Second-order topological insulators and loop-nodal semimetals in Transition Metal Dichalcogenides XTe₂ (X = Mo, W)

Abstract

Transition metal dichalcogenides XTe₂ (X = Mo, W) have been shown to be second-order topological insulators based on first-principles calculations, while topological hinge states have been shown to emerge based on the associated tight-binding model. The model is equivalent to the one constructed from a loop-nodal semimetal by adding mass terms and spin-orbit interactions. We propose to study a chiral-symmetric 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 second-order topological semimetals and two-dimensional second-order 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 second-order 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 in-plane 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

Higher-order topological insulators (HOTIs) are generalization of topological insulators (TIs). In the second-order 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 two-dimensional (2D) SOTI¹⁶. A three-dimensional (3D) SOTI is experimentally realized in rhombohedral bismuth¹⁷, where topological quantum chemistry is used for the material prediction¹⁸. Transition metal dichalcogenides XTe₂ (X = Mo, W) are also theoretically shown to be 3D SOTIs^19,20.

The tight-binding model for transition metal dichalcogenides has already been proposed, which is closely related to a type of loop-nodal semimetals²⁰. A loop-nodal semimetal is a semimetal whose Fermi surfaces form loop nodes^{21,22,23,24,25}. Especially, the Hopf semimetal is a kind of loop-nodal semimetal whose Fermi surfaces are linked and characterized by a nontrivial Hopf number^{26,27,28,29,30}. There is another type of loop nodal-semimetals characterized by the monopole charge²¹. An intriguing feature is that loop nodes at the zero-energy and another energy form a linked-loop structure. The proposed model²⁰ may be obtained by adding certain mass terms to this type of loop-nodal 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 out-of-plane magnetization, while the Dirac cone shifts for in-plane magnetization. Similar phenomena also occur in 2D TIs, which can be used as a giant magnetic resistor³⁵. Recently, a topological switch between a SOTI and a topological crystalline insulator (TCI) was proposed³⁶, 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 chiral-symmetric limit of the original model²⁰ 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 second-order 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 Hamiltonian²⁰ which describes the topological properties of transition metal dichalcogenides β-(1T′-)MoTe₂ and γ-(Td-)XTe₂ (X = Mo, W), we propose to study a simplified model Hamiltonian,

$${H}_{{\rm{SOTI}}}={H}_{0}+{H}_{{\rm{SO}}}+{V}_{{\rm{Loop}}}+{V}_{{\rm{SOTSM}}},$$

(1)

with

$$\begin{array}{ccc}{H}_{0} & = & [m+{\sum }_{i=x,y,z}{t}_{i}\,\cos \,{k}_{i}]{\tau }_{z}\\ & & +\,{\lambda }_{x}\,\sin \,{k}_{x}{\tau }_{x}+{\lambda }_{y}\,\sin \,{k}_{y}{\tau }_{y}{\mu }_{y},\end{array}$$

(2)

$${H}_{{\rm{SO}}}={\lambda }_{z}\,\sin \,{k}_{z}{\tau }_{y}{\mu }_{z}{\sigma }_{z},$$

(3)

$${V}_{{\rm{Loop}}}={m}_{{\rm{Loop}}}{\tau }_{z}{\mu }_{z},\,{V}_{{\rm{SOTSM}}}={m}_{{\rm{SOTSM}}}{\mu }_{x},$$

(4)

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 loop-nodal 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 tight-binding model.

Figure 1

(a) Brillouin zone and high symmetry points. (b–d) Bulk band structures along the Γ-X-S-Y-Γ-Z-U-R-T-Z-Y-T-U-X-S-R-Γ line (b) for loop-nodal semimetal, (c) for SOTSM and (d) for SOTI. There are four bands in each phase. The dashed magenta curves represent the band structure of the chiral-symmetric model, while the dashed cyan curves represent that of the chiral-nonsymmetric model. They are indistinguishable in these figures. We have chosen t_x = t_y = 1, t_z = 2, λ_x = λ_y = 1, λ_z = 1.2, m = −3, m₂ = 0.3, m₃ = 0.2, m_mv1 = −0.4, m_mv2 = 0.2, m_Loop = 0.3 and m_SOTSM = 0.3.

The original Hamiltonian contains two extra mass parameters and given by

$${H}_{{\rm{SOTI}}}^{^{\prime} }={H}_{0}+{H}_{{\rm{SO}}}+{V}_{{\rm{Loop}}}^{^{\prime} }+{V}_{{\rm{SOTSM}}}^{^{\prime} }$$

(5)

with

$${V}_{{\rm{Loop}}}^{^{\prime} }={m}_{2}{\tau }_{z}{\mu }_{x}+{m}_{3}{\tau }_{z}{\mu }_{z},$$

(6)

$${V}_{{\rm{SOTSM}}}^{^{\prime} }={m}_{mv1}{\mu }_{z}+{m}_{mv2}{\mu }_{x}\mathrm{.}$$

(7)

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 XTe₂.

Figure 2

(a1–d1) Bird’s eye’s views of the LDOS of the zero-energy states: (a1) for H₀ with surface zero-energy states on the four side surfaces; (b1) for H_Loop with surface zero-energy states on the two side surfaces; (c1) H_SOTSM with hinge-arc states at two pillars; (d1) H_SOTI with hinge states at two pillars. (a2–d2) Top view of the LDOS corresponding to (a1–d1). (a3–d3) Bulk band structures of valence bands along k_x = 0 plane for these Hamiltonians. (a4–d4) Band structures of the square rod along z direction for these Hamiltonians. (a5–d5) Corresponding rod band structures for the chiral-nonsymmetric Hamiltonian H′. In these two sets of figures red curves represent topological boundary states. The Parameters are the same as in Fig. 1.

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_yμ_zσ_x or C = τ_yμ_zσ_y. Let us call the original model a chiral-nonsymmetric model and the simplified model a chiral-symmetric 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 time-reversal symmetry T = iτ_zσ_yK with K the complex conjugation operator. Inversion symmetry P acts on H_SOTI as P⁻¹H_SOTI(k)P = H_SOTI(−k), while time-reversal symmetry T acts as T⁻¹H_SOTI(k)T = H_SOTI(−k). Accordingly, the Hamiltonian has the PT symmetry (PT)⁻¹H_SOTI(k)PT = H_SOTI(k), which implies that H^* = H. Second, the z-component of the spin is a good quantum number σ_z = s_z. Since we may decompose the Hamiltonian into two sectors,

$${H}_{{\rm{SOTI}}}={H}_{{\rm{SOTI}}}^{\uparrow }\oplus {H}_{{\rm{SOTI}}}^{\downarrow },$$

(8)

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 Kane-Mele model is decomposed into the up-spin and down-spin 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 H₀ 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

$${H}_{{\rm{Loop}}}={H}_{0}+{V}_{{\rm{Loop}}}$$

(9)

in Fig. 2(b1), where the topological surface states appear only on the two side surfaces parallel to the y-z plane. We will soon see that a loop-nodal semimetal is realized in H_Loop. Next, we show the LDOS for the Hamiltonian

$${H}_{{\rm{SOTSM}}}={H}_{0}+{V}_{{\rm{Loop}}}+{V}_{{\rm{SOTSM}}}$$

(10)

in Fig. 2(c1), where a SOTSM is realized with two topological hinge-arcs. 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 two-hinge state.

Topological phase diagram

The chiral-symmetric Hamiltonian H_SOTI is analytically diagonalizable. The energy dispersion is given by

$$E=\pm \sqrt{F\pm \sqrt{G}}$$

(11)

with

$$\begin{array}{ccc}F & = & {M}^{2}+{m}_{{\rm{Loop}}}^{2}+{m}_{{\rm{SOTSM}}}^{2}\\ & & +\,{\lambda }_{x}^{2}{\sin }^{2}{k}_{x}+{\lambda }_{y}^{2}{\sin }^{2}{k}_{y}+{\lambda }_{z}^{2}{\sin }^{2}{k}_{z},\end{array}$$

(12)

$$G={({m}_{{\rm{SOTSM}}}{\lambda }_{x}\sin {k}_{x}-{m}_{{\rm{Loop}}}{\lambda }_{y}\sin {k}_{y})}^{2}$$

(13)

$$+\,2{M}^{2}({m}_{{\rm{Loop}}}^{2}+{m}_{{\rm{SOTSM}}}^{2}),$$

(14)

and

$$M=m+{\sum }_{i=x,y,z}{t}_{i}\,\cos \,{k}_{i}\mathrm{.}$$

(15)

The topological phase diagram is determined by the energy spectra at the eight high-symmetry 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 time-reversal inversion symmetry. The energies at these high-symmetry points (k_x, k_y, k_z) are analytically given by

$$E({k}_{i})={\eta }_{a}M({k}_{i})+{\eta }_{b}\sqrt{{m}_{{\rm{Loop}}}^{2}+{m}_{{\rm{SOTSM}}}^{2}},$$

(16)

where η_a = ±1 and η_b = ±1. The phase boundaries are given by solving the zero-energy condition (E = 0),

$${(m+{\eta }_{x}{t}_{x}+{\eta }_{y}{t}_{y}+{\eta }_{z}{t}_{z})}^{2}={m}_{{\rm{Loop}}}^{2}+{m}_{{\rm{SOTSM}}}^{2},$$

(17)

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.

Loop-nodal semimetals

We first study the loop nodal phase described by the Hamiltonian H_Loop. The energy spectrum is simply given by

$$E=\pm \,\sqrt{{\lambda }_{x}^{2}{\sin }^{2}{k}_{x}+{(\sqrt{{\lambda }_{y}^{2}{\sin }^{2}{k}_{y}+{M}^{2}}\pm |{m}_{{\rm{Loop}}}|)}^{2}}.$$

(18)

The loop-nodal Fermi surface is obtained by solving E(k) = 0. It follows that k_x = 0 and

$${\lambda }_{y}^{2}{\sin }^{2}{k}_{y}+{M}^{2}(\mathrm{0,}\,{k}_{y},{k}_{z})={m}_{{\rm{Loop}}}^{2}.$$

(19)

Loop nodes at zero energy exist in the k_x = 0 plane. They are protected by the mirror symmetry M_x = τ_zμ_zσ_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 drum-head 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

$$H=(\sqrt{{\lambda }_{y}^{2}{\sin }^{2}{k}_{y}+{M}^{2}}\pm |{m}_{{\rm{Loop}}}|){\sigma }_{z}+{\lambda }_{x}\,\sin \,{k}_{x}{\sigma }_{x},$$

(20)

where σ is the Pauli matrix for the reduced two bands.

Figure 3

(a1–d1) Loop-nodal zero-energy Fermi surfaces for (a1) t_z = t₁, (b1) t₁ < t_z < t₂, (c1) t_z = t₂ and (d1) t₁ < t_z < t₂. (a2–d2) Band structures along k_x = 0 plane. (a3–d3) Drum-head surface states of the valence band along the y-z plane. t_x = t_y = 1, λ_x = λ_y = 1; m = −3, m_Loop = 0.75. In (a2–d3), only the valence bands are shown for clarity.

In addition, there are loop nodes on the k_y = 0 plane at E = −m_Loop, which are determined by

$${\lambda }_{x}^{2}{\sin }^{2}{k}_{x}+{(M({k}_{x},\mathrm{0,}{k}_{z})-{m}_{{\rm{Loop}}})}^{2}={m}_{{\rm{Loop}}}^{2}.$$

(21)

We find the two loops determined by Eqs (19) and (21) are linked, as shown in Fig. 4.

Figure 4

Evolution of linking structures for various t_z. (a) t_z = t₁, (b) t₁ < t_z < t₂, (c) t_z = t₂, (d) t₂ < t_z < t₃, (e) t_z = t₃, (f) t₃ < t_z < t₄, (g) t_z = t₄, (h) t₄ < t_z < t₅, (i) t_z = t₅, (j) t₅ < t_z < t₆, (k) t_z = t₆ and (l) t_z > t₆. (a1–l1) Loop-nodal Fermi surfaces at the zero-energy (magenta) and at E = −m_Loop (cyan). They are linked, whose linking number N is shown in figures. (a2–l2) Band structure along the Γ-Z line (red), the X-U and Y-T lines (thick blue curves representing double degeneracy) and the S-R line (green). Only the valence bands are shown for 0 ≤ k_z ≤ π. Cross section of the loop nodes are marked in circles. The Parameters are the same as in Fig. 3.

The system is a trivial insulator for 0 ≤ t_z < t₁. One loop emerges for t₁ < t_z < t₂ [Fig. 3(b1)], which splits into two loops for t₂ < t_z < t₃, as shown in Fig. 3(d1). Correspondingly, drum-head 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 loop-nodal 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 Γ-Z line, a single loop node appears [Fig. 4(b1)]. When two bands are inverted along the Γ-Z line, two loop nodes appear [Fig. 4(d1)]. In the similar way, additional loops appear when additional bands are inverted along the X-U and Y-T 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 S-R line [Fig. 4(j1)], which splits into two loops [Fig. 4(l1)].

It has been argued^20,40 that a new topological nature of loop-nodal semimetals becomes manifest when we plot the loop-nodal 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

$$|{k}_{z}|=\arccos \,[({m}_{{\rm{Loop}}}-m-2t)/{t}_{z}].$$

(22)

Along the X-U and Y-T lines, the band crossing occurs also at ±m_Loop with

$$|{k}_{z}|=\text{arccos}\,[\,-\,m/{t}_{z}].$$

(23)

Along the S-R line, the band crossing occurs also at ±m_Loop with

$$|{k}_{z}|=\arccos \,[(\,-\,m+2t)/{t}_{z}].$$

(24)

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

$${(m+{\eta }_{x}{t}_{x}+{\eta }_{y}{t}_{y})}^{2}={m}_{{\rm{Loop}}}^{2}+{m}_{{\rm{SOTSM}}}^{2},$$

(25)

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

$${\kappa }_{1}\equiv \frac{1}{4}{\sum }_{K\in {\rm{TRIMs}}}({n}_{K}^{+}-{n}_{K}^{-}),$$

(26)

where ${n}_{K}^{\pm }$ is the number of occupied band with the parity ±. There is a relation^41,42,43

$${{\rm{mod}}}_{2}{\kappa }_{1}=\nu ,$$

(27)

where ν is the ${{\mathbb{Z}}}_{2}$ index characterizing the time-reversal invariant TIs. We find from Fig. 5(c1) that κ₁ = 0, 2 in the TI phase, which implies that it is trivial in the viewpoint of the time-reversal invariant topological insulators.

Figure 5

(a1–c1) Energy spectrum as a function of m/t for TI, TCI and SOTI phases. (a2–c2) Corresponding ${{\mathbb{Z}}}_{4}$ index. (a3–c3) Corresponding mirror-symmetry detector χ. It follows that χ = 1 for the TI and the insulating phase of the TCI, and that χ ≠ 1 for the SOTI since the mirror symmetry is broken.

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 κ₁ = 2, where topological edge states appear for all edges [See Fig. 6(a)]. We show the energy spectrum and the Z₄ index in Fig. 5(a1,a2), respectively. The energy spectrum is two-fold 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_zμ_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 Z₄ index are shown in Fig. 5(b1,b2). The symmetry $P\bar{T}$ is broken for m_Loop ≠ 0 and the two-fold 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 Z₄ 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 κ₁ = 1, 3. However, in this region, the system is semimetallic and the κ₁ index has no meaning.

Figure 6

(a1–c1) Eigenvalues of the sample in a square geometry, where the insets show the zero-energy states in red. The vertical axis is the energy. (a2–c2) corresponding LDOS of the zero-energy states. The amplitude is represented by the radius of the circles. We have set t_x = t_y = m = λ = 1 and m_Loop = m_SOTSM = 0.3.

The Z₄ 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 them⁴⁴, the symmetry indicator cannot change its value⁴³. 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 two-fold degenerate due to the symmetry $P\bar{T}$ in the TI phase, where we can define a topological index by

$$\zeta ={{\rm{Mod}}}_{4}\sum _{K\in {\rm{TRIMs}}}\,\frac{{\rm{Pf}}[w]}{\sqrt{{\rm{\det }}[w]}}$$

(28)

with

$${w}_{ij}=\langle {\psi }_{i}(\,-\,K)|P\bar{T}|{\psi }_{j}(K)\rangle ,$$

(29)

where i and j are the two-fold degenerated band index. It is only defined for the TI phase, where it gives the same result as κ₁. On the other hand, it is ill-defined for the TCI and SOTI phases since there is no band degeneracy.

The TCI and SOTI are differentiated by the mirror-symmetry detector defined by

$$\chi \equiv {\chi }_{0}^{+}{\chi }_{\pi }^{+}{\chi }_{0}^{-}{\chi }_{\pi }^{-},$$

(30)

where

$${\chi }_{\alpha }^{\pm }\equiv \frac{-i}{2\pi }{{\int }_{0}^{2\pi }\langle \psi |{M}_{x}|\psi \rangle d{k}_{y}|}_{{k}_{x}=\alpha }$$

(31)

is the mirror symmetry indicator³⁶ 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

$${M}^{2}({k}_{z})={m}_{{\rm{Loop}}}^{2}+{m}_{{\rm{SOTSM}}}^{2}.$$

(32)

On the other hand, there emerge hinge-arc states connecting the two gap closing points. Accordingly, the topological corner states in the 2D SOTI model evolves into hinge-states, 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 two-dimensional 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

$${H}_{{\rm{Z}}}={B}_{x}{\sigma }_{x}+{B}_{y}{\sigma }_{y}+{B}_{z}{\sigma }_{z},$$

(33)

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)].

Figure 7

Band structures for hinge states (a1) without magnetic field, (b1) with magnetic field along the z direction and (c1) with magnetic field along the x direction for the chiral-symmetric Hamiltonian H_SOTI. Hinge states are depicted in red. (a2–c2) Corresponding ones for the chiral-nonsymmetric Hamiltonian ${H}_{{\rm{SOTI}}}^{^{\prime} }$. (a2–c2) and (a′2–c′2) The conductance is quantized proportional to the number of bands in various cases.

For comparison, we also show the hinge states calculated from the chiral-nonsymmetric 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

$$H=\hslash {v}_{{\rm{F}}}{k}_{z}{\sigma }_{z}.$$

(34)

In the presence of the external magnetic field, it is modified as

$$H=\hslash {v}_{{\rm{F}}}{k}_{z}{\sigma }_{z}+{B}_{x}{\sigma }_{x}+{B}_{y}{\sigma }_{y}+{B}_{z}{\sigma }_{z},$$

(35)

which is easily diagonalized to be

$$E=\pm \,\sqrt{{(\hslash {v}_{{\rm{F}}}{k}_{z}+{B}_{z})}^{2}+{B}_{x}^{2}+{B}_{y}^{2}}.$$

(36)

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 e²/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 single-particle Green’s functions, the conductance σ(E) at the energy E is given by^45,51

$$\sigma (E)=({e}^{2}/h){\rm{Tr}}[{{\rm{\Gamma }}}_{{\rm{L}}}(E){G}_{{\rm{D}}}^{\dagger }(E){{\rm{\Gamma }}}_{{\rm{R}}}(E){G}_{{\rm{D}}}(E)],$$

(37)

where ${{\rm{\Gamma }}}_{R(L)}(E)=i[{{\rm{\Sigma }}}_{R(L)}(E)-{{\rm{\Sigma }}}_{R(L)}^{\dagger }(E)]$ with the self-energies Σ_L(E) and Σ_R(E), and

$${G}_{{\rm{D}}}(E)={[E-{H}_{{\rm{D}}}-{\Sigma }_{{\rm{L}}}(E)-{\Sigma }_{{\rm{R}}}(E)]}^{-1},$$

(38)

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 in-plane magnetization, the conductance is switched off since the hinge states are gapped. It is a giant magnetic resistor³⁵, where the conductance is controlled by the magnetization direction.

Conclusion

We have studied chiral-symmetric models to describe SOTIs and loop-nodal semimetals in transition metal dichalcogenides. The Hamiltonian is analytically diagonalized due to the chiral symmetry. We have obtained analytic formulas for various phases including loop-nodal 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.