
(y > 0),when m" < 0. The wavefunction of this spin up state appears
near the edge consisting of A-atoms (y < 0) in the valence

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J. Phys. Soc. Jpn., Vol. 77, No. 5

K. S ASAKI and R. SAITO

band (E < 0): px ;" ðy < 0Þ / expðjyj=Þt ð1; 0Þ. The wavefunction at the edge consisting of B-atoms (y > 0) is
pseudo-spin unpolarized and the amplitude is negligible dueto the normalization of the wavefunction. Similarly, form# > 0, we have8(y < 0)><0 Z
g
g# ðyÞ 
ð26Þ
Aqx ðy0 Þ dy0 (y > 0).
>
:
g

The corresponding wavefunction of spin down state ispseudo-spin down state appearing only near the edge
consisted of B-atoms (y > 0). It is noted that the spin for
a conduction edge state (E > 0) is opposite to that of avalence edge state. Thus, we obtain local ferrimagnetism
near the zigzag edge.
In Fig. 6, we show how the magnetism appears around thezigzag edge. In Fig. 6(a), if U ¼ 0, the opposite directionof pseudo-spin (the edge state) appears both for y > 0(z ¼ 1) and y < 0 (z ¼ 1). However when U 6¼ 0, due
to the mass term the edge state exists only one of the twosides in Fig. 6(a). When pseudo-spin order is suppressed(hpn i ¼ 0) and antiferromagnetic order appears (hpm i 6¼ 0),
we get a situation of a diﬀerent sign for m" and m# (m" ¼
m# ) from eq. (13). In this case, up (down) spin edge state
appears for y < 0 (y > 0) for the valence band according toeqs. (25) and (26), while down (up) spin edge state appears
for y < 0 (y > 0) for the conduction band. It is consistent
with the numerical result of Fujita et al. for EF ¼ 0,4) inwhich the electrons are occupied only for the valence band.
If EF is shifted from EF ¼ 0 then hpm i will disappear.
When pseudo-spin order appears (hpn i 6¼ 0) and antiferromagnetic order is suppressed (hpm i ¼ 0), we get anothersituation that the sign of m" and m# are the same (m" ¼ m# )from eq. (13). In this case, the energy levels for up anddown spin edge states are degenerate even for U 6¼ 0 [seeFig. 6(c)]. Thus the ground state is not spin polarized but
pseudo-spin polarized. Even in case that m" 6¼ m# , spin up
and down edge states both appear below the Fermi level so
(b)
A B

(a)

–

A B

–

σ
σ

σ

–

σ

that the ground state is still spin unpolarized if signðm" Þ ¼signðm# Þ. It is interesting to note that, in Fig. 6(c), theenergy level position for y < 0 can appear above (below)
EF ¼ 0 when ms > 0 (ms < 0). The sign of ms , that is, ms <0 (ms > 0) for y < 0 and ms > 0 (ms < 0) for y > 0, is
consistent with the numerical results given in §3.
As we have shown in §3, the magnetism of the edge states
is aﬀected by the nnn hopping that can stabilize hpn i. In thecontinuous model, we showed in the previous paper that thennn perturbation works as
Hnnn ¼ n

5.

Discussion

A magnetism of the edge states would give rise to two
LDOS peaks since only spin up (or down) edge states arelocated below the Fermi energy to give a spin polarization in
the ground state. Although we have examined this mechanism using the Hubbard model, the appearance of two
peaks seems to be a model independent consequence of themagnetism of the edge states. The LDOS near the zigzag
edge of graphite has been measured by STS,11–15) but noexperimental group has observed the two peaks in the STS
data. It is possible that the position of the Fermi energy inthese experiments is not suitable for the occurrence of themagnetism (see Fig. 5). Thus if we change of the Fermi
energy, LDOS will give a split of the peak, which is anevidence that the edge states form a magnetism.
We explained the magnetism of the edge states in terms of
the spin dependent mass terms and the deformation induced
gauge ﬁeld. It is known that the mass and a gauge ﬁeld in the
Weyl equation induce the parity anomaly in the groundstate.25–27) The mass term in eq. (12) changes its sign underK
spatial parity with respect to y ! y and K
px ;s ! x px ;s .
The mass term violates the spatial parity and can induce aquantum anomaly in the ground state, which is referred to as
the parity anomaly. By applying the formula of the parity
anomaly25) to our case, we obtainhns ðrÞi ¼

(c)
A B

σ

σ

–

Fig. 6. (a) Deformation-induced gauge (magnetic) ﬁeld produces asymmetry of pseudo-spin. The pseudo-spin polarized localized state (i.e., theedge states) appears in pair with respect to y ¼ 0. (b) Due to the massterm, this symmetric character with respect to y ¼ 0 is lost and the edge
state can appear independently at y > 0 and y < 0 (spatial parity withrespect to y ! y is broken by the mass term). When the sign of themass is opposite with respect to spin, namely when m" ¼ m# , spin
becomes asymmetric with respect to E ¼ 0, and a local ferrimagnetism
appears. (c) In case that m" ¼ m# , spin up and down states are degenerate
and both appear below the Fermi level when ms < 0 (ms > 0) for y < 0(y > 0) so that the ground state is spin unpolarized.

ð27Þ

for the edge states where ‘  3acc =2.5) Hnnn is proportionalto z so that Hnnn appears as an additional term for the
mass. If n is suﬃciently large then we have m" ¼ m# andmagnetism disappears. This is consistent with the numericalresults given in §3.

–

Zigzag‘2 q
B ðrÞz ;h z

1 Bqz ðrÞsignðms Þ þ    ;
2 0

ð28Þ

where 0 ¼ 2h is the ﬂux quantum and correction may
arise due to higher order derivatives of Bqz ðrÞ. In the case ofm" ¼ m# , we have
R magnetism, i.e., hn" ðrÞ  n# ðrÞi 6¼ 0.
Moreover, using unit cell Bqz ðrÞd2 r ¼ 0 =4 that will bederived in the following, the magnetization at the edge isestimated by hmi ¼ 1=4, which is good agreement with
our numerical result, hmi0;L  0:25, shown in Fig. 4. Thus,
we think that there is a close relationship between theedge states magnetism and the parity anomaly. In fact, theanomaly survives even in the massless limit ms ! 0, whichis consistent to the fact that an inﬁnitesimal value of U gives
a ﬁnite magnetism if we do not consider the nnn interaction[see Pedge
in Fig. 5(a) and Fig. 5 in ref. 4]. A graphene withmthe zigzag edge and the Klein edge can be used to know that054703-6

J. Phys. Soc. Jpn., Vol. 77, No. 5

K. S ASAKI and R. SAITO

the deformation induced magnetic
ﬁeld in a unit cell at one
Rside of the edge is given by unit
Bqz ðrÞd 2 r ¼ 0 =4 [seeR cellqqFig. 2(c)]. In this case,  ¼ Bz ðrÞ d2 r is nonzero and theindex theorem28) can be used to know Bqz ðrÞ. The theorem
states that HK possesses jq =0 j zero energy edge states.
Since it is known that the number of the zero energy states ina ðn; 0Þ nanotube is given by n,29) then we can know that theﬂux in a unit cell at the zigzag edge, u , is given by 0 =4.
Here, we used 2 ð2nu Þ ¼ n0 . The factor 2 comesfrom the time-reversal symmetry (the K and K0 points) andthe factor 2n is the total number of edge sites at the zigzag
edge and the Klein edge. ju j ¼ 0 =4 is consistent with thenumerical result by Nakada et al.30) who demonstrate that anedge shape with three or four zigzag sites per sequence is
suﬃcient to show an edge state.
In summary, we have shown that the instability of thepseudo-spin order of the edge states induces ferrimagnetic
order in the presence of the Coulomb interaction. The nnn
hopping can stabilize the pseudo-spin order, but a larger
value of U makes the pseudo-spin order unpolarized andgives rise to a ferrimagnetic order. The ferrimagnetic orderis sensitive to the Fermi energy position. In case that thepseudo-spin order is realized one peak appears in the LDOS
near the zigzag edge, which is consistent to the existing
experimental results. Using a continuous model of the Weyl
equation, we showed that the deformation-induced gauge
ﬁeld gives rise to the magnetism of the edge states if themass terms have diﬀerent sign for diﬀerent spin edge states.
Acknowledgments
Authors would like to thank M. Suzuki and K. Nomura for
fruitful discussion. R.S. acknowledges a Grant-in-Aid for
Scientiﬁc Research (No. 16076201) from the Ministry of
Education, Culture, Sports, Science and Technology.

1) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I.
Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov: Nature
438 (2005) 197.

2) Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim: Nature 438 (2005)
201.
3) H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K.
Vandersypen, and A. F. Morpurgo: Nature 446 (2007) 56.
4) M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe: J. Phys.
Soc. Jpn. 65 (1996) 1920.
5) K. Sasaki, S. Murakami, and R. Saito: J. Phys. Soc. Jpn. 75 (2006)
074713.
6) K. Sasaki, M. Suzuki, and R. Saito: Phys. Rev. B 77 (2008) 045138.
7) K. Sasaki, J. Jiang, R. Saito, S. Onari, and Y. Tanaka: J. Phys. Soc.
Jpn. 76 (2007) 033702.
8) S. Okada and A. Oshiyama: J. Phys. Soc. Jpn. 72 (2003) 1510.
9) K. Kusakabe and M. Maruyama: Phys. Rev. B 67 (2003) 092406.
10) Y.-W. Son, M. L. Cohen, and S. G. Louie: Nature 444 (2006) 347.
11) Z. Klusek, Z. Waqar, E. A. Denisov, T. N. Kompaniets, I. V.
Makarenko, A. N. Titkov, and A. S. Bhatti: Appl. Surf. Sci. 161 (2000)
508.
12) Y. Kobayashi, K. Fukui, T. Enoki, K. Kusakabe, and Y. Kaburagi:
Phys. Rev. B 71 (2005) 193406.
13) Y. Kobayashi, K. Fukui, T. Enoki, and K. Kusakabe: Phys. Rev. B 73
(2006) 125415.
14) Y. Niimi, T. Matsui, H. Kambara, K. Tagami, M. Tsukada, and H.
Fukuyama: Appl. Surf. Sci. 241 (2005) 43.
15) Y. Niimi, T. Matsui, H. Kambara, K. Tagami, M. Tsukada, and H.
Fukuyama: Phys. Rev. B 73 (2006) 085421.
16) T. Ando, T. Nakanishi, and R. Saito: J. Phys. Soc. Jpn. 67 (1998) 2857.
17) T. Ando: J. Phys. Soc. Jpn. 74 (2005) 777.
18) K. Sasaki, Y. Kawazoe, and R. Saito: Prog. Theor. Phys. 113 (2005)
463.
19) D. J. Klein: Chem. Phys. Lett. 217 (1994) 261.
20) D. Klein and L. Bytautas: J. Phys. Chem. A 103 (1999) 5196.
21) R. Saito, G. Dresselhaus, and M. Dresselhaus: Physical Properties ofCarbon Nanotubes (Imperial College Press, London, 1998).
22) K. Sasaki, S. Murakami, and R. Saito: Appl. Phys. Lett. 88 (2006)
113110.
23) D. Porezag, T. Frauenheim, T. Köhler, G. Seifert, and R. Kaschner:
Phys. Rev. B 51 (1995) 12947.
24) C. L. Kane and E. J. Mele: Phys. Rev. Lett. 78 (1997) 1932.
25) G. W. Semenoﬀ: Phys. Rev. Lett. 53 (1984) 2449.
26) R. Jackiw: Phys. Rev. D 29 (1984) 2375.
27) A. N. Redlich: Phys. Rev. D 29 (1984) 2366.
28) R. A. Bertlmann: Anomalies in Quantum Field Theory (Oxford
University Press, Oxford, U.K., 2000).
29) K. Sasaki, S. Murakami, R. Saito, and Y. Kawazoe: Phys. Rev. B 71
(2005) 195401.
30) K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus: Phys.
Rev. B 54 (1996) 17954.

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