Abstract
Transition metal dichalcogenides XTe_{2} (X = Mo, W) have been shown to be secondorder topological insulators based on firstprinciples calculations, while topological hinge states have been shown to emerge based on the associated tightbinding model. The model is equivalent to the one constructed from a loopnodal semimetal by adding mass terms and spinorbit interactions. We propose to study a chiralsymmetric model obtained from the original Hamiltonian by simplifying it but keeping almost identical band structures and topological hinge states. A merit is that we are able to derive various analytic formulas because of chiral symmetry, which enables us to reveal basic topological properties of transition metal dichalcogenides. We find a linked loop structure where a higher linking number (even 8) is realized. We construct secondorder topological semimetals and twodimensional secondorder topological insulators based on this model. It is interesting that topological phase transitions occur without gap closing between a topological insulator, a topological crystalline insulator and a secondorder topological insulator. We propose to characterize them by symmetry detectors discriminating whether the symmetry is preserved or not. They differentiate topological phases although the symmetry indicators yield identical values to them. We also show that topological hinge states are controllable by the direction of magnetization. When the magnetization points the z direction, the hinges states shift, while they are gapped when it points the inplane direction. Accordingly, the quantized conductance is switched by controlling the magnetization direction. Our results will be a basis of future topological devices based on transition metal dichalcogenides.
Introduction
Higherorder topological insulators (HOTIs) are generalization of topological insulators (TIs). In the secondorder topological insulators (SOTIs), for instance, topological corner states emerge though edge states do not in two dimensions, while topological hinge states emerge though surface states do not in three dimensions^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. The emergence of these modes is protected by symmetries and topological invariants of the bulk. Hence, an insulator so far considered to be trivial due to the lack of the topological boundary states can actually be a HOTI. Indeed, phosphorene is theoretically shown to be a twodimensional (2D) SOTI^{16}. A threedimensional (3D) SOTI is experimentally realized in rhombohedral bismuth^{17}, where topological quantum chemistry is used for the material prediction^{18}. Transition metal dichalcogenides XTe_{2} (X = Mo, W) are also theoretically shown to be 3D SOTIs^{19,20}.
The tightbinding model for transition metal dichalcogenides has already been proposed, which is closely related to a type of loopnodal semimetals^{20}. A loopnodal semimetal is a semimetal whose Fermi surfaces form loop nodes^{21,22,23,24,25}. Especially, the Hopf semimetal is a kind of loopnodal semimetal whose Fermi surfaces are linked and characterized by a nontrivial Hopf number^{26,27,28,29,30}. There is another type of loop nodalsemimetals characterized by the monopole charge^{21}. An intriguing feature is that loop nodes at the zeroenergy and another energy form a linkedloop structure. The proposed model^{20} may be obtained by adding certain mass terms to this type of loopnodal semimetals.
It is intriguing that topological boundary states can be controllable externally. Magnetization is an efficient way to do so. Famous examples are surface states of 3D magnetic TIs^{31,32,33,34}, where the gap opens for outofplane magnetization, while the Dirac cone shifts for inplane magnetization. Similar phenomena also occur in 2D TIs, which can be used as a giant magnetic resistor^{35}. Recently, a topological switch between a SOTI and a topological crystalline insulator (TCI) was proposed^{36}, where the emergence of topological corner states is controlled by magnetization direction. We ask if a similar magnetic control works in transition metal dichalcogenides.
In this paper, we investigate a chiralsymmetric limit of the original model^{20} constructed in such a way that the simplified model has almost identical band structures and topological hinge states as the original one. Alternatively, we may consider that the original model is a small perturbation of the chiral symmetric model. A great merit is that we are able to derive various analytic formulas because of chiral symmetry, which enable us to reveal basic topological properties of transition metal dichalcogenides. We find that a linking structure with a higher linking number is realized in the 3D model. We also study 2D SOTIs and 3D secondorder topological semimetals (SOTSMs) based on this model. Depending on the way to introduce mass parameters there are three phases, i.e., TIs, TCIs and SOTIs in the 2D model. We find that topological phase transitions occur between these phases without band gap closing. Hence, the transition cannot be described by the change of the symmetry indicators. We propose symmetry detectors discriminating whether the symmetry is preserved or not. They can differentiate these three topological phases. Furthermore, we show that the topological hinge states in the SOTIs are controlled by magnetization. When the magnetization direction is out of plane, the topological hinge states only shift. On the other hand, when the magnetization direction is in plane, the gap opens in the topological hinge states.
Result
Hamiltonians
Motivated by the model Hamiltonian^{20} which describes the topological properties of transition metal dichalcogenides β(1T′)MoTe_{2} and γ(Td)XTe_{2} (X = Mo, W), we propose to study a simplified model Hamiltonian,
with
where σ, τ and μ are Pauli matrices representing spin and two orbital degrees of freedom. It contains three mass parameters, m, m_{Loop} and m_{SOTSM}. The role of the term m_{Loop} is to make the system a loopnodal semimetal, and that of the term m_{SOTSM} is to make the system a SOTSM. The Brillouin zone and high symmetry points are shown in Fig. 1(a). Although the band structure of the transition metal dichalcogenides is chiral nonsymmetric, the topological nature is well described by the above simple tightbinding model.
The original Hamiltonian contains two extra mass parameters and given by
with
The simplified model H_{SOTI} captures essential band structures of the original model \({H}_{{\rm{SOTI}}}^{^{\prime} }\). Indeed, the bulk band structures are almost identical, as seen in Fig. 1(b–d). The rod band structures are also very similar, as seen in Fig. 2(a4–d4,a5–d5), where the bulk band parts are found almost identical while the boundary states (depicted in red) are slightly different. Moreover, the both models have almost identical hinge states, demonstrating that they describe SOTIs inherent to transition metal dichalcogenides XTe_{2}.
A merit of the simplified model is the chiral symmetry, {H_{SOTI}(k_{x}, k_{y}, k_{z}), C} = 0, which is absent in the original model, \(\{{H}_{{\rm{SOTI}}}^{^{\prime} }({k}_{x},{k}_{y},{k}_{z}),C\}\ne 0\). Accordingly, the band structure of H is symmetric with respect to the Fermi level. Moreover, the bulk band structure is analytically solved. Here, the chiral symmetry operator is C = τ_{y}μ_{z}σ_{x} or C = τ_{y}μ_{z}σ_{y}. Let us call the original model a chiralnonsymmetric model and the simplified model a chiralsymmetric model.
The common properties of the two Hamiltonians H_{SOTI} and \({H}_{{\rm{SOTI}}}^{^{\prime} }\) read as follows. First, they have inversion symmetry P = τ_{z} and timereversal symmetry T = iτ_{z}σ_{y}K with K the complex conjugation operator. Inversion symmetry P acts on H_{SOTI} as P^{−1}H_{SOTI}(k)P = H_{SOTI}(−k), while timereversal symmetry T acts as T^{−1}H_{SOTI}(k)T = H_{SOTI}(−k). Accordingly, the Hamiltonian has the PT symmetry (PT)^{−1}H_{SOTI}(k)PT = H_{SOTI}(k), which implies that H^{*} = H. Second, the zcomponent of the spin is a good quantum number σ_{z} = s_{z}. Since we may decompose the Hamiltonian into two sectors,
it is enough to diagonalize the 4 × 4 Hamiltonians. All these relations hold also for \({H}_{{\rm{SOTI}}}^{^{\prime} }\). The relation (8) resembles the one that the KaneMele model is decomposed into the upspin and downspin Haldane models on the honeycomb lattice^{37,38,39}.
A convenient way to reveal topological boundary states is to plot the local density of states (LDOS) at zero energy. First, we show the LDOS for the Hamiltonian H_{0} in Fig. 2(a1). It describes a Dirac semimetal, whose topological surfaces appear on the four side surfaces. Then, we show the LDOS for the Hamiltonian
in Fig. 2(b1), where the topological surface states appear only on the two side surfaces parallel to the yz plane. We will soon see that a loopnodal semimetal is realized in H_{Loop}. Next, we show the LDOS for the Hamiltonian
in Fig. 2(c1), where a SOTSM is realized with two topological hingearcs. Finally, by including H_{SO}, we show the LDOS for the Hamiltonian H_{SOTI} in Fig. 2(d1), where a SOTI is realized with topological twohinge state.
Topological phase diagram
The chiralsymmetric Hamiltonian H_{SOTI} is analytically diagonalizable. The energy dispersion is given by
with
and
The topological phase diagram is determined by the energy spectra at the eight highsymmetry points Γ = (0, 0, 0), S = (π, π, 0), X = (π, 0, 0), Y = (0, π, 0), Z = (0, 0, π), R = (π, π, π), U = (π, 0, π) and T = (0, π, π) with respect to timereversal inversion symmetry. The energies at these highsymmetry points (k_{x}, k_{y}, k_{z}) are analytically given by
where η_{a} = ±1 and η_{b} = ±1. The phase boundaries are given by solving the zeroenergy condition (E = 0),
where η_{x} = ±1, η_{y} = ±1 and η_{z} = ±1. There are 16 critical points apart from degeneracy. When t_{x} = t_{y}, the critical points are reduced to be 12 since E(X) = E(Y) and E(U) = E(T). Hence, solving E = 0 for t_{z}, there are 6 solutions for t_{z} > 0, which we set as t_{n}, n = 1, 2, 3, …, 6 with t_{i} < t_{i+1}.
Loopnodal semimetals
We first study the loop nodal phase described by the Hamiltonian H_{Loop}. The energy spectrum is simply given by
The loopnodal Fermi surface is obtained by solving E(k) = 0. It follows that k_{x} = 0 and
Loop nodes at zero energy exist in the k_{x} = 0 plane. They are protected by the mirror symmetry M_{x} = τ_{z}μ_{z}σ_{x} with respect to the k_{x} = 0 plane and the PT symmetry^{21,40}. We show the band structure along the k_{x} = 0 plane in Fig. 3(a2–d2). We see clearly that the loop node structures are formed at the Fermi energy in Fig. 3(b2–d2). These loop nodes are also observed as the drumhead surface states, which are partial flat bands surrounded by the loop nodes as shown in Fig. 3(b3–d3). The low energy 2 × 2 Hamiltonian is given by
where σ is the Pauli matrix for the reduced two bands.
In addition, there are loop nodes on the k_{y} = 0 plane at E = −m_{Loop}, which are determined by
We find the two loops determined by Eqs (19) and (21) are linked, as shown in Fig. 4.
The system is a trivial insulator for 0 ≤ t_{z} < t_{1}. One loop emerges for t_{1} < t_{z} < t_{2} [Fig. 3(b1)], which splits into two loops for t_{2} < t_{z} < t_{3}, as shown in Fig. 3(d1). Correspondingly, drumhead surface states, which are partial flat band within the loop nodes, appear along the [100] surface [see Fig. 3(b3,c3 and d3)].
The emergence of the loopnodal Fermi surface is understood in terms of the band inversion^{20,40}, as shown in Fig. 4. The number of the loops are identical to the number of circles at the Fermi energy as in Fig. 4(a2–l2). When only one band is inverted along the ΓZ line, a single loop node appears [Fig. 4(b1)]. When two bands are inverted along the ΓZ line, two loop nodes appear [Fig. 4(d1)]. In the similar way, additional loops appear when additional bands are inverted along the XU and YT lines [Fig. 4(f1)], and it is split into two loops [Fig. 4(h1)] as t_{z} increases. In the final process, a loop appears along the SR line [Fig. 4(j1)], which splits into two loops [Fig. 4(l1)].
It has been argued^{20,40} that a new topological nature of loopnodal semimetals becomes manifest when we plot the loopnodal Fermi surfaces at the band crossing energies, where one is at the Fermi energy and the other is at E = −m_{Loop} in the occupied band. We show them in Fig. 4. Along the Γ−Z line, the other band crossing occurs at ±m_{Loop} with
Along the XU and YT lines, the band crossing occurs also at ±m_{Loop} with
Along the SR line, the band crossing occurs also at ±m_{Loop} with
As a result, it is enough to plot the Fermi surfaces at E = 0 and E = −m_{Loop}. The linking number N increases as t_{z} increases, where even the linking number N = 8 is realized as in Fig. 4(l1).
2D TI, TCI and SOTI
At this stage it is convenient to study the 2D models by setting t_{z} = λ_{z} = 0. It follows from (17) that the 2D topological phase boundaries are given by
where η_{x} = ±1 and η_{y} = ±1. Depending on the way to introduce the mass parameters there are three phases, i.e., TIs, TCIs and SOTIs.
The topological number is known to be the \({{\mathbb{Z}}}_{4}\) index protected by the inversion symmetry in three dimensions^{20,41,42,43}. This is also the case in two dimensions. It is defined by
where \({n}_{K}^{\pm }\) is the number of occupied band with the parity ±. There is a relation^{41,42,43}
where ν is the \({{\mathbb{Z}}}_{2}\) index characterizing the timereversal invariant TIs. We find from Fig. 5(c1) that κ_{1} = 0, 2 in the TI phase, which implies that it is trivial in the viewpoint of the timereversal invariant topological insulators.
We show the LDOS for TI, TCI and SOTI in Fig. 6. (i) When m_{Loop} = m_{SOTSM} = 0 and m < 2t, the system is a TI with κ_{1} = 2, where topological edge states appear for all edges [See Fig. 6(a)]. We show the energy spectrum and the Z_{4} index in Fig. 5(a1,a2), respectively. The energy spectrum is twofold degenerate since there is the symmetry \(P\bar{T}={\mu }_{y}\) such that \({(P\bar{T})}^{1}{H}_{0}(k)P\bar{T}={H}_{0}(k)\). Furthermore, there is the mirror symmetry M_{x} = iτ_{z}μ_{z} such that \({M}_{x}^{\,1}{H}_{{\rm{Loop}}}({k}_{x},{k}_{y}){M}_{x}={H}_{{\rm{Loop}}}(\,\,{k}_{x},{k}_{y})\). (ii) When m_{Loop} ≠ 0 and m_{SOTSM} = 0, the system is a TCI, where topological edge states appear only for two edges [See Fig. 6(b)]. The energy spectrum and the Z_{4} index are shown in Fig. 5(b1,b2). The symmetry \(P\bar{T}\) is broken for m_{Loop} ≠ 0 and the twofold degeneracy is resolved. On the other hand, the mirror symmetry M_{x} remains preserved. (iii) Finally, when m_{Loop} ≠ 0 and m_{SOTSM} ≠ 0, the system is a SOTI, where two corner states emerge [See Fig. 6(c)]. The energy spectrum and the Z_{4} index are shown in Fig. 5(c1,c2). The mirror symmetry is broken in the SOTI phase. In TCI and SOTI phases, there are regions where κ_{1} = 1, 3. However, in this region, the system is semimetallic and the κ_{1} index has no meaning.
The Z_{4} index takes the same value for the TI, TCI and SOTI phases, and hence it cannot differentiate them. Indeed, because there is no band gap closing between them^{44}, the symmetry indicator cannot change its value^{43}. A natural question is whether there is another topological index to differentiate them. We propose the symmetry detector discriminating whether the symmetry is present or not.
The TI and TCI are differentiated whether the symmetry \(P\bar{T}\) is present or not. The band is twofold degenerate due to the symmetry \(P\bar{T}\) in the TI phase, where we can define a topological index by
with
where i and j are the twofold degenerated band index. It is only defined for the TI phase, where it gives the same result as κ_{1}. On the other hand, it is illdefined for the TCI and SOTI phases since there is no band degeneracy.
The TCI and SOTI are differentiated by the mirrorsymmetry detector defined by
where
is the mirror symmetry indicator^{36} along the axis k_{x} = α with α = 0,π, and ± indicates the band index under the Fermi energy. It is χ = 1 when there is the mirror symmetry. On the other hand, it is χ ≠ 1 when there is no mirror symmetry since \(\psi \rangle \) is not the eigenstate of the mirror operator. In addition, it is χ ≠ 1 when the system is metallic since \(\langle \psi {M}_{x}\psi \rangle \) changes its value at band gap closing points. See Fig. 5(a3–c3). In Fig. 5(a3), we find always χ = 1 since the mirror symmetry is preserved, where we cannot differentiate the topological and trivial phases. On the other hand, in Fig. 5(b3), there are regions with χ ≠ 1 where the system is metallic. Finally, we find χ ≠ 1 in Fig. 5(c3) since the mirror symmetry is broken.
SOTSM
A 3D SOTSM is constructed by considering k_{z} dependent mass term in the 2D SOTI model^{10,12,13}. We set t_{z} ≠ 0, while keeping λ_{z} = 0 in the 2D SOTI model. The properties of the SOTSM are derived by the sliced Hamiltonian H(k_{z}) along the k_{z} axis, which gives a 2D SOTI model with k_{z} dependent mass term M(k_{z}). The bulk band gap closes at
On the other hand, there emerge hingearc states connecting the two gap closing points. Accordingly, the topological corner states in the 2D SOTI model evolves into hingestates, whose dispersion forms flat bands as shown in Fig. 2(c4).
Magnetic control of hinges in SOTI
Hinge states are analogous to edge states in twodimensional topological insulators. Without applying external field, spin currents flow. On the other hand, once electric field is applied, charge current carrying a quantized conductance flows. We show that the current is controlled by the direction of magnetization as in the case of topological edge states.
With the inclusion of the H_{SO}, the system turns into a SOTI, which has topological hinge states. We study the effects of the Zeeman term, where the Hamiltonian is described by H_{SOTI} together with the Zeeman term
which will be introduced by magnetic impurities, magnetic proximity effects or applying magnetic field.
We show the hinge states in the absence and the presence of magnetization in Fig. 7. Helical hinge states appear in its absence [see Fig. 7(a1)]. They are shifted in the presence of the B_{z} term [see Fig. 7(b1)]. On the other hand, they are gapped out when the B_{x} or B_{y} term exists [see Fig. 7(c1)].
For comparison, we also show the hinge states calculated from the chiralnonsymmetric Hamiltonian \({H}_{{\rm{SOTI}}}^{^{\prime} }\) [see Fig. 7(a2–c2)]. The band structure is almost symmetric with respect to the Fermi energy.
By taking into the fact that the σ_{z} is a good quantum number, the low energy theory of the hinge states is well described by
In the presence of the external magnetic field, it is modified as
which is easily diagonalized to be
It well reproduces the results based on the tight binding model shown in Fig. 7.
One of the intrinsic features of a topological hinge state is that it conveys a quantized conductance in the unit of e^{2}/h. We have calculated the conductance of the hinge states in Fig. 7 based on the Landauer formalism^{45,46,47,48,49,50,51}. In terms of singleparticle Green’s functions, the conductance σ(E) at the energy E is given by^{45,51}
where \({{\rm{\Gamma }}}_{R(L)}(E)=i[{{\rm{\Sigma }}}_{R(L)}(E){{\rm{\Sigma }}}_{R(L)}^{\dagger }(E)]\) with the selfenergies Σ_{L}(E) and Σ_{R}(E), and
with the Hamiltonian H_{D} for the device region. The self energies Σ_{L}(E) and Σ_{R}(E) are numerically obtained by using the recursive method^{45,46,47,48,49,50,51}.
The conductance is quantized, which is proportional to the number of bands. When there is no magnetization or the magnetization is along the z axis, the conductance is 2 since there are two topological hinges. On the other hand, once there is inplane magnetization, the conductance is switched off since the hinge states are gapped. It is a giant magnetic resistor^{35}, where the conductance is controlled by the magnetization direction.
Conclusion
We have studied chiralsymmetric models to describe SOTIs and loopnodal semimetals in transition metal dichalcogenides. The Hamiltonian is analytically diagonalized due to the chiral symmetry. We have obtained analytic formulas for various phases including loopnodal semimetals, 2D SOTIs, 3D SOTSMs and 3D SOTIs. We have proposed the symmetry detector discriminating whether the symmetry is present or not. It can differentiate topological phases to which the symmetry indicator yields an identical value. Furthermore, we have proposed a topological device, where the conductance is switched by the direction of magnetization. Our results will open a way to topological devices based on transition metal dichalcogenides.
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Acknowledgements
The author is very much grateful to N. Nagaosa for helpful discussions on the subject. This work is supported by the GrantsinAid for Scientific Research from MEXT KAKENHI (Grant Nos JP17K05490, JP15H05854 and No. JP18H03676). This work is also supported by CREST, JST (JPMJCR16F1).
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Ezawa, M. Secondorder topological insulators and loopnodal semimetals in Transition Metal Dichalcogenides XTe_{2} (X = Mo, W). Sci Rep 9, 5286 (2019). https://doi.org/10.1038/s41598019417465
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DOI: https://doi.org/10.1038/s41598019417465
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