Stability of edge states in strained graphene
Pouyan Ghaemi,1 Sarang Gopalakrishnan,1, 2 and Shinsei Ryu1

arXiv:1211.1675v1 [cond-mat.mes-hall] 7 Nov 2012

1

Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
2
Department of Physics, Harvard University, Cambridge, MA 02138, USA

Spatially inhomogeneous strains in graphene can simulate the eﬀects of valley-dependent magnetic
ﬁelds. As demonstrated in recent experiments1,2 , the realizable magnetic ﬁelds are large enough to
give rise to well-deﬁned ﬂat pseudo-Landau levels, potentially having counter-propagating edge
modes. In the present work we address the conditions under which such edge modes are visible. We
ﬁnd that, whereas armchair edges do not support counter-propagating edge modes, zigzag edges do
so, through a novel selective-hybridization mechanism. We then discuss eﬀects of interactions on
the stability of counter-propagating edge modes, and ﬁnd that, for the experimentally relevant case
of Coulomb interactions, interactions typically decrease the stability of the edge modes. Finally, we
generalize our analysis to address the case of spontaneous valley polarization, which is expected to
occur in charge-neutral strained graphene3,4 .

I.

INTRODUCTION

Among condensed-matter systems, graphene is unique
in being a ﬂexible two-dimensional membrane whose
electronic properties are tunable through deformations
and strains. Spatially varying deformations aﬀect the
band structure of graphene by modulating the hopping
amplitudes between lattice sites. Indeed, certain spatially varying deformation patterns can mimic the effects of uniform “pseudo-magnetic ﬁelds,” which have
opposite signs in the two low-energy valleys5 (i.e., in
the neighborhoods of the K and K Dirac points) of
graphene. (Unlike real magnetic ﬁelds, therefore, pseudomagnetic ﬁelds preserve overall time-reversal invariance.)
To date, such pseudo-magnetic ﬁelds have been realized
using two distinct experimental approaches1,2 ; in both
experiments, the ﬁelds realized were strong enough to
drive the electronic structure deep into the quantum
Hall regime in each valley, and—in line with theoretical predictions—the electronic structure measured by
scanning-tunneling microscopy (STM) was clearly seen
to consist of well-spaced pseudo-Landau levels (PLLs)
in each valley. While the PLLs in each valley, considered separately, have nontrivial topological invariants3,6 ,
these are opposite for the two PLLs in the K and K
valleys; thus, the fate of various topological features in
the full system is not completely understood.
In the present work, we address the nature of one of the
crucial topological features of strained graphene, namely
its edge states. We ﬁrst show, by analyzing the noninteracting system, that the Landau levels of strained
graphene are very sensitive to the distinction between
zigzag and armchair edges. For zigzag edges, the Landau
levels (though ﬂat in an inﬁnite system) acquire an appreciable linear dispersion due to their hybridization with
the pre-existing, non-topological “surface” states. (We
shall use this term to distinguish these states from the
topological “edge” states.) On the other hand, for armchair edges, where such boundary states do not exist, the
Landau levels do not disperse even near the edge; this is
in sharp contrast with the case of a real magnetic ﬁeld7,8 ;

as we discuss, the physics behind this diﬀerence is that
the lowest PLLs in the two valleys of strained graphene
live on the same sublattice, whereas in the presence of a
real magnetic ﬁeld the corresponding Landau levels live
on opposite sublattices. Thus, on a typical sample with
rough edges, the edge states are expected to be localized
in the zigzag regions and as a consequence should not
contribute to transport; however, two-terminal measurements on zigzag nanoribbons should reveal the presence
of multiple edge states. We then address the possibility that Luttinger-liquid eﬀects stabilize the edge states
against disorder, but ﬁnd that they do not for the case of
a Coulomb interaction. Finally, we generalize our analysis to the case of edge states in spontaneously valleypolarized states, and argue that, in general, these should
not exhibit protected edge states. (One further mechanism for the destabilization of edge states, for instance
in the experiments of Ref. 1, is their hybridization with
the Dirac sea in the surrounding, non-strained region;
however, for undoped or weakly doped graphene this hybridization is presumably weak, owing to the vanishing
density of states at the Dirac point.)
Our paper is organized as follows. In Sec. II we introduce the microscopic Hamiltonian for strained graphene
that was used in our numerical work, as well as an eﬀective low-energy description that enables us to arrive at an
analytic understanding of the eﬀects discussed here. We
then consider the physics of an armchair edge in Sec. III
and that of a zigzag edge in Sec. IV. In Sec. V we extend
our analysis to include interaction eﬀects and their inﬂuence on edge stability. In Sec. VI we generalize our arguments, in the speciﬁc case of an armchair edge, to the
case of a valley-polarized state. Finally, Sec. VII summarizes our results and discusses possible experimental
signatures of edge-state physics in strained graphene.

II.

MODEL AND BULK PROPERTIES

In what follows, we shall chieﬂy consider the following noninteracting, nearest-neighbor Hamiltonian for

2
Strain generates a pseudo vector potential given by
A0 + iA3 = a=1,2,3 δta (r)e±iK·δ a near the Dirac points
x
y
±K5,9 . Note that
δt (r)e±iK·δ a is complex bea=1,2,3

a

cause the nearest-neighbor hoppings are not symmetric
under inversion. The real part of the strain gauge ﬁeld A0
x
is the same in both valleys and so couples to Q0 = σ0 τ0
and can be gauged away assuming time-reversal symmetry holds; whereas the imaginary part iA3 , has opposite
y
sign in the two valleys and couples with10 Q3 = σ0 τ3
leading to the valley-dependent magnetic ﬁelds realized
in the experiments of Ref. 1.
For the purposes of the present work, we shall consider
ribbon geometries, in which the strain-induced ﬁeld is
taken to realize the Landau “gauge” (A = (0, Bx)). A
concrete lattice realization of this gauge ﬁeld is shown in
Fig. 1. In this realization of strain, exploiting translation
invariance in y direction, the Dirac Hamiltonian can be
written in the form:
FIG. 1. Schematic pattern of change in hopping matrix element on red bonds and varying along x direction as δt3 =
evF Bx which leads to the pseudo-vector potential Ax = 0
and Ay = Bx. The dotted green square is the corresponding
four cite unite cell and the green arrows are primitive lattice
vectors.

strained graphene:
H0 =

(t + δta (ri ))(a† (ri )b(ri + δ a ) + h.c.), (1)
ri a=1,2,3

where δta (ri ) is, the strain-induced, nearest neighbour
hopping amplitude modulation between the A-sublattice
site at ri and the B-sublattice site at ri + δ a of the bipartite honeycomb lattice9 . The bond δ a connects any
A-sublattice atom to its three nearest neighbors on the
B-sublattice. In the absence of strain, the low-energy excitations have a linear dispersion around the two Dirac
√
points at momenta ±K with K = (4π/3 3a0 )ex , a0 being the carbon-carbon bond length9 . Near the Dirac
points K and K one can write the Bloch states in
terms of a four-component spinor, as follows: Ψ ≡
(ψA,K , ψB,K , ψA,K , ψB,K ) where the ﬁrst index denotes
the component of the wavefunction on the A(B) sublattice of the honeycomb unit cell, and the second index denotes the component of the state that is associated with the K (K ) valley. (In what follows, we shall
write ψA,K(K ) ≡ uK(K ) and ψB,K(K ) ≡ vK(K ) , in order to make contact with the standard notation for Dirac
fermions.) The low energy eﬀective Hamiltonian close to
the Dirac points then reads as:
H0 = vF [ˆx Γx + py Γy ]
p
ˆ

(2)

where Γx = τ3 σ1 , Γy = τ0 σ2 , vF is the Fermi velocity,
and the σ and τ operators are Pauli matrices acting on
sublattice and valley indices respectively.

e
H0 = vF −i∂x τ3 σ1 + py τ0 σ2 − Bxτ3 σ2
ˆ
c

(3)

where vF is the Fermi velocity, and the σ and τ operators are Pauli matrices acting on sublattice (u and v)
and valley (K and K ) indices respectively. As this strain
pattern respects translation invariance in the y direction,
it is suitable for a strained ribbon extended in the y direction. Notice that the Hamiltonian for valley K associated
with py is the same as the Hamiltonian for valley K with
momentum −py .
In this Landau “gauge”, the wavefunctions in the
zeroth PLL have the four-component form ΨK y =
0,p
(φ0,py , 0, 0, 0) and ΨK y = (0, 0, φ∗ y , 0), respectively,
0,p
0,p
where φ0,py is the mth Landau orbital in the lowest (nonrelativistic) Landau level, namely φpy ∝ exp(−ipy y−(x−
2
2
py lM )2 /2lM ) where lM is the magnetic length associated
with the strain induced pseudo-magnetic ﬁeld. In the
higher PLLs, the wavefunctions take the form ΨK y =
n,p
(φn,py , φn−1,py , 0, 0) and ΨK y = (0, 0, φ∗ y , φ∗
n,p
n,p
n−1,py ).
These forms should be contrasted with those for graphene
in a real magnetic ﬁeld; in the case of a real ﬁeld, the
Landau level wavefunctions in the two valleys have opposite sublattice structure, whereas in the case of a straininduced ﬁeld, the wavefunctions have the same sublattice
structure.
In our treatment of the noninteracting problem we
have suppressed the physical spin, and thus ignored the
(weak) intrinsic spin-orbit coupling in graphene. However, the spin index can be included trivially. We shall
consider the consequences of the physical spin in Sec. V,
as interaction eﬀects in the spinless and spinful cases are
diﬀerent; moreover, a combination of interaction eﬀects
and spin-orbit coupling was argued to lead to stabilization of fractional phases4 . (The spinless situation can be
experimentally realized by applying a large ﬁeld parallel to the graphene sheet, and thus spin-polarizing the
electrons.)

3
III.

ARMCHAIR EDGE

Having discussed the bulk properties, we now turn to
the case of a graphene ribbon with armchair edges along
the y direction. Owing to the translational invariance
along y as well as our choice of the Landau “gauge”
the momentum in y direction is a good quantum number. On the other hand, in the ribbon geometry, translation invariance in x direction is broken. For our analytic calculations we consider the case of a semi-inﬁnite
graphene ribbon covering the region x < 0, with an armchair boundary at x = 0.

If we make the transformations uK → −uK and
vK → −vK , the Hamiltonian in the two valleys reads
as:
e
HK = vF −i∂x σ1 + py σ2 − Bxσ2 ,
c
e
HK = vF −i∂x σ1 − py σ2 − Bxσ2 .
c

In order to get the spectrum, we square these Hamiltonians, leading to:
2
e
e
2
2
vF −∂x + py − Bx − Bσ3 Ψk = E 2 Ψk , (6)
c
c
2
e
e
2
2
vF −∂x + py + Bx − Bσ3 Ψk = E 2 Ψk , (7)
c
c

uK (py ) = uK (py )
-0.5

0

0.5

1

0.5

1

Py

(a)

E0

-1

(5)

which is deﬁned for x < 0. In the zeroth Landau level, in
which (for a strain-induced ﬁeld) the wave functions in
both valleys are based on the same sublattice, so that the
σ matrix is trivial. The boundary condition then reduces
to the wave function vanishing on the last row. In terms
of the sublattice wave functions this reads as:

E0

-1

(4)

-0.5

0

Py

(b)

FIG. 2. Energy dispersion for a graphene nanoribbon with
armchair edges as a function of the momentum along the edges
(py ); (a) in the presence of a strain-induced pseudo magnetic
ﬁeld and (b) in the presence of a real magnetic ﬁeld. The
center of mass position of the Landau orbitals is correlated
with their momentum py i.e. py = 0 is localized in the center
of the ribbon and as |py | increases, they get closer to the edge.
For the strain induced magnetic ﬁeld, there is no dispersion
in the zeroth PLL, whereas for real magnetic ﬁeld there are
dispersing edge states associated to the topological character
of the bulk (i.e., the quantum Hall eﬀect).

(8)

If we change x → −x for K , the two equations (6) and
(7) become identical. We can join the two equations and
the boundary condition above will then correspond to
continuity of the wave function at x = 0. The Hamiltonian then reduces to the simple harmonic oscillator
Hamiltonian and the boundary question is trivially satisﬁed. The salient property of this Hamiltonian is the
fact that the energy eigenvalues are independent of py ,
and thus the states near the edge do not disperse. This
is in sharp contrast with the case of a regular magnetic
ﬁeld7,8 , in which states near the edge do disperse.
One can understand the nondispersing nature of the
edge states intuitively as follows, for the case of a semiinﬁnite ribbon with an edge at x = 0. Because both the
(K, ky ) and (K , ky ) wavefunctions are located on the
same sublattice, and their guiding centers are at ky and
−ky respectively. One can always construct equal-weight
superpositions of the two wavefunctions, which automatically satisfy the boundary condition. (By contrast, in
the case of a regular magnetic ﬁeld, the two states live on
opposite sublattices and therefore cannot be superposed
to meet the boundary conditions; instead, the boundary condition must be satisﬁed by introducing a slowly
varying envelope function7,8 , and this costs additional kinetic energy for states near the edge.) Note that these
arguments extend straightforwardly to the case of higher
PLLs.
As Fig. 2(a) shows, a numerical computation of the
band structure is consistent with the argument above.
An important implication of this argument is that there
should be no counter-propagating edge modes on armchair edges; i.e., if the chemical potential lies between
two PLLs, it should not cross any states at all.

4
IV.

ZIGZAG EDGE

We now turn to the case of nanoribbons with zigzag
edges. Even in the absence of a magnetic ﬁeld, these
edges support non-dispersing modes localized at the
edges of the sample; these “surface” states occupy the
A sublattice at one end of the system, and the B sublattice at the other end. Because these states are conﬁned
on scales smaller than the magnetic length, they are expected to be unaﬀected by a magnetic or strain ﬁeld7,8 .
In the case of a real magnetic ﬁeld, the dispersion of
the topological edge states7,8 occurs chieﬂy by intrinsic,
universal means, and is therefore only slightly modiﬁed
by the presence of zigzag surface states. The case of a
strain-induced magnetic ﬁeld is fundamentally diﬀerent
in this respect: as we have argued above, there is no intrinsic dispersion mechanism, and thus the primary contribution to the Landau level dispersion is the diﬀerential
hybridization between Landau levels and the zigzag edge
(or “surface”) states.
We now address the eﬀects of this hybridization, focusing at ﬁrst on the lowest PLL. In the lowest PLL,
all Landau orbitals are situated on either the A or the
B sublattice. Thus, they hybridize only with the zigzag
surface states at one edge of the system; the strength of
this hybridization depends on the distance of the guiding center of the Landau orbital from the “Hybridizing”
edge.
Assuming the hybridizing edge is located at x = 0,
this hybridization changes the energy of each Landau orbital by a factor ∼ exp[−(py lM )2 ]. Because this energy
shift is larger for levels situated near the edge, it causes
these states to disperse. Moreover, because of the valleydependence of the pseudo-magnetic ﬁeld, the states in the
K valley that are near the edge have opposite momenta
to the corresponding states in the K valley. Near the
“non-hybridized” edge, by contrast, there is no hybridization and thus no dispersion at all. The size of this eﬀect
is quite strong in mesoscopic systems (such as all existing experimental realizations of strain-dependent gauge
ﬁelds), as one can see from diagonalization on graphene
nanoribbons (Fig. 3(a)).
Two important observations can be made about this
surface-induced dispersion eﬀect.
(1) The hybridization with the zigzag edge state plays
a role similar to an electric ﬁeld perpendicular to the
edges of the ribbon; thus, the dispersion generated by the
zigzag surface states can presumably be either enhanced
or compensated by the application of such an electric
ﬁeld. In particular, the number of surface states crossing the chemical potential might be detectable by these
means. (2) Unlike armchair edges, zigzag edges with surface states do not mix the valleys. This can easily be seen
from Fig. 3(a)—the level crossings between PLLs from
opposite valleys are not avoided, as they would be in the
presence of valley mixing. The physical reason for this is
that the Landau orbitals in the two valleys—since they
are counter-propagating—couple to orthogonal, counter-

E0

-1

-0.5

0

0.5

1

0.5

1

Py
(a)

E0

-1

-0.5

0

Py
(b)

FIG. 3. Energy dispersion for a graphene nanoribbon with
zigzag edges as a function of the momentum along the edges
(py ); (a) in the presence of inversion symmetry and (b) in the
presence of a inversion symmetry breaking perturbation.

propagating linear combinations of the zigzag edge states.
Thus, for a clean zigzag edge, there should be counterpropagating edge states, which can be detected (e.g.) in
two-terminal transport measurements. Note that these
counterpropagating edge states can be gapped out by
adding a valley-mixing term such as the Kekul´ distore
tion (see Fig. 3(b)).
In practice, perhaps the most promising candidate for
realizing a system with a clean zigzag edge is molecular graphene. In this system, one can tunably create a
Kekul´ distortion2 and thus study the transition between
e
Fig. 3(a) and Fig. 3(b).

5
V.

STABILITY UNDER INTERACTIONS

The considerations discussed so far apply to the case
of noninteracting edge modes. We now address the issue of whether the edge modes continue to carry current
in a disordered interacting system, using the approach
of Ref.11 . (Similar questions were addressed for quantum spin-Hall insulators with multiple edge modes in
Ref. 12.) It is clear that, in the absence of interactions,
backscattering due to disorder localizes the edge modes,
leading to an insulating edge. However, there are certain
regimes11–13 in which interactions prevent the localization of edge states, by making backscattering processes
“irrelevant” in the renormalization-group sense. (Qualitatively, this eﬀect can be understood in terms of interactions screening out the impurity potential; this would
happen, e.g., if the electron-electron interactions were attractive14 .)
For repulsive interactions, as discussed in Ref. 11, the
low-temperature behavior of the conductance is given
by the strength of the density-density interaction between the left-moving and right-moving edge states (i.e.,
forward-scattering) compared with back-scattering processes involving momentum-exchange. For the conductance to stay ﬁnite as T → 0, the above criterion implies
that the Fourier components of the interaction potential
1
satisfy V (0) ≤ 2 V (2|G|), where G is a reciprocal lattice
vector. (This is an approximate expression; the precise
momentum transfer involved in backscattering is the momentum diﬀerence between the two states on the edge,
which is ∼ |G| + 2kF , which we approximate as |G| using the assumption that the chemical potential is near the
Dirac point.) This criterion is not met by realistic interaction potentials. However, it has been shown11 that for
V (0) ≤ 2V (2|G|) (as in, e.g., a contact potential, or a
suﬃciently short-ranged potential achieved via gating),
the conductance exhibits non-monotonic behavior and
increases with decreasing temperature until a crossover
scale

T ∗ ∼ D exp −

2V (2|G|) − V (0)
,
V (2|G|)(2V (0) − V (2|G|))

(9)

where D is the bandwidth of the Dirac band (for molecular graphene this is ∼ 200 meV). The physics of this
crossover can be explored in strained graphene by tuning
the range of the inter-electron interaction via gating.

VI.

EFFECTS OF VALLEY POLARIZATION

Thus far, we have considered the edge states of strained
graphene under the assumption that time-reversal invariance is preserved. We now turn to situations in which
time-reversal invariance is spontaneously broken; such
situations naturally arise in the case of half-ﬁlled PLLs3 ,
in which the valley-polarized state naively appears to be

Φ

CDW order
at domain wall

τz
τx
−τz

FIG. 4. Corbino disk with broken valley degeneracy

a topologically protected state. We argue in this section
that such states do not generically have topologically protected edge modes.
Within the low-energy theory, there are two valleypolarizing perturbations, given by τz σ0 (i.e., simple
valley-polarization) and τz σz (i.e., the Haldane mass
term15 ). The former opens up a gap in any half-ﬁlled
PLL, whereas the latter opens up a gap only in the zeroth
PLL. We shall consider these terms separately, beginning
with the simple valley-polarization term. As discussed in
Ref. 10, this term anticommutes with two charge-densitywave perturbations that take the form τx σ0 and τy σ0 in
the low-energy theory. Being spatial modulations, these
terms are naturally generated at sharp edges; thus, the
valley polarization gap in the bulk can be rotated into the
valley-mixing gap at the edge without closing any gaps,
thus leading to the absence of edge states. We only consider the case of uniform charge-density-wave terms on
the edge, and neglect the question of whether disordered
or local charge-density-wave terms can gap out the edges.
One expects this mechanism to prevent edge states from
being present at half-ﬁlling of any PLL, as long as (a) the
edges mix the valleys, or (b) the charge-density-wave is
dynamically generated by interactions at the edge. It is
of course possible that very smooth edges, which do not
mix the valleys, can support chiral edge states, but the
fragility toward valley-mixing indicates that these edge
states are not in fact protected.
Before we proceed to the case of the zeroth PLL, we
brieﬂy explain how our remarks above can be related
to the conventional ﬂux-insertion argument16 , which was
used4 to argue for topologically protected edge modes in
half-ﬁlled PLLs. Consider half-ﬁlled PLLs in a Corbino
disk geometry as shown in Fig. 4, in which the left half of
the disk has a perturbation of the form τz and the right
half has a perturbation of the form −τz ; this situation,
which corresponds to magnetic domains, should typically
arise in experiments, given the Ising nature of the valley
polarization4 . We assume that the gauge realized by the
strain has the property that py = 0 is near the center of
the disk, and then add a perturbation of the form τx in

6
the medial region of the Corbino disk. (This assumption
can easily be relaxed, if one then chooses a perturbation
that is modulated along y.) Now suppose one inserts a
(real) magnetic ﬂux through the center of the Corbino
disk: if the regions having a τz perturbation were indeed
topological, one would expect the ﬂux insertion to pump
two charges from the edges of the disk to its center (given
the opposite nature of the strain-induced magnetic ﬁeld
in the two regions). However, this is impossible because
the center of the disk is fully gapped and therefore incompressible; thus, we conclude that the ﬂux insertion
argument fails, and that half-ﬁlled PLLs do not in general have a quantized Hall conductance.
The argument given above holds for all the PLLs; however, an additional subtlety arises in the case of the zeroth PLL, owing to the possibility of a Haldane mass
gap, τz σz 15 . In any PLL other than the zeroth, this term
does not open a gap. However, within the zeroth PLL,
it does open up a gap, which leads to the following apparent paradox. As the Haldane mass term generates
a gap both with and without the strain-induced ﬁeld,
one can imagine adding it before adding strain; in this
case, the gap generated is a topologically nontrivial gap,
accompanied by chiral edge states. The strain-induced
ﬁeld does not compete with this gap (because the lowest
PLL is gapped out by the Haldane mass), thus this topologically trivial gap and the corresponding edge states
must continue to exist even in strained graphene with
a Haldane gap. On the other hand, the Haldane gap—
when projected onto the zeroth PLL—can be rotated into
the charge-density-wave gaps discussed in Ref. 10; thus,
its projection is topologically trivial and cannot generate
edge states. Our numerical diagonalization studies, plotted in Fig. 5, suggest that the resolution is as follows—the
topologically protected Haldane edge state is connected
to higher-order PLLs, and crosses the zeroth PLL without hybridizing with it. Thus, although these edge states
give rise to a robust, quantized Hall eﬀect, the mechanism is unrelated to the low-energy PLL structure. Indeed, in the limit where the Haldane mass term is much
smaller than the other perturbations, we ﬁnd that these
edge states move far away from the Dirac points, demonstrating that they are unrelated to the PLLs. In addition, when the electron density is tuned to be at the
charge neutral point, and the valley polarization (hence
the Haldane mass) is generated spontaneously by interactions, the interaction eﬀects are operative nominally
only for the half-ﬁlled lowest PLL, but not for the other
PLLs that are totally empty or occupied. I.e., the Haldane mass is selectively generated for the lowest PLL,
but not for the other PLLs. We would then expect there
is no edge state at all in this case.

VII.

CONCLUSIONS AND OUTLOOK

In the present work we have addressed the properties
of the edge states of strained graphene in the quantum

-1

-0.5

0

0.5

1

Py

FIG. 5. Pseudo-Landau level (PLL) structure in the presence
of a large Haldane mass gap. As one can see, the topologically
protected edge modes guaranteed by the Haldane mass term
do not mix with the PLLs, but appear to originate at much
higher energies.

Hall regime. We have argued that the edge physics is not
universal—owing to the lack of topological protection—
but is strongly dependent on the nature of the edge.
Whereas armchair edges do not support edge states
at all, zigzag edges are expected to support counterpropagating edge states. Furthermore, in the case of
zigzag edges, the hybridization between the PLL states
and the zigzag “surface” states gives rise to dispersion
of the PLLs; for short-range interactions, these counterpropagating edges are expected to manifest themselves
via nonmonotonic temperature-dependence of the conductance. Finally, we considered the case of valleypolarized edges (e.g., a quantum Hall ferromagnet in
charge-neutral strained graphene) and argued that, in
general, valley-polarization does not imply a ﬁnite Hall
conductance, because (within the zeroth PLL) valley polarization can be continuously transformed into a chargedensity wave, which is evidently non-topological.
Our results have several experimental implications.
Most notably, the diﬀerence between the dispersion near
zigzag and armchair edges can easily be detected using
the spectroscopic methods of current experiments. In addition, transport experiments would—assuming the contribution of the PLLs could be isolated—provide several
clear signatures. For example, for strained graphene rib-

7
bons having clean zigzag or armchair edges, the striking diﬀerence between the former case (with multiple
counterpropagating edge modes) and the latter (no edge
modes at all) should be easily detectable via two-terminal
measurements. Similarly, a nonmonotonic temperaturedependence of the two-terminal conductance would provide a signature of interaction eﬀects near the edges. Finally, either spectroscopy or transport can address our
prediction that, for half-ﬁlled PLLs, the regions between
magnetic domains should (a) contain no propagating
modes, and (b) exhibit spontaneously broken transla-

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ACKNOWLEDGMENTS

The authors are grateful to Dmitry Abanin, Jeﬀrey
Teo, Eduardo Fradkin, Taylor Hughes and Ashvin Vishwanath for helpful discussions. This work was supported by NSF DMR-1064319 (P.G.) and DOE DEFG02-07ER46453 (S.G.).

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