Abstract
As the three-dimensional analogs of graphene, Weyl semimetals display signatures of chiral anomaly which arises from charge pumping between the lowest chiral Landau levels of the Weyl nodes in the presence of parallel electric and magnetic fields. In this work, we study the pseudo chiral anomaly and its transport signatures in graphene ribbon with zigzag edges. Here "pseudo" refers to the case where the inverse of width of zigzag graphene ribbon plays the same role as magnetic field in three-dimensional Weyl semimetals. The valley chiral bands in zigzag graphene ribbons can be introduced by edge potentials, giving rise to the nonconservation of chiral current, i.e., pseudo chiral anomaly, in the presence of a longitudinal electric field. Further numerical results reveal that pseudo magnetoconductivity of zigzag graphene ribbons is positive and has a nearly quadratic dependence on the pseudofield, which is regarded as the transport signature of pseudo chiral anomaly. arXiv:1907.00517v1 [cond-mat.mes-hall] 1 Jul 2019
I. INTRODUCTION
Graphene, a celebrated semimetal with nearly vanishing band gaps, holds massless Dirac fermions in two dimensions (2D) [1] . The Dirac fermions in graphene behave in unusual ways leading to remarkable transport properties such as anomalous integer quantum Hall effect [2, 3] and minimum conductivity [4, 5] . As analogs of graphene in 3D, topological Dirac and Weyl semimetals have attracted intense research interest over the past years [6, 7] . The unique topological nature of Dirac and Weyl semimetals indicates novel properties such as Fermi arc states on their surface [8] [9] [10] . Also the chiral anomaly, a well-known phenomenon in quantum field theory [11] , may show its existence in topological Dirac and Weyl semimetals through characteristic transport properties such as chiral magnetic effect and negative magnetoresistance [12] [13] [14] [15] [16] .
I. INTRODUCTION
Graphene shares several characters with Weyl semimetals. First, the lower-energy physics near the graphene nodes (band touching points K and K ) and Weyl nodes in Weyl semimetals are governed by Weyl equation. The graphene nodes, which are dubbed Dirac points, nevertheless, are actually described by Weyl Hamiltonian in 2D [17] . Second, topological invariants can be defined from the Weyl nodes as well as graphene nodes. For Weyl semimetals, the integration of Berry curvature on a simple closed surface enclosing a Weyl node gives the topological charge (magnetic monopole charge) that works as a topological invariant. Similarly, the graphene nodes give rise to a topological charge [18] whereŜ is the chiral (sublattice) symmetry operator and H is the effective Hamiltonian of graphene nodes. In addition, the Weyl nodes, as well as graphene nodes, come in pairs with opposite topological charges in the whole Brillouin zone according to the no-go theorem [19, 20] . Third, there are surface states connecting two nodes with opposite topological charges both in graphene and Weyl semimetals. The Fermi arcs surface states in Weyl semimetals are associated with a well-defined Chern number [6] , and the surface (edge) states in graphene ribbons with zigzag edges correspond to winding numbers [21] .
I. INTRODUCTION
The similarities between graphene and Weyl semimetals naturally lead to a question: Is it possible to reveal some properties of Weyl semimetals, such as chiral anomaly and its related transport signatures, on graphene basis? The chiral anomaly, also called Adler-Bell-Jackiw anomaly, is a quantum anomaly which states the violation of chiral symmetry of a classical action in the corresponding quantized theory [11, 22, 23] . The chiral anomaly in Weyl semimetals is readily understood in terms of Landau levels in a crystal [24] , and it promises novel transport properties including the negative magnetoresistance [24] [25] [26] and chiral magnetic effect [27, 28] . Generally, chiral anomaly only exists in (1 + 1)D and (3 + 1)D [11] , and it should be absent in graphene. Whereas the graphene ribbons, synthesized with atomical precision [29, 30] , serve as quasi-1D systems. Besides endowed with many interesting properties and applications, graphene ribbons are even designed to mimic topological quantum phases in 1D [31, 32] . Indeed, the spectrum of graphene ribbon with zigzag edges closely resembles the Landau level of Weyl semimetals in the presence of a magnetic field (consider the dispersive direction) [6, 33] . Thereby the answer to the aforementioned question may be affirmative [34] .
I. INTRODUCTION
In this work we extend the chiral anomaly to graphene ribbon with zigzag edges in a phenomenological way, and investigate its transport related signatures numerically. Confining zigzag graphene ribbons by edge potentials realizes the truly chiral bands. Further application of an external electric field parallel to the ribbon drives the chiral bands with opposite chirality to give rise to chiral anomaly. This pseudo chiral anomaly in graphene lead to the finite-size conductivity bearing a positive dependence on effective magnetic field B ≡ 1/W with W the width of ribbon. With the help of numerical calculations, we show that the pseudo magnetoconductivity indeed has a nearly quadratic dependence on effective magnetic field B.
I. INTRODUCTION
The rest of this paper is structured as follows. In Sec. II, the topological nature of surface states in zigzag graphene ribbons is revisited in terms of Su-Schriffer-Heeger (SSH) model. Then in Sec. III we extend the concept of chiral anomaly to graphene by constructing gapless chiral bands utilizing the flat band surface states. In Sec. IV, the transport properties associated with pseudo chiral anomaly in zigzag graphene ribbons are investigated numerically. Finally, Sec. V is devoted to discussion and conclusions.
II. ZIGZAG GRAPHENE RIBBON AS SU-SCHRIFFER-HEEGER MODEL
The existence of surface states in graphene is determined by the geometry of its edges [35, 36] . Typically, the graphene ribbons with zigzag (beared) edges exhibit localized surface states while the ribbons with armchair edges have no such surface states. It is possible to characterize surface states of graphene ribbons with different edge structures, namely, zigzag, beared, and armchair edges, even arbitrary edge geometry, topologically [21, [37] [38] [39] [40] .
II. ZIGZAG GRAPHENE RIBBON AS SU-SCHRIFFER-HEEGER MODEL
We find it is more concise and straightforward to understand the surface states by identifying the graphene ribbon as an SSH model in the reduced 1D parameter space. SSH model describes spinless electrons sitting in a 1D dimer chain, and it has been extensively studied as a prototype for associated topological properties [41] [42] [43] . In the following we make a simple connection between the SSH model and graphene ribbons. Graphene has a honeycomb lattice structure as shown in Fig. 1(a) . The three nearest-neighbor vectors in real space are denoted as δ 1 = a 2 ( √ 3, 1), δ 2 = a 2 (− √ 3, 1), δ 3 = a(0, −1) with a the carbon-carbon distance. Considering only nearestneighbor hopping, the tight-binding Hamiltonian for electrons in graphene reads
II. ZIGZAG GRAPHENE RIBBON AS SU-SCHRIFFER-HEEGER MODEL
where C † R (C R ) creates (annihilates) an electron at site R. Here the spin degree of freedom is suppressed and t is the nearest-neighbor hopping energy. In the following, we first focus our attention on graphene ribbon with zigzag edges and then analyze ribbons with beared and armchair edges. After performing partial Fourier transformation in x direction, the zigzag ribbon is reduced to 1D fictitious chain along y direction, in which each unit cell contains two sites. The hopping energy in and between unit cells of the chain is different, i.e., 2t cos( √ 3ak x /2) for intracell hopping and t for intercell hopping. Therefore, the reduced zigzag ribbon is identical to SSH model of dimerized 1D lattice. The spectrum of graphene ribbons with zigzag edges obtained in this way is shown in Fig. 1(b) , in which the flat band surface states connect K and K valleys. Imposing periodic boundary condition along y direction further, the system is described by the Hamiltonian
II. ZIGZAG GRAPHENE RIBBON AS SU-SCHRIFFER-HEEGER MODEL
where σ ≡(σ x , σ y ) is the Pauli matrix vector, and d(k) = (2 cos( √ 3k x a/2) + cos(3k y a/2), sin(3k y a/2)). The wave numbers are normalized with corresponding primitive translation vectors, and the energy is scaled by hopping energy t. Note that the primitive lattice spacings in x and y directions are √ 3a and 3a/2, respectively. Diagonalizing the Hamiltonian of Eq. 2, we obtain the spectrum that is consistent with the very one of zigzag graphene [40, 44] . It should be noted that Eq. 2 is exactly the SSH Hamiltonian if we redefine d x (k y ) = v + w cos k y and d y (k y ) = w sin k y with v ≡ 2 cos(k x /2) and w ≡ 1 [42, 43] . The topology of SSH model is characterized by the winding number defined as ν = 1 2πi¸q −1 dq with q = v + we −iky . Since ν = 1 for |v| < |w| in SSH model, the zigzag graphene is topologically nontrivial with wave vector k x locating in the region ( 2π 3 , 4π 3 ). From the bulkboundary correspondence, there are surface states connecting K and K valleys, as shown in Fig. 1(b 
II. ZIGZAG GRAPHENE RIBBON AS SU-SCHRIFFER-HEEGER MODEL
, the winding number is zero and no surface states are expected.
II. ZIGZAG GRAPHENE RIBBON AS SU-SCHRIFFER-HEEGER MODEL
The vector d(k) provides a more vivid way to explore the topology of Eq. 2. The path of the end point of vector d(k), as the wave number k y goes through the 1D Brillouin zone [0, 2π], is a unit circle centered at (2 cos(k x /2), 0) on the d x -d y plane, as shown in Fig. 1(c) .
II. ZIGZAG GRAPHENE RIBBON AS SU-SCHRIFFER-HEEGER MODEL
Increasing k x moves the unit circle left. Once the circle cuts the origin, it gives Dirac cone. Varying the wave vector k x from 0 to 2π, the gap of the reduced 1D system will close at k x = 2π/3 then reopen, and similar process happens again around k x = 4π/3. This gap closereopen process may indicate topological phase transition. Whether or not the system is topologically nontrivial is determined by the origin is enclosed by the unit circle or not. Similar analysis can also be applied to graphene ribbon with beared and armchair edges. For example, the reduced Hamiltonian for the beared edge case is
II. ZIGZAG GRAPHENE RIBBON AS SU-SCHRIFFER-HEEGER MODEL
For the ribbons with armchair edges, there are no such surface states.
III. PSEUDO CHIRAL ANOMALY IN ZIGZAG GRAPHENE RIBBONS
We demonstrated in the previous section that the surface states in the zigzag graphene ribbons can be interpreted by SSH model. In this section we show how pseudo chiral anomaly is possible in graphene ribbon with zigzag edges by constructing genuinely chiral bands using these surface states. The wave functions of surface states at k x = π with energy E = 0 are entirely localized at the edges of the ribbon. Applying edge potentials U edge at the outmost sites of two edges, intuitively, drags the surface states to the states with energy E = U edge [45] . Considering the band structures as shown in Fig. 1(b) , in which the energies of all other bands coincide at E = ±1 for k x = π, we choose the confining edge potentials to be U edge = ±1. For the case of U edge = −1, gapless chiral bands, labeled as n = 0, cross the valleys, as seen in Fig. 2(a) . Focusing on the left (right) valley, modes n = 1, 2, · · · , N lie above the n = 0 chiral band, whereas modes n = −1, −2, · · · , −(N − 1) lie below the n = 0 chiral band, and the gapless chiral band cross the valley with group velocity v F = − 3 2 ta (v F = + 3 2 ta ). The two n = 0 chiral bands should be stable, as will be shown, since they originate from the topologically protected surface states. The wave numbers k x = 2π/3 and k x = 4π/3 are unique points at which the system has special energy and wave functions. For k x = 2π/3 and k x = 4π/3, the Hamiltonian for the zigzag graphene ribbons confined by the edge potentials turns out to be
III. PSEUDO CHIRAL ANOMALY IN ZIGZAG GRAPHENE RIBBONS
under the basis (C ηA1 , C ηB1 , C ηA2 , C ηB2 , · · · , C ηAN , C ηBN ) T .
III. PSEUDO CHIRAL ANOMALY IN ZIGZAG GRAPHENE RIBBONS
Here η = −1 for k x = 2π/3 and η = +1 for k x = 4π/3, respectively. For U edge = −1, the eigen values of Eq. 3 read E η,j = 2 cos[jπ/(2N )], j = 1, 2, ..., 2N.
III. PSEUDO CHIRAL ANOMALY IN ZIGZAG GRAPHENE RIBBONS
The zero-energy modes, corresponding to the chiral bands n = 0, always exist when j = N . By Taylor expansion and elimination of N from the width W = (3N/2 − 1)a, the mode spacing under the limit W 1 is
III. PSEUDO CHIRAL ANOMALY IN ZIGZAG GRAPHENE RIBBONS
which is proportional to 1/W . The energy separation of the conduction and valence bands, as shown in Fig. 2(a) , is 2∆ 0 now and different from 3∆ 0 as in Fig. 1(b) [46] . The corresponding eigenvectors for n = 0 chiral bands are
III. PSEUDO CHIRAL ANOMALY IN ZIGZAG GRAPHENE RIBBONS
It is interesting to note from Fig. 2(b) that the wave functions of the n = 0 chiral bands are not localized at edges but distribute equally on each site along y direction (it is even true at k x = η), which is different from the case where the bulk gap is opened [45] . The spectrum in Fig. 2(a) is very similar to that of Weyl fermions in the presence of a strong magnetic field [6] , and it provides the possibility to realize chiral anomaly in graphene ribbons. Crucially, the chiral bands have definite chiralities just as the chiral Landau levels of Weyl fermions. Suppose the temperature is at zero and only the chiral bands are relevant, thus the system is in the so called 'quantum limit' regime. Applying an external electric field with strength E along the ribbon will drive electrons flowing from one valley to the other. The change rate of charge carrier density at one valley η(= ±) in the quasi-1D system is
III. PSEUDO CHIRAL ANOMALY IN ZIGZAG GRAPHENE RIBBONS
dkx dt , and the momentum vector will obey the equation dkx dt = eE . Therefore, we have dρη dt = η e h E W . Similar to chiral anomaly in Weyl semimetals, here we can also define an equation for divergence free current j µ 5 as
III. PSEUDO CHIRAL ANOMALY IN ZIGZAG GRAPHENE RIBBONS
where µ = 0, 1, ∂ 0 ≡ ∂ t and ∂ 1 ≡ ∂ x , and B ≡ 1/W . Physically, Eq. 7 states nonconservation of the chiral charge in the presence of electric field and effective magnetic field B. Thus, by the same mechanism that is operative in 3D Weyl semimetals [24, 47] , the chiral anomaly can be generalized to quasi-1D graphene ribbons. Comparing with the current conservation equation of Weyl semimetals, B here takes the role of a magnetic field. This analogy also makes sense in terms of physical picture of Landau levels: In graphene ribbons, the energy band spacing ∆ 0 decreases with decreasing the effective magnetic field B, i.e., ∆ 0 ∝ B; and likewise the Landau level separation also decreases with decreasing the real magnetic field strength. It should be noted here we mainly focus on the orbital effect of a magnetic field imposed on electrons. The difference between the effective magnetic field B and a real magnetic field, however, lies in several aspects: First, the effective magnetic field B preserves time reversal symmetry; second, the unit of effective magnetic field B is not as well-defined as the real magnetic field; third, the effective magnetic field B has no Zeeman effect. For these reasons above, the chiral anomaly presented here in graphene ribbons is therefore referred to as pseudo chiral anomaly.
IV. TRANSPORT SIGNATURES OF THE PSEUDO CHIRAL ANOMALY
Due to the interplay of chiral magnetic effect and intervalley scattering, the pseudo magnetoconductivity σ(B) should be positive and have quadratic dependence on B in the diffusive transport regime [34, 48] , which is regarded as the signature of pseudo chiral anomaly in graphene. In this section, we try to reveal this anomalous scaling behavior of finite-size conductivity numerically. From Fig. 2(a) , the left valley locates at k − = 2π/3 and the right valley locates at k + = 4π/3. The rather large momentum difference ∆k = 2π/3 between the two valleys indicates that the transport properties depend on impurity ranges heavily. As mentioned above, the signature of chiral anomaly will only be revealed with intervalley scattering. Thereby two typical kinds of disorders, namely, onsite disorders and Gaussian type disorders, are considered. The onsite disorders are considered by independently adding every lattice site a potential drawn from uniform distribution [−U/2, U/2] with U the disorder strength. For Gaussian type disorders uniformly distributed in the 2D real space, the potential on a lattice site with position R has the form [49] 
IV. TRANSPORT SIGNATURES OF THE PSEUDO CHIRAL ANOMALY
where d is the disorder range and r i is the disorder position. Here u is the disorder strength uniformly distributed within [−u m , u m ] constrained by the normalization condition
IV. TRANSPORT SIGNATURES OF THE PSEUDO CHIRAL ANOMALY
R∈full space
IV. TRANSPORT SIGNATURES OF THE PSEUDO CHIRAL ANOMALY
The Landau-Büttiker formalism provides an efficient way to calculate the transport properties of electrons.
IV. TRANSPORT SIGNATURES OF THE PSEUDO CHIRAL ANOMALY
Here we consider a two-terminal setup with disordered central region connected to two clean leads. With the help of recursive Green's function techniques, the conductance of two-terminal device can be evaluated as
IV. TRANSPORT SIGNATURES OF THE PSEUDO CHIRAL ANOMALY
where Γ L,R are the line-width functions coupling to left lead and right lead, respectively, and G r (G a ) is the retarded (advanced) Green's function of the disordered region [50] . Note that the conductivity meets σ = G for a square sheet [5, 51] . Here the width W = (3N/2 − 1)a and length L = √ 3M a with M the number of defined unit cells in Fig. 1(a) . Here 5000 disorder configurations are taken for all.
IV. TRANSPORT SIGNATURES OF THE PSEUDO CHIRAL ANOMALY
Let us now consider the transport properties of the n = 0 chiral bands. For a clean sample, the relevant chiral bands should underlie a quantized conductance 2e 2 /h. To verify the validity of chiral bands induced by edge potentials, the onsite disorders are added at the outmost sites of the two edges first. As seen in Fig. 3(a) , the quantized conductance from chiral bands persists until edge disorder strength reaches critical strength U c 2.5. Therefore, the mechanism of confining edge potentials still works as long as the disorder strength does not exceed the critical value U c . While for the case where disorders are on all sites, the conductance decreases quickly as shown by blue line in Fig. 3(a) . Seen from Fig. 3 (b) (also in Fig. 4(a) ), the n = 0 chiral bands are more stable than other bands: The conductance plateau from n = 0 chiral bands is nearly unchanged as increasing the disorder strength. The scaling behavior of conductance with respect to sample length also shows the robustness of the n = 0 chiral bands (see Fig. 3(c) ).
IV. TRANSPORT SIGNATURES OF THE PSEUDO CHIRAL ANOMALY
For onsite disorders, the pseudo magnetoconductivity as function of B 2 with different disorder strengths is presented in Fig. 3(d) : Focusing on the curve with disorder strength U = 1, the conductivity drops from a constant value linearly with respect to B 2 as soon as the system is in laterally diffusive regime where the width exceeds mean free path; with appropriate disorder strength, the pseudo magnetoconductivity has a nearly quadratic dependence on effective magnetic field B and it is treated as the transport signature of pseudo chiral anomaly. Here the Fermi energy shifts away from zero slightly to avoid strong fluctuations [52] , and we keep the assumption that the Fermi energy only crosses the two n = 0 chiral bands. For Gaussian disorders, we fix the disorder strength at u 0 = 1 and vary the interacting range d. For a shorter interacting range d, the intervalley scattering is relatively stronger. It is shown from Fig. 4(b) that the pseudo magnetoconductivity shares similar behavior as in the onsite disorder case, i.e., it shows nearly quadratic scaling behavior on effective magnetic field B. Comparison of Fig.  3(d) and Fig. 4(b) indicates that transport properties of chiral bands exhibit little dependence on the disorder types but much on disorder strengths.
V. DISCUSSION AND CONCLUSIONS
Varying the termination direction away from zigzag case will gradually weak the pseudo chiral anomaly and finally it disappears at armchair case. Thus we mainly focus on zigzag type graphene ribbons, and it is mostly relevant for experiments. One possible concern about the analogy between graphene and Weyl semimetals is that the stability of graphene nodes, unlike Weyl nodes, needs to be protected by inversion symmetry [53] . Consider staggered potentials U A = ∆ for A sites and U B = −∆ for B sites universally in the graphene lattice. At this moment, the bulk bands are gaped with amount 2∆. The winding number is not well-defined now due to the existence of mass term in the bulk Dirac Hamiltonian. However, there are still flat bands with energy E = ±∆ [38, 45] . After applying proper edge potentials, the chiral bands crossing two valleys persist as in Fig. 2(a) , thereby the physics of pseudo chiral anomaly remains. Besides, the magnetic order at zigzag edges of graphene ribbons may have influence on the bulk states [54] . While the band gap opened by the magnetic order is negligible as long as the zigzag ribbons wider than 8 nanometers [55] , which is satisfied in our considerations, i.e., N = 40 ∼ 300 and W = 8.5 ∼ 64 nm.
V. DISCUSSION AND CONCLUSIONS
In summary, we show that pseudo chiral anomaly may exist in graphene ribbon with zigzag edges. We first show that the zigzag graphene ribbon can be mapped to SSH model, and then we turn the flat band surface states of zigzag graphene ribbons to chiral bands, based on which the pseudo chiral anomaly can be realized. This pseudo chiral anomaly underlies anomalous scaling behavior of pseudo magnetoconductivity on the effective magnetic field B. Further numerical calculations on the transport properties indicates the pseudo magnetoconductivity has a positive and nearly quadratic dependence on B. (A2)
V. DISCUSSION AND CONCLUSIONS
The Zak phase is defined as [56] γ(k x ) =ˆπ −π A kx (k y )dk y .
V. DISCUSSION AND CONCLUSIONS
Here the Berry connection is A kx (k y ) = s∈occ ψ s (k y )|i∂ ky |ψ s (k y ) . Explicitly,
V. DISCUSSION AND CONCLUSIONS
A kx (k y ) = 1 2 1 + 2 cos(k x /2) cos k y [2 cos(k x /2) + cos k y ] 2 + sin 2 k y .
V. DISCUSSION AND CONCLUSIONS
Taking k x as the parameter and dk y for the integral measure, the Zak phase γ(k x ) takes value π for k x ∈ ( 2π 3 , 4π 3 ), and takes 0 for k x ∈ (0, 2π
V. DISCUSSION AND CONCLUSIONS
3 ) ∪ ( 4π 3 , 2π), corresponding to winding numbers in the main text. Similar calculations can also be applied to graphene ribbon with armchair and beared edges.

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