Topological Properties of Electrons in Honeycomb Lattice with Detuned Hopping Energy

Honeycomb lattice can support electronic states exhibiting Dirac energy dispersion, with graphene as the icon. We propose to derive nontrivial topology by grouping six neighboring sites of honeycomb lattice into hexagons and enhancing the inter-hexagon hopping energies over the intra-hexagon ones. We reveal that this manipulation opens a gap in the energy dispersion and drives the system into a topological state. The nontrivial topology is characterized by the index associated with a pseudo time-reversal symmetry emerging from the C6 symmetry of the hopping texture, where the angular momentum of orbitals accommodated on the hexagonal “artificial atoms” behaves as the pseudospin. The size of topological gap is proportional to the hopping-energy difference, which can be larger than typical spin-orbit couplings by orders of magnitude and potentially renders topological electronic transports available at high temperatures.

where c i is the annihilation operator of electron at atomic site i with on-site energy ε 0 satisfying the anti-commutation relation, 〈 i , j〉 and 〈 i ′ , j′ 〉 run over NN sites inside and between hexagonal unit cells with hopping energies t 0 and t 1 respectively [see Fig. 1(a)]. The orbitals are considered to be the simplest one without any internal structure, such as the π electron of graphene. Below we are going to detune the hopping energy t 1 while keeping t 0 constant, and elucidate possible changes in the electronic state. In this case, the pristine honeycomb lattice of individual atomic sites is better to be considered as a triangular lattice of hexagons, with the latter characterized by C 6 symmetry. Let us start with the Hamiltonian within a single hexagonal unit cell   with eigen energies 2t 0 , t 0 , t 0 , − t 0 , − t 0 and − 2t 0 respectively, up to normalization factors. As shown in Fig. 1(b), the emergent orbitals accommodated on the hexagonal "artificial atom" take the shapes similar to the conventional s, p, d and f atomic orbitals. It is easy to check that the wave functions x y x y xy 2 2 are related each other by the operator T UK , where σ z is the Pauli matrix. Therefore, the operator  can be taken as a pseudo-TR operator, and the orbital angular momentum plays the role of a pseudospin. The relation  = −1 2 yields the Kramers doubling, a property originating from the C 6 symmetry. It is noticed that the high-energy states |s〉 and f are singlets, and thus the pseudospin and pseudo-TR symmetry are valid only for low-energy physics, which however is sufficient for realizing nontrivial topological properties in the present system (see also ref. 21).
Distinguished from the intrinsic spin, the pseudospin is directly related to the chiral current density on the hexagon. For a lattice model, the current density between two sites is given by . The current distributions evaluated using wave functions in Eq. (4) for the pseudospin-up and -down states are shown in Fig. 2(a,b) with anticlockwisely and clockwisely circulating currents. By considering the hexagonal artificial atoms composed by six sites in honeycomb lattice, one harvests states with angular momenta merely from simple orbitals, such as π electrons in graphene. The pseudo-TR symmetry is supported by the C 6 crystal symmetry, sharing the same underlying physics with the topological crystalline insulator 24 . However, for crystal-symmetry-protected topological insulators addressed so far, strong SOCs are required to achieve band inversions [25][26][27] , which is different from the present approach as revealed below.
Topological phase transition. We calculate the energy dispersion of Eq. (1) for several typical values of t 1 (hereafter the on-site energy is put as ε 0 = 0 without losing generality). As shown in Fig. 2, there are two two-fold degeneracies at the Γ point corresponding to the two two-dimensional (2D) representations of C 6 point group. Projecting the wave functions for t 1 = 0.9t 0 onto the orbitals given in Fig. 1(b), it is found that the topmost two valance bands show the character of d orbitals whereas the lowest two conduction bands behave like p orbitals [see Fig. 2 , which are equivalent to the ones at K and K′ points in the unfolded Brillouin zone of honeycomb lattice with the rhombic unit cell of two sites. When t 1 increases further from t 0 , a band gap reopens at the Γ point. As shown in Fig. 2(e) for t 1 = 1.1t 0 , the valence (conduction) bands are now occupied by p (d) orbitals around the Γ point, opposite to the order away from the Γ point and to that before gap closing. Therefore, a band inversion between p and d orbitals takes place at the Γ point when the inter-hexagon hopping energy is increased across the topological transition point t 1 = t 0 , namely the pristine honeycomb lattice.
We can characterize the topological property of the gap-opening transition shown in Fig. 2 by a low-energy effective Hamiltonian around the Γ point. Since the bands near the Fermi level are predominated by p and d orbitals, it is sufficient to downfold the six-dimensional Hamiltonian H(k) associated with the tight-binding model (1) into the four-dimensional subspace [p + , d + , p − , d − ]. The second term in Eq. (1) is then simply given by Contributions from the third term in Eq. (1) to the effective Hamiltonian can be evaluated in the following way 28 . First, we list the inter-hexagon hoppings in terms of 6 × 6 matrices † † h h h h h , , , , With Fourier transformations of matrices in Eqs. (5) and (6), one obtains the effective low-energy Hamiltonian H′ (k) on the basis [p + , d + , p − , d − ] in the momentum space. Expanding H′ (k) around the Γ point up to the lowest-orders of k, one arrives at is a 2 × 2 zero matrix, and a 0 is the lattice constant of the triangular lattice. For δt = 0, the Hamiltonians H + (k) and H − (k) in Eq. (8) are the same as the well-known one for honeycomb lattice around the K and K′ points 29 , where the quadratic terms of momentum in the diagonal parts can be neglected.
For δt > 0, however, the quadratic terms are crucially important since they induce a band inversion 30 , resulting in the orbital hybridization in the band structures shown in Fig. 2(e). Associated with a skyrmion in the momentum space for the orbital distributions in the individual pseudospin channels, a topological state appears characterized by the  2 topological invariant 10,11,21,31 . It is clear that for δt < 0 there is no band inversion taking place and thus the band gap is trivial as shown in Fig. 2(c).
The pseudo-TR symmetry satisfying = −1 2  is preserved at the Γ point. Going away from the Γ point, the C 6 symmetry is gradually broken, so does the pseudo-TR symmetry, and the two pseudospin channels start to mix with each other. However, this mixing is weak around the Γ point where the topological property of the system is determined. This can be seen directly from the band structure shown in Fig. 2(e) [as well as in Fig. 2(c,d)], where a gap between p + and p − (d + and d − ) is hardly observed around the Γ point. Analytically, the matrix elements between the two pseudospin channels in the Hamiltonian (7) are quadratic in momentum, which can be neglected as high-order corrections when the topological property is addressed.
It is worthy noticing that, comparing with the Kane-Mele model for the honeycomb lattice 10,11 , the mass term δt(> 0) in Eq. (8) can be considered as an effective SOC associated with the pseudospin, namely λ eSOC = δt. For δt = 0.1t 0 , a moderate texture in hopping energies, the effective SOC is approximately 3000 times larger than the real SOC in magnitude in graphene where λ SOC ≃ 0.1 meV and t 0 = 2.7 eV. The huge effective SOC is due to its pure electronic character as compared with the intrinsic SOC originated from the relativistic effect. This is one of the fantastic aspects of the present approach, which renders a topological gap corresponding to temperature of thousands of Kelvin.
Scientific RepoRts | 6:24347 | DOI: 10.1038/srep24347 Topological edge states and associated conductances. We consider a ribbon of hexagonal unit cells of t 1 = 1.1t 0 with its two edges cladded by hexagonal unit cells of t 1 = 0.9t 0 . As can be seen in Fig. 3(a), additional states appear in the bulk gap as indicated by the red solid curves carrying double degeneracy. Plotting the spatial distribution of the corresponding wave functions, we find that the in-gap states are localized at the two interfaces between topological and trivial regions [see Fig. 3(b)]. As displayed in Fig. 3(c,d), there is an excess upward (downward) edge current in the pseudospin-up (-down) channel associated with the state indicated by the red (green) dot in Fig. 3(a).
At the interface between topological and trivial regimes, the crystal symmetry is reduced from C 6 to C 2 , which breaks the pseudo-TR symmetry in contrast to the real TR symmetry. As the result, a mini gap of ~0.01t 0 [unnoticeable in the scale of Fig. 3(a)] opens in the edge states at the Γ point due to the coupling between two pseudospin channels. In order to quantitatively check possible backscatterings caused by this mini gap, we perform calculations on the longitudinal and Hall conductances based on a Hall bar system as sketched in Fig. 4(a). It is clear that the current I injected from the left electrode divides itself into two branches according to the pseudospin states, namely pseudospin-up (-down) electrons can flow only in the upper (lower) edge of the Hall bar. By matching wave functions at the interfaces between the six semi-infinite electrodes and the topological scattering region 32,33 , one can evaluate the transmissions of plane waves scattered among all the six leads, and then the longitudinal and Hall conductances,  34 , where ρ xx and ρ xy are the longitudinal and transverse resistances, respectively. Similar to the case of QSHE with magnetic impurities 35 , the values of conductivity in the present system deviate from the quantized ones when the Fermi level falls in the mini gap of ~0.01t 0 as shown in Fig. 4(b). It is noticed, however, that both G xx and G xy heal quickly after several periods of oscillations that come from interferences between the two pseudospin channels. It is emphasized that almost perfectly quantized conductances G xx = 2e 2 /h and G xy = 0 30,36 are achieved for the Fermi level beyond 0.04t 0 up to the bulk gap edge at 0.1t 0 , where the edge states with almost perfect linear dispersions hardly feel the existence of the mini gap and essentially no appreciable backscattering exists. On the other hand, if the inter-hexagon hopping energy is put far away from the intra-hexagon one in topological and/or trivial regimes, edge states may hardly be noticed 37 .
Now we investigate the hopping-energy dependence of the longitudinal conductance. The size of scattering region is same as in Fig. 4(a) and fixed for all cases. As displayed in Fig. 5(a), G xx saturates to the quantized value 2e 2 /h as expected for a  2 topological state for all the cases with t 1 = 1.05t 0 , 1.1t 0 and 1.2t 0 in the topological region (whereas 0.95t 0 , 0.9t 0 and 0.8t 0 in the trivial region correspondingly) when the Fermi level is set away from the mini gaps, accompanied by oscillations due to interferences between the two pseudospin channels.
We then check the sample-size dependence of the longitudinal conductance. We fix inter-hexagon hopping integrals at 1.1t 0 and 0.9t 0 in the topological and trivial regions respectively. As displayed in Fig. 5(b), G xx saturates in all cases to the quantized value 2e 2 /h when the Fermi level is shifted away from the mini gap. The topological edge transports remain unchanged when the size of the topological region becomes large. Real spin and QSHE. In addition to the pseudospin, the real spin degree of freedom also contributes to transport properties. In absence of the real SOC, the results presented in Fig. 4 remain exactly the same, with an additional double degeneracy due to the two spin channels and thus G xx = 4e 2 /h. An intrinsic SOC is induced when next-nearest-neighbor hoppings in honeycomb lattice are taken into account 10,11 . The low-energy Hamiltonian around the Γ point in Eq. (7) is then modified to with  where ν = 1 and −1 stand for spin-up and -down states respectively. Therefore, in the spin-up channel SOC enhances (suppresses) the topological gap in the pseudospin-up (-down) channel presuming λ > 0 [see the left and central panels of Fig. 6(a)]. As far as λ < δt, the system remains the  2 topological state associated with the pseudospin, where electrons with up pseudospin and down pseudospin counter propagate at edges, both carrying on up and down spins. The longitudinal conductance G xx saturates to 4e 2 /h as displayed in Fig. 6(b). When SOC is increased to λ = δt, the pseudospin-down (-up) channel with up (down) spin is driven into a semi-metallic state with zero band gap and the Dirac dispersion appears at the Γ point. When SOC goes beyond δt, this Dirac dispersion opens a gap accompanied by a topological phase transition. The system now takes a QSHE state where at edges electrons with up spin and pseudospin propagate oppositely to electrons with down spin and pseudospin. Evaluating the longitudinal conductance, one finds that G xx is quantized exactly to 2e 2 /h [see Fig. 6(b)], and as shown in the right panel of Fig. 6(a) there is no mini gap in the edge states, as protected by real TR symmetry 10,11 .

Discussions
A tight-binding model for spinless fermions on honeycomb lattice was considered in a previous work with a Kekulé pattern in nearest-neighbor hopping integrals 23 . It was revealed that fractional charges can be achieved when the mass gap contains a vortex, which requires the hopping integral depending on position in a special way. In contrast, in the present model Hamiltonian (1) there are only two values of hopping integral, and when the inter-hexagon one is larger than the intra-hexagon one a  2 topological state characterized by emergent pseudospin degree of freedom appears. We then discuss possible experimental realization of our theoretical proposal. Much effort has been devoted towards realizing the Dirac-like energy dispersion in artificial honeycomb lattices 38 , ranging from optical lattices 39,40 to 2D electron gases modulated by periodic potentials 13,41,42 and In 2 Te 2 /graphene bilayers 43 . All these systems provide promising platforms for realizing topological properties by detuning effective hopping energy among NN sites either by modulating muffin-tin potentials or bond lengthes periodically. To be specific, here we focus on how to achieve a topological state on the Cu [111] surface modulated by triangular gates of carbon monoxide (CO) molecules 13 . When extra CO molecules are placed at specific positions over the pristine molecular graphene, the bonds of the hexagons surrounding them are elongated since the CO clusters enhance local repulsive potentials and push electrons away from them, which reduces the corresponding electron hopping energies 42 . It is extremely interesting from the present point of view that hopping textures with C 6 symmetry have already been achieved in experiments 13 . We propose to place extra CO atoms in the pattern displayed in Fig. 7(a), where the intra-hexagon hopping energy t 0 (green thin bonds) surrounding the CO clusters is reduced and the inter-hexagon hopping energy t 1 (red thick bonds) is enhanced relatively. According to the above discussions, the system displayed in Fig. 7(a) with t 1 > t 0 should take a topological state. The hopping texture in Fig. 7(b), dual to that shown in Fig. 7(a), was realized in experiments 13 , where the system takes a topologically trivial state because t 1 < t 0 (see also ref. 44).
The underlying idea of the present scheme for achieving the  2 topological state is to create artificial orbitals carrying on opposite orbital angular momenta and parities with respect to spatial-inversion symmetry, and to induce a band inversion between them by introducing the hopping texture with C 6 symmetry on honeycomb lattice. In the sense that it does not require SOC, the present state may be understood as a quantum orbital Hall effect. The topological properties can also be extended to photonic crystals 21 , cold atoms, physical systems of exciton, polariton, surface plasmon, and phonon.
In conclusion, we propose to derive topological properties by modulating electron hopping energy between nearest-neighbor sites of honeycomb lattice. Because of the hopping texture with C 6 symmetry, atomic-like orbitals emerge, which carry a pseudospin degree of freedom characterizing a pseudo time-reversal symmetry and rendering Kramers pairs. We reveal that the effective spin-orbit coupling associated with the pseudospin degree of freedom can be larger than the intrinsic one by orders of magnitude because of the pure electronic origin. The present work offers a new possibility for achieving topological properties and related novel quantum properties and functionalities at high temperatures.  surface with a triangular lattice of CO molecules: (a) t 1 > t 0 generating topological state, (b) t 1 < t 0 for trivial state. Gray balls are CO molecules decorated by STM techniques, and red thick bonds are shorter than green thin ones which generates the hopping textures.