Generalized Chiral Symmetry and Stability of Zero Modes for Tilted Dirac Cones

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by  Tohru Kawarabayashi

Generalized Chiral Symmetry and Stability of Zero Modes
for Tilted Dirac Cones
Tohru Kawarabayashi,1 Yasuhiro Hatsugai,2 Takahiro Morimoto,3 and Hideo Aoki3
1Department of Physics, Toho University, Funabashi, 274-8510 Japan
2Institute of Physics, University of Tsukuba, Tsukuba, 305-8571 Japan
3Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033 Japan
(Dated: January 25, 2011)
1 While it has been well-known that the chirality is an important symmetry for Dirac-fermion
1 systems that gives rise to the zero-mode Landau level in graphene, here we explore whether this
0 notion can be extended to tilted Dirac cones as encountered in organic metals. We have found
2 that there exists a ”generalized chiral symmetry” that encompasses the tilted Dirac cones, where
n a generalized chiral operator γ, satisfying γ†H +Hγ = 0 for the Hamiltonian H, protects the
a zero mode. We can use this to show that the n = 0 Landau level is delta-function-like (with no
J broadening) by extending the Aharonov-Casher argument. We have numerically confirmed that a
2 lattice model that possesses the generalized chirality has an anomalously sharp Landau level for
2 spatially correlated randomness.

]
l Introduction — Since the seminal observation of the function-like) zero Landau levelaccompaniedby a sharp
l
a quantum Hall effect in graphene,[1, 2] the zero modes (step-function-like)quantumHallstep,ifthedisorderre-
h
of the massless Dirac fermion, which is essential for the spectsthechiralsymmetry. Wecanthusposeaquestion:
-
s characteristic quantum Hall effect in graphene, has been what is the effect of disorder for the zero Landau level
e
intensively discussed. Specifically, the zero mode is sta- in tilted Dirac cones, where the conventionalchiralsym-
m
bleagainstripplesingraphene,whichhasbeendiscussed metry is absent. This is exactly the motivation of the
. in terms of the index theorem, or more explicitly for presentwork. Inparticular,weshalllookathowthesta-
t
a wave functions with the argument due to Aharonov and bilityofthezeromodespossessingananomalouscritical-
m
Chasher [1, 3–8]. For the stability of zero modes, a cru- ity at the n=0 Landau level in the presence of disorder
- cial ingredient is the chiral symmetry defined as the ex- (e.g., random components in magnetic fields) is affected
d
istence of an operator Γ that anti-commutes with the bythebreakdownoftheconventionalchiralsymmetryin
n
o effectiveHamiltonianH, Γ,H =0,withΓ2 =1,which tilted Dirac cones. Curiously, the study leads us to find
{ }
c is more relevant than the Dirac-cone dispersion per se, a “generalized chiral symmetry”, which is then shown
[ since the stability is also shown for a chiral symmetric to give rise to an even wider stability of the zero modes

bilayer graphene with a quadratic dispersion [9, 10]. In persisting in tilted Dirac cones.
1
v the case of graphene, the Dirac cone is vertical, so that Thus we shall first generalize the conventional chiral
3 thechiraloperatorhasasimplestpossibleformofΓ=σz. symmetry so that the tilted and untilted Dirac disper-
7 Evenwhenthe systemis disordered,the anomalouscrit- sions can be captured in a unified manner. The gener-
2 icality atthe n=0 Landaulevelis retainedasfar as the alized chiral symmetry is defined by the existence of an
4
randomness respects the chiral symmetry [11–14]. The operatorγ,whichisnotnecessarilyHermitian,satisfying
.
1 stabilityofzeromodeshasbeenobservedexperimentally the relation γ†Hγ = H and γ2 = 1. This is consistent
0 as well for a mono-layergraphene [15]. with the condition fo−r the Dirac operator to be elliptic
1
that is required for the index theorem [26]. It enables
1 Ontheotherhand,weencounterDiracconesinawider
us to show a topological stability of zero modes that is
: classofmaterials,wheretheeffectivetheoryismoregen-
v present in tilted Dirac cones. With this generalized chi-
erally describedby tilted Dirac cones, as is the case with
i ral symmetry, we reformulate the eigenvalue problem so
X an organic material α-(BEDT-TTF) I [16–22]. Since
2 3 that the Aharonov-Casherargument[3] for counting the
r theconventionalchiralsymmetryisbrokenintiltedDirac
a number of zero modes is extended to tilted cones. This
cones, it becomes an essential question to ask whether
implies the zero Landau level is indeed delta-function-
(i) the symmetry is entirely broken,and (ii) whether the
like. We then numerically confirm how these field theo-
anomaly in the systems with usual chiral symmetry is
retictreatmentsonthestabilityofzeromodesappearsin
washed out in tilted Dirac cones. In the absence of dis-
alatticefermionmodelthathastiltedDiracconeswhere
order,the eigenvaluesandeigenfunctionsfortilted Dirac
the titling is varied continuously.
cones in magnetic fields have already been obtained in
Formalism — The effective Hamiltonian for a tilted
existing literatures[23–25],which indicate thatthe zero-
Dirac cone can be generically expressed as [8, 23–25]
energy modes themselves persist in clean tilted cones.
Effects of disorder on the zero modes is indeed an im- H =σ (W π/~)+(σ X)π /~+(σ Y)π /~, (1)
0 x y
· · ·
portant issue, since, for graphene with the conventional
chiral symmtry, it has been established that the disor- where (σ ,σ ,σ ) are Pauli matrices while σ is a 2 2
x y z 0
×
der has an anomalous effect of retaining a sharp (delta- unit matrix, and X,Y,W are real. The Dirac cone is
2
tilted when tW = (W ,W ) is nonzero, while tX = Schr¨odinger equation for the E = 0 states (zero modes)
x y
(X ,X ,X ) and tY = (Y ,Y ,Y ) characterize the therefore amounts to that for the zero modes of the un-
x y z x y z
anisotropy of the Dirac cone. Here π = p+eA (e > 0) tilted (but can be anisotropic) Dirac cones (with zero
is the dynamical momentum with tp = i~(∂ ,∂ ) diagonal elements). We see that the zero mode is then
x y
−
and the vector potential tA = (A ,A ) for the mag- given by
x y
netic field B = ∂ A ∂ A perpendicular to the x-
x y y x
y plane. Note that t−he dynamical momentum satis- α πψ− =0 and α∗ πψ+ =0. (2)
· ·
fies the commutation relation [π ,π ] = i~eB. For
x y
− As we shall see in the following, the chirality has to
the conventional Dirac cone (W = 0), the Hamilto- ±
be assigned to (normalizable) wave functions ψ . We
nian has the chiral symmetry associated with an oper- ±
canalso note that the equations for zero modes apply to
ator defined (and generalized to anisotropic cases) by
spatially varying magnetic fields B(x,y) as well, which
Γ = σ (X Y)/X Y , which anti-commutes with
the Ham· ilton×ian H| =×(σ |X)π /~+(σ Y)π /~ with can even be random.
Γ2 =σ [8]. · x · y Aharonov-Casher argument extended to the general
0 chirality — Is the stability of zero modes inherited by
When the Dirac cone is tilted (with H containing σ
0
forW =0),the conventionalchiralsymmetryis broken. the general chiral symmetry? For this we can look at
6 the zero modes for the tilted Dirac cone in a spatially
However,here we find that a generalizedsymmetry does
varying magnetic field B(x,y), which are therefore de-
exist, which can be seen if we introduce a new operator
terminedby eq.(2). Here itis convenientto adopt“prin-
γ defined by
cipalcoordinates”by rotatingαwithanorthogonalma-
γ =σ [(X Y) i(W X W Y)]/∆ trix T (with detT = 1), so that the complex numbers
y x
· × − − (z ,z ) = (α ,α )T−1 become orthogonal with each
x y X Y
with∆2 = X Y 2 (WyX WxY)2. Thisoperatorγ other on the complex plane. It is indeed possible to do
is non-Herm|iti×an fo|r−W = 0,−but its eigenvalues are 1 this with (z /z )/z /z = sgn(Im(α α∗))i iχ.
since γ2 = (γ†)2 = σ0. W6 e can show that, if ∆2 > 0±, γ With the tryansxform| eyd dxy|nam−ical momeXntuYm Π≡ −Tπ
satisfies a relation with the Hamiltonian, and the coordinates R t (X,Y) = Tx, Eq.(2) f≡or ψ
−
≡
reads
γ†Hγ = H,
−
α πψ =z λ−1(λΠ iχλ−1Π )ψ =0,
which we call the generalized chiral symmetry. This · − x X − Y −
symmetry reduces to the conventional chiral symmetry where the ”ellipticity” λ = z /z > 0 is a positive
x y
(sγect→ionΓ)offotrheWtil=ted0.DIitraschocounldebweitnhoatedcotnhsatatnttheencerrogsys irχeaλl−n1uΠmb)eψr. T=he0.eqTuahtiiso,nafloopnrg|ψ−w|itt|hhena| bseimcoimlarese(qλuΠatXio−n
Y −
plane is an ellipse as long as ∆2 > 0 (while a hyperbola for ψ , reduces to
+
when ∆2 <0).
Can we say anything about the wave functions as a (λΠ iχλ−1Π )ψ =0. (3)
X Y ±
±
direct consequence of this generalized chiral symmetry?
For this purpose it is instructive to choose the (right-) With these equations, it is straightforward to general-
eigenvectors of the opeator γ (with γ = ) ize the analytic Aharonov-Casher argument[3] for the
in the spinor|s±piace as a basis of the eigenv|a±luie pro±b|l±emi zero modes to the present case. We use the “Coulomb
for H. If we express the (normalized) wave function as gauge”, λ2∂XAX +λ−2∂YAY = 0, which is automati-
ψ = + ψ + ψ ,theSchr¨odingerequationHψ =Eψ callysatisfiedif weintroduce a “scalarpotential”ϕ with
+ −
reduc|esito |−i (AX,AY) = ( λ−2∂Yϕ,λ2∂Xϕ). Then Eq. (3) simpli-
−
fies to
+H + +H ψ
+
h | | i h | |−i
(cid:20) h−|H|+i h−|H|−i(cid:21)(cid:20) ψ− (cid:21) i~ χ2π ϕ ψ =0
± ± ±
1 + ψ − (cid:20)D ∓ φ0(cid:18)D (cid:19)(cid:21)
=E h |−i + .
(cid:20) h−|+i 1 (cid:21)(cid:20) ψ− (cid:21) with ± (λ∂X iχλ−1∂Y) and φ0 = h/e the flux
D ≡ ±
Note that the operator γ, being non-Hermitian, has quantum. Putting ψ = exp( 2πχϕ/φ )ψ˜ , we finally
± 0 ±
±
eigenvectors that are in general not orthogonal with obtain
each other with β = + = 0. When the gener-
alized chiral symmetry,hγ†|−Hiγ6 = −H, holds, we have D±ψ˜± =(∂X˜ ±iχ∂Y˜)ψ˜± =0
+H + = H = 0. The Schr¨odinger equation
hthe|n b|eciomesh−| |−i with R˜ (X˜,Y˜) = (λ−1X,λY). Namely, the function
≡
ψ˜ isanentire functionofZ =X˜ iχY˜ overthewhole
0 α π ψ 1 β ψ ± ± ±
(cid:20) α∗·π 0· (cid:21)(cid:20) ψ−+ (cid:21)=E(cid:20)β∗ 1 (cid:21)(cid:20) ψ−+ (cid:21), tcioomnpϕleixs dpelatnerem, innaemdeblyy Bazp=oly∂nXoAmYia−l i∂nYZA±X. =T(hλe2∂fuX2nc+-
where we have introduced a complex tα = (αX,αY) λ−2∂Y2)ϕ = (∂X2˜ + ∂Y2˜)ϕ, which implies that ϕ(R˜) =
~−1( +W σ + X σ , +W σ + Y σ ). Th≡e dR˜′G(R˜ R˜′)B (R˜′) with G(R˜) = (1/2π)log(r/r )
x 0 y 0 z 0
h | · |−i h | · |−i −
R
3
and r2 = R˜2. When the magnetic field is nonzero in the first Brillouin zone has a pair of Dirac cones at
only in a finite region, we have an asymptotic behavior, (k ,k ) = (π/2, π/2) and ( π/2,π/2) with E = 0 as
1 2
ϕ (Φ/2π)log(r/r ),forr ,whereΦ= dR˜B = long as t′/t < 0−.5, where th−e tilting becomes stronger
0 z
d→RBz is the total flux. Th→en∞we obtain R with t′ |(Fig|. 1). We can then show that the effective
| |
R Hamiltonian around these Dirac cones is reduced to the
ψ± ψ˜±(r/r0)±χ(Φ/φ0) Hamiltonian (1) with tW = (0, 4t′), tX = (0,2t,0)
→ ±
and tY = ( 2t,0,0). The model reduces to the π-flux
for r → ∞, in which ±χΦ < 0 is necessary for ψ± to model[27] wh∓en t′ =0 (Fig.1(b)).
be normalizable. The normalizability of ψ 2 indicates
±
| |
that the degeneracy of zero modes is Φ/φ [3]. This is
0
exactly equal to the total number of energy levels in the
Landau level, which implies a remarkable property that
no broadening occurs for the n=0 Landau level.
Higher Landau levels — We can also note that the
present formulation provides a simple algebraic repre-
sentation of the Landau levels, including higher ones,
in a uniform magnetic field B(x,y) = B > 0, where
the Schr¨odinger equation reads α π˜ψ = Eψ and
− +
α∗ π˜ψ = Eψ . Here π˜ = π q· with a real q sat-
+ −
· −
isfying α q = Eβ, which amounts to taking the origin
·
to be the center of the cross section (an ellipse) of the
tilted Dirac cone on a constant-energy plane. Since we
have [π˜ ,π˜ ]=[π ,π ], an annihilation operator a˜ sat-
x y x y E
isfying [a˜ ,a˜† ] = 1 can be defined as a˜ = α π˜/Λ
E E E · B
with Λ 2Im(α α∗)~eB when χ = 1. With a
B ≡ X Y
basis f =p(a˜† )ℓf /√ℓ! for positive integers ℓ where
ℓ,E E 0 FIG. 1. (Color Online) (a) A lattice model possessing tilted
a˜†E acting as the usual raising operator with a˜Ef0 = 0, Dirac cones. Hopping energies are t: solid thick lines, −t:
the eigenstate Hψn = Enψn with a nonzero energy solid thin lines, and t′: dotted lines. A unit cell is indicated
En = sgn(n) |n|ΛB, found in Refs.[23–25], can be bytheprimitivevectorse1ande2. EnergydispersionsE(k)/t
algebraically epxpressed as ψn = sign(n)|+if|n|−1,En + for (b) t′ =0, (c) t′/t=0.2 and (d) t′/t=0.4.
f for n= 1, 2,....
|−i |n|,En ± ±
A lattice model — In actual materials with general Weapplyamagneticfieldtothismodeltoexaminethe
band structures, the above argument based on the effec- stability of zero modes against the disorder in the mag-
tive Hamiltonian (1) and the generalized chiral operator netic field. The magnetic field is taken into account by
γ holds only as a low-energy effective model. In order the Peierls substitution t te−2πiθ(r), t′ t′e−2πiθ′(r)
to confirm whether the anomalously sharp zero-Landau suchthatthesummationo→fthephasesθ(r)→,θ′(r)around
level discussed above appears as well in a lattice model a loop is equal to the encircled magnetic flux in units
thatpossessestiltedDiraccones atlowenergies,wecon- of the flux quantum. Here we have adopted the string
sider a two-dimensional lattice model with a Hamilto- gauge[28] to treat smaller magnetic fields. Disorder is
nianhavingnearest(t)andsecondneighbor(t′)hoppings introduced here as a random component, δφ(r), in the
(Fig.1) on a square lattice, magnetic flux φ(r) = φ + δφ(r) piercing each plaque-
tte, where φ is the uniform part. The random part
H = −t(c†r+eycr+h.c.)+(−1)x+yt(c†r+excr+h.c.) δφ(r) is assumed to have a gaussian distribution with
Xr
a variance σ and a spatial correlation length η with
+ t′(c†r+ex+eycr+c†r+ex−eycr+h.c.), hδφ(r)δφ(r′)i = hδφ2iexp(−|r−r′|2/4η) [29]. We have
Xr chosen this disorder since it restores, for large enough η,
thegeneralizedchiralsymmetryofthe effectiveHamilto-
where r = (x,y) denotes a lattice point in units of the
nian at tilted Dirac cones.
lattice constant, and e (e ) the unit vector in x(y)-
x y Figure2displaysthedensityofstatesfort′/t=0.4,for
direction. The primitive vectors for the present lattice
which the tilting is significant (Fig.1(d)). The result is
system can be chosen as e =e e and e =e +e
1 x y 2 x y
− obtained by the exact diagonalizationof a finite system,
(Fig.1). In the absence of magnetic fields, the Hamilto-
with an average over 5000 samples is performed. It is
nian in the momentum space becomes
clearlyseenthatthen=0Landaulevelbecomesanoma-
2t′(cosk +cosk ) ∆(k) lously sharp even in random magnetic fields as soon as
H(k)= 1 2
(cid:20) ∆∗(k) 2t′(cosk1+cosk2)(cid:21) the correlation length η of the random flux exceeds the
lattice constant a, while other Landau levels are broad-
with ∆(k) = t( 1 + eik1 + eik1+ik2 + eik2), where enedinausualfashion. Thisanomaly,appearingonlyfor
− −
k = k e and k = k e . The band dispersion the n=0 Landau level, suggests that the n=0 Landau
1 1 2 2
· ·
4
ral symmetry can be extended to a generalized chiral
symmetry that encompasses the models having tilted
Dirac cone, so that the untilted and tilted Dirac cones
can be treated in a unified way. The stability of zero
modes can be proved under this generalized chiral sym-
metrywithanAharonov-Casherargument. Wehavefur-
ther shown, numerically for a lattice model, that topo-
logically protected zero modes of tilted Dirac fermions
survive even in random magnetic fields correlated over
a few lattice constants. These results suggest that the
anomaly at n = 0 Landau level can be observed gener-
ally in systems with tilted Dirac dispersions.
As a significance of this, we can finally note that the
existence of the generalized chiral symmetry (∆2 > 0)
is equivalent to the ellipticity of the Hamiltonian (1) as
a differential operator, under which the index theorem
[26] can be applied. It is an interesting future problem
to extend the notion of the generalized chiral symmetry
FIG. 2. (Color Online) Density of states for the model with
t′/t = 0.4 depicted in the inset is plotted against the spatial to a broader class of Dirac-cone systems [30].
correlation length of the random component of the magnetic
field η for a uniform magnetic field φ/φ0 = 1/100 and the
ACKNOWLEDGMENTS
amplitude of the random magnetic field σ/φ0 =0.0029. The
resultisanaverageover5×103 sampleswiththesystem-size
30a by 30a. We wish to thank Yoshiyuki Ono and Tomi Ohtsuki
for useful discussions andcomments. The workwassup-
ported in part by Grants-in-Aid for Scientific Research,
statesaredegeneratedatE =0,endorsesthestabilityof Nos. 20340098 (YH and HA) and 22540336(TK) from
zero modes for tilted Dirac cones. JSPS and No. 22014002 on Priority Areas from MEXT
In summary, we have found that the conventionalchi- for YH.
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