Intrinsic topological metal state in T-graphene

An intrinsic topological metal (TM) state is found in the T-graphene, a monolayer with both the time-reversal symmetry and the four-fold symmetry. The state distinguishes itself by the nontrivial electric polarization from the ordinary metals and features with two local edge states in the corresponding nanoribbons. The TM state is confirmed as a transition state bridging the ordinary metal state and the topological insulator state when the relative neighboring hoppings change in the lattice. The topological nature is further verified by checking the robustness of transport property against randomly-introduced strong disorders. The fact that the multiple topological states indexed by different parameters coexist in such a practical system shows a broad prospect in versatile topological transport devices.


Introduction
The development of topological theory has greatly deepened our understanding of the matter phases and many exotic topological phases are elutriated out from their corresponding traditional classifications through nontrivial topological invariants [1][2][3]. The foremost one is the Chern insulator, which breaks the time-reversal symmetry and distinguishes itself by a nonzero Chern number [4][5][6][7]. When keeping the time-reversal symmetry, people find the topological insulators (TIs) which are characterized by the Z 2 number, alternatively [8][9][10][11][12]. These TIs, as well as topological superconductors [13], all possess the gaped energy structures and symmetry-protected edge states falling totally inside the gap. The accompanied topological phase transitions happen sharply just when the energy band gaps are closed and reopened at certain condition. Later the studies of topological semimetals tell us the topological phase transition can be fulfilled in the gapless states [14]. The topological semimetals are distinctive by their unique nodal points [15][16][17][18]. People still pursue whether the topological phase transition could exist within any metallic states, with gapless states but different from semimetals.
Recently, this kind of topological metal (TM) is proposed by Ying and Kamenev in a two-dimensional p + ip superconductor model through adjusting the magnetic flux [19,20]. They extended the sharply defined topological quantum phase transitions to states of matter with gapless electronic spectra. TMs are also studied in the one-dimensional chain [21,22] and the time-reversal symmetric two-dimensional lattice [23]. In these models the nontrivial edge states are predicted within the gapless bulk states. However, one expects a practical material to be an intrinsic TM, and also wants to know the role of the topologically-protected nontrivial edge states hidden within the gapless bulk states in its metallicity, such as the transport properties.
In the paper, we propose a concrete two-dimensional material called T-graphene which is a TM in its intrinsic state. The topological nature of the T-graphene are characterized by the nontrivial electronic polarization which is protected by both the time-reversal symmetry and the D 4 point group symmetry. The TM phase is found to be a transition state bridging the ordinary metal (OM) phase and the TI phase if the lattice parameters are adjusted. The topological state is also verified by its nontrivial edge states and robust transport properties against disorders in corresponding T-graphene nanoribbons (TGNRs).
The paper is organized as follows. In section 2, we present the model and the energy band structure of the system. Based on this, the topological invariant is introduced to classify the different topological phases. Also the method to calculate the transport property is also given. In section 3, we check the robustness of the TGNRs against the disorders, which further confirms the topological nature of the system. The influence of single point defect on the conductance is studied in section 4. The conclusion is drawn in section 5.

Model and methods
The T-graphene is a kind of graphene allotrope. It was theoretically predicted by Liu et al [24], and its industrial synthesis route is given [25]. It has been reported excellent application prospects in hydrogen storage [26], ion battery [27], and carbon monoxide detection [28]. So that the study on the exotic topolgical phases in such practical system not only extends the topological phase family in theory but also shows prospective applications in versatile topological transport devices.
As shown in figure 1(a), the structure of planar T-graphene is a two-dimensional monolayer composed of the vertices of four carbon rings and connected together by sp 2 orbital hybridization. T-graphene belongs to a square lattice, with four atoms in one primary cell, labelled by the red square in figure 1(a). The length of C-C bond inside the tetragonal carbon ring is l 1 = 1.468 Å, whereas that between the nearest-neighboring tetra-rings is l 2 = 1.383 Å. Both the lattice and its reciprocal one (the first Brillouin zone (BZ) is drawn in figure 1(b)) respect the D 4 symmetry.
The paper is organized as follows. In section 2, we present the model of T-graphene and the energy band structure of the system. Based on this, the topological invariant is pointed out to classify different topological phases. The method to calculated the transport property is also given here. In section 3, the robustness of topological state against strong disorder is checked, which further confirms the topological nature of the system. The influence of single point defect on the conductance is studied in section 4. The conclusion is in section 5.

The tight-binding model
The electronic properties in the T-graphene can be well described by the tight-binding Hamiltonian, which reads where t ij is the hopping of electrons through π-bonding from site i to j, c † (c) is the creation (annihilation) operator, and ε i is the on-site energy. Neglect the on-site energy, the Hamiltonian can be concisely expressed in the momentum space as where α = exp ik y a, β = exp ik x a and a is the lattice constant. For the high symmetric points Γ, M, X, Y, the energy values are solved as: The energy eigenvalues E(k) for other points can be solved numerically. Adopting two typical values of t 1 = −2.525 eV and t 2 = −2.835 eV [29], the energy band structure is obtained as figure 1(c) shows. It is obvious that the T-graphene is inherently a metal as a whole, and the energy band structure embodies its symmetry. It seems a trivial metallic material, however, the lattice holds nontrivial topological properties in its intrinsic state. Intuitively, we can make a judge by comparing the inter-and intra-cell hopping of the T-graphene. The Su-Schrieffer-Heeger (SSH) model [30][31][32] states the topological phase becomes nontrivial when the inter-cell hopping is over the intra-cell one [33,34], and so does the two-dimensional SSH model. This hints that the energy band structure in figure 1(c) maybe more than an ordinary metallic state. We changed the hoppings and see the evolution of the energy band structure. The subtle changes of the energy band structure, although all metallic, can be obviously seen in figure 2(a). When |t 2 /t 1 | < 1, there exists a band gap between bands I(II)and band III(IV). These band gaps collapse and finally close when |t 2 /t 1 | = 1. When |t 2 /t 1 | > 1, a new local band gap forms between bands II and III.

The topological invariant
Due to the time-reversal symmetry, the Berry curvature disappears everywhere in BZ so that we have to characterize the possible nontrivial topological phase by a topological invariant other than Chern number, which is zero here [8][9][10]. We employ the two-dimensional Zak phase together with the fractional wave polarization [30,35] to characterize the topological phases here. The polarization P is given by: where A = ⟨ψ|i∂ k |ψ⟩. A is the Berry connection, and the integration is over the first BZ. In addition, the lattice also has the inversion symmetry, which leads to P i = 0 or 1 2 . When P i = 1 2 (i = x, y), the state is nontrivial.
For our structure, calculations of P have observably simplification, thanks to its point group symmetry. We can check the following relationship [36]: i is the ith energy band, ϱ is the parity in high symmetrical points. Moreover, P x = P y for the T-graphene lattice, so that we only need to calculate P x . In another words, we only need the parities of X and Γ to determine the topological states. Taking account of C 4V point group symmetry, the parity can be easily obtained through the distribution of wave function in the primary cell. In figures 2(d) and (e) we plot the normalized wave functions ψ n=I(orII),Γ(orX) for |t 2 /t 1 | < 1 and |t 2 /t 1 | > 1, respectively. The parity of band is mainly determined by the real part of wave function. It is seen that the parity of ψ II,Γ is changed from odd to even while others keep unchanged. By this method, we calculated the polarization of each band, which are listed in figure 2(f). It is seen that the case of |t 2 /t 1 | > 1 is really a topological nontrivial state, distinguishing itself from the case of |t 2 /t 1 | < 1 by the polarization contributed by band II and band IV. The case of |t 2 /t 1 | = 1 is a critical point standing for a TM phase transition.

The conductance and the current distributions
Then we calculated the conductance and the microscopic current distributions of the TGNRs to verify the existence of various topological phases of T-graphene and the robustness of its topological states. The conductance of TGNRs can be otained by the nonequilibrium Green's function. The zero-temperature conductance G of the T-graphene is given by [37,38]  In there I is the identity matrix. The local current density at the Fermi level E between two neighboring sites i and j can be expressed as [39] where H ij is the corresponding matrix element of the Hamiltonian.

The robustness of topological state
The topological state can also be verified by edge states in the corresponding TGNRs. We engineer the T-graphene to TGNR by cutting it in the y direction, i.e. it is still infinite in x direction, but only has 20 unit cells in y direction. The energy band structure of the TGNRs can be calculated in the k x momentum space. In figures 3(a)-(c) the energy band structures of TGNRs are given in three cases corresponding to that of figures 2(a)-(c), respectively. As we stated before, the case of |t 2 /t 1 | > 1 is the topological nontrivial state. It is clearly seen that two local edge states (plotted in red lines) appear inside the gap, but is absent in the case of |t 2 /t 1 | < 1. Although the energy band structure tells it is metal as a whole, the edge states separate the energy bands above from that below locally. These two local edge states features the nontrivial topological nature of the metal, and would result in the robust transport properties against disorders as we will check soon. The different topological nature when |t 2 /t 1 | crossing 1 can also be seen in the armchair TGNRs. However, the topological phase transitions in zigzag TGNRs are clearer, so we focus on the zigzag TGNRs in the following calculations.
If we call the case of |t 2 /t 1 | < 1 the OM state, then this exotic topological state for |t 2 /t 1 | > 1 could be named as a TM state. The case of |t 2 /t 1 | = 1 is just the transition point from OM to TM. The local edge states can be transferred to a global one if we continue to increase the value of |t 2 /t 1 |. Calculations show that when   figure 3(e), which gives all the possible states in the system as the value of |t 2 /t 1 | increasing. With respect to the overlapping of the conduction and valence bands, the TGNRs undergoes not only a topological transition, but also a Wilson transition [23] from |t 2 /t 1 | < 1 to |t 2 /t 1 | > 2. Within the range where the conduction and valence band overlap, the material is metallic, but the electronic polarization has changed sharply and thus leads to two different metal phases.
We have checked the wave functions of the edge states. In the case of TI phase, it is well known that they are localized at two boundaries. In the case of TM, the wave functions are not strictly distributed at the boundary. Or we can say the edge states coexist with the bulk states. The edge states are protected topologically, and show wonderful robustness against disorders. Actually, it is more valuable to see the transport for the TM phase because it is metallic after all. The metallicity is accompanied with certain transport channels, which are expected to be immunity to disorders due to its nontrivial topology. Consequently, we manage to find the microscopic current distributions inside the TGNRs, especially in case of TM. With the help of nonequilibrium Green's function method, we can calculate the conductance and the current distribution [39]. The robustness of the edge states can be checked by its corresponding current distribution when certain disorders are introduced. Here a global disorder is considered by shifting the hopping randomly, i.e. the hopping is changed as t → t + αt, where α is a random number, representing the degree of disturbance to the system and ranging within [−δ, δ] in our calculation. It should be noted that we keep the Hermitian symmetry of the Hamiltonian. We consider the transport of a finite TGNRs with the width of N = 4 primary cells, which is sandwiched by the source and drain parts. The microscopic current distribution can be obtained by mimicking the transport from source to drain. In contrast, both the current distributions in TGNRs without disorder (figures 4(a), (c) and (e)) and those with 20% disorder(δ = 0.2) (figures 4(b), (d) and (f)) are calculated for OM, TM, and TI phases, respectively.
The results show that when |t 2 /t 1 | < 1 (OM phase, corresponding to figures 4(a) and (b)) the disorder has a significant influence on the current distribution. Several vortex currents are formed somewhere so that the otherwise perfect transport paths are destroyed. As the value of |t 2 /t 1 | increasing across 1, i.e. the system going into the phase of TM, the current densities at two edges begin to increase and exceed those inside the TGNRs as figure 4(c) shows. In this case, transport channels at the edges and those inside the TGNRs coexist. Then turn on the disorder, it is found (see figure 4(d)) that the currents inside the TGNRs almost disappeared as expected, however, the currents at two edges are still present, although its intensity is modified to some extent. The current channels along two edges still support better transport, which means the robustness against disorders for TM. When |t 2 /t 1 | > 2, the TGNRs becomes the well-known TI state. The  edge state evolves to a global one, totally separating from the bulk states in energy. Thus, when the Fermi level pinpoints near the edge state, the corresponding current is strictly distributed on both edges of the nanoribbons, as seen in figure 4(e). After the disorders is turned on, the current paths are immunity to the disorder and the edge state behaves strong robustness, which is impressively demonstrated in figure 4(f).

Effect of point defects in TGNRs
We also considered the influence of point defects on the conductance in TGNRs. For a perfect TGNRs, the quantum conductance is consistent with its energy band structure. For example, the conductance of the intrinsic TGNRs, with the typical width of four primary cells, is calculated as shown in figure 5. It is seen that there is no zero conductance because of its metallicity.
not occur near the Γ point, the Fermi surface, in (a) and (b), because the band gap is completely closed at the moment. Whereas from (c) to (d), i.e. from TM to TI phase, the energy band structure experiences a fundamental change: the separation of the conduction band and valence band.
When a single point defect is introduced, we consider the location of the single defect both at the edge and inside the TGNRs (intra defect) and we only focus on TM and TI phases, in comparison of the otherwise perfect TGNRs. The results are given in figure 7. It is seen that the single point defect positioned at the edge has a destructive influence on the current distributions in both cases. The otherwise perfect edge current channel has been destroyed completely by the point defect. But it has little effect on the current inside and the current channel along the other edge.
If we introduce the single point defect inside the TGNRs other than at the edge, the influence of the current distribution would be different in contrast. In figures 7(c) and (f), we plot the microscopic current distribution of the TGNRs when the single point defect locates inside the TGNRs, as the blue circle denotes. It is obvious that the current channels at two edges have no influence. Although the current distribution inside the TGNRs has changed to some extent, it does not cause important influence on the total conductance. This case is obvious especially for that of TI phase, where the edge states are only possible channels. It is concluded that only the defects at the edges would significantly affect the edge currents. The defects inside the TGNRs only affect the magnitude and the direction of the internal current, and have little effect on its edge currents. This further confirms the robustness of edge states.

Conclusion
To sum up, we investigated the multiple topological phases in the T-grpahene lattice and concluded that the T-graphene is an intrinsic TM, which is protected by both the time-reversal symmetry and the point group symmetry. The TM phase is characterized by the electronic polarization, and features with two local edge stats in corresponding nanoribbons. The topological nature is also verified by the robustness of the edge states against disorders in the transport. From OM to TM and finally to TI, all these phases could be realized in one lattice by adjusting the interactions. This means T-graphene is a perfect platform both to study the exotic topological states and to design the versatile transport devices.

Data availability statement
The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.