Zeeman-field-induced valley-dependent topological phase transitions on the surface of a topological crystalline insulator SnTe

Mirror-symmetric (001) surfaces of a topological crystalline insulator SnTe host an even number of Dirac cone structures of surface states. A Zeeman field generically gaps the surface states, leading to a 2D topological insulator. By symmetry analysis and calculation of spin-Chern numbers, we show that with varying the direction of the Zeeman field, the system displays a rich phase diagram, consisting of a quantum anomalous Hall (QAH) phase with Chern number C = 2, a QAH phase with C = 1, a quantum pseudospin Hall phase, and an unusual insulator phase. In the QAH phase with C = 1 and the insulator phase, the two valleys X and Y are in different topological states. These valley-dependent topological phases provide a new pathway to potential applications of valleytronics.


Introduction
A topological insulator is one of the most fascinating concepts found in this decade [1,2]. Recent flourish in the study of topological insulators is based on the finding of a time-reversal symmetry-protected topological insulator [3][4][5]. Most recently, a new class of topological insulator, a topological crystalline insulator (TCI) [6], has attracted much attention. Different from the Z 2 topological insulator, the metallic boundary states of a TCI are protected by crystal symmetry rather than time-reversal symmetry. This demonstrates the rich interplay between electronic topology and crystal symmetry in the TCI, and has advanced our understanding of the topological insulator. Since the concept of TCI was proposed by Fu [6], on the theoretical frontier, intensive efforts have been devoted to classifying topological phases in different crystal symmetry classes [7][8][9]. On the other hand, there is great interest in seeking TCI materials both theoretically and experimentally. It has been predicted theoretically that a TCI state can be realized in the IV-VI semiconductor SnTe material class [10] and anti-perovskite material family A 3 BX [11]. The former has been confirmed experimentally by angle-resolved photoemission spectroscopy [12][13][14] and scanning tunneling microscopy measurements [15,16].
The first material realization of TCI are the IV-VI semiconductors SnTe and related alloys Pb x Sn 1−x Te(Se), in which there are three types of the surface states, the (001), (111) and (110) surfaces [10,17], which have reflection symmetry with respect to the (110) plane. For the most interesting (001) surface, there exist four gapless Dirac cones [12][13][14], two of them, denoted by L X 1,2 , are located along ΓX, close to and symmetric about X; two others, denoted by L Y 1,2 , are located along ΓY, close to and symmetric about Y, shown in figure 1(a). Because the gapless Dirac cones are protected by crystalline symmetry, it is proposed that they can be gapped through some symmetry-breaking perturbations such as Zeeman field and structure distortion [10,18,19]. It is predicted that each gapped Dirac cone contributes s =  H e h 2 2 to the Hall conductance [19,20]. However, in the previous paper [19], only a minimal model for each Dirac cone located at L X Y , 1,2 is considered, which is a two-band · k p model and inevitably drops some information about spin or pseudospin. On the other hand, as pointed out in [21], although the low-energy band structures are largely determined by the gapless states at L X 1,2 (L Y 1,2 ), the topological properties of the TCI surfaces are dictated by these gapped surface states at X(Y). This prompts us to Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. consider a full four-band model containing the spin or pseudospin degree of freedom to investigate the topological property near X(Y) point.
It is interesting to note that the Dirac cones near X and Y points are protected by different reflection mirrors, thus applying different symmetry-breaking perturbations can open gaps at the Dirac cones near X and Y points separately, which opens up the possibility of the valley-dependent topological phase transitions [22,23]. In this paper, we investigate how to use a Zeeman field to achieve the valley-dependent topological phase transitions, as the Zeeman field orienting in different directions corresponds to different symmetry-breaking perturbations. We find that when the direction of a Zeeman field with a fixed magnitude B varies in space, a rich topological phase transition can occur. Using the pseudospin Chern numbers to classify the topological phases, we obtain a phase diagram on a sphere of radius B, consisting of a quantum anomalous Hall (QAH) phase with Chern number C = 2, a QAH phase with C = 1, a quantum pseudospin Hall (QPSH) phase, and an insulator phase. Among these phases, the QAH phase with Chern number C = 1 and the insulator phase are very interesting, as the X and Y points are in different topological states. For the QAH phase with Chern number C = 1, one valley is in the QAH phase, while the other is in the QPSH phase. The insulator phase is unusual in that both valleys are in the QPSH phase, but their pseudospin Chern numbers have opposite signs, although the total Chern number and total pseudospin Chern number of the two valleys are zero. These valley-dependent topological phases provide a new pathway to potential application of the TCI in valley-based electronics.

Model Hamiltonian
We adopt the four-band · k p model for the (001) surface states of the TCI near X point derived in [17]: where  k is the momentum with respect to X point, v 1 , v 2 are the velocities along x and y direction, σ α and τ α (α = x, y, z) are the Pauli matrices for the spin and pseudospin, respectively. The pseudospin represents the cation-anion degree of freedom, and m and δ describe the pseudospin mixing. The first two terms describe two degenerate Dirac cones. The Dirac points of these cones are located precisely at X point and at zero energy. The first pseudospin mixing term mτ x shifts the energy of the two Dirac cones from zero to positive and negative energies m and -m. The two Dirac cones overlap on a ring in  k -space at zero energy. The second pseudospin mixing term δσ x τ y lifts this degeneracy everywhere except for two points on the axis k x = 0, where two bands with opposite mirror eigenvalues ±i (associated with the reflection M x ) cross each other. The band hybridization generates a pair of Dirac points at energy E = 0 located on opposite sides of X at momenta ( ) , as illustrated in figure 1(b).
The X point is invariant under three point group symmetry operations C 2 , M x , and M y , which represent the two-fold rotation around surface normal, and mirror symmetry with respect to reflections of x and y axes,  -M x x : x and  -M y y : y . In addition, the time-reversal symmetry Θ is present. The Hamiltonian is invariant under the corresponding symmetry operations. When various perturbations are added, they have to satisfy a certain symmetry constraint, which has been summarized in [18]. Here, we only consider the effect of a Zeeman field on the surface of the TCI. It is apparent that the out-of-plane field B z breaks both mirror symmetries, while an in-plane field B x parallel to the x axis preserves the mirror symmetry M x but breaks the mirror symmetry M y , and vice versa for B y . According to symmetry analysis, the allowable coupling terms for the three components of the Zeeman field are given by [18] ( ) In the next section, we study the topological phase transitions with varying the directions of the Zeeman field.

Phase diagram
As mentioned above, the gapless Dirac cones located at L X 1,2 are protected by mirror symmetry M x , so the terms in V B x , which are invariant under M x , do not open up an energy gap, but just shift the positions of degenerate points. To our surprise, although all the terms in V B y and V B z break the mirror symmetry M x , not every one of them could open up a gap. Our calculation shows that among the mirror symmetry M x breaking perturbations, only σ z , σ z τ z , σ y τ z , σ z τ x could open up a gap. From figure 2(a), one can see that among the four gap-opening terms, an infinitesimal s z term or σ y τ z term is sufficient to open up a gap. The σ z τ z and σ z τ x terms open up a gap only when they reach certain critical values. The σ y and σ y τ x terms cannot open up a gap. They just shift the positions of the degenerate points, the same as V B x , and their respective influences on the band structures are shown in figures 2(b) and (c). In general, in two dimensions the band degeneracy is protected by certain symmetry [24], e.g., the point group or time-reversal symmetry. Sometimes the symmetry is not so apparent, as it may consist of a translation, a complex conjugation and a gauge transformation. Owing to the difficulty to find it, such a symmetry may be called a hidden symmetry [25,26]. Different from the point group and time-reversal symmetry, the hidden-symmetry-invariant points are not always at the high-symmetry points in the Brillouin zone, and their positions may also depend on the parameters [26]. In the present system, we conjecture that while σ y and σ y τ x break the mirror symmetry M x , a hidden symmetry still exists in the system. Similarly, for the terms σ y τ z and σ z τ x , when their strengths are smaller than the corresponding critical values, the hidden symmetry keeps protecting the band degeneracy. When their strengths exceed the critical values, the hidden symmetry is broken, leading to the gap opening.
Due to the presence of a pseudospin degree of freedom, the pseudospin Chern numbers can be defined when the energy gap is opened. Applying the standard method to calculate the pseudospin Chern numbers [27][28][29], we can derive the pseudospin Chern numbers C X± for the X valley. The values of the pseudospin Chern numbers characterize the topological phases of the X valley. If is in the QAH phase. Among the four gap-opening terms, σ z and σ y τ z will induce the QAH phase. If = -¹ + - is in the QPSH phase. When σ z τ z and σ z τ x exceed the critical values and open an energy gap, the system enters the QPSH phase. If C X+ = C X− = 0, X is in the topologically trivial phase, but none of the terms in equation (2) could induce this phase. When the terms inducing the QPSH effect and inducing the QAH effect coexist, along with the change in the strengths of the terms, the energy gap may close and then reopen. Accompanied with the energy gap closing, topological phase transitions occur. The terms σ z τ z and σ z τ x have a similar effect, favoring the QPSH phase, and σ z and σ y τ z have a similar effect, favoring the QAH phase. For example, we may consider first that the two terms σ z τ z and σ z coexist, which favor the QPSH and QAH effect, respectively, to study the topological phase transitions. In other words, we assume that the perpendicular Zeeman field only induces the coupling term σ z , and the in-plane Zeeman field only induces σ z τ z . For definiteness and without loss of generality, we fix the amplitude of σ z , and then increase the strength of σ z τ z from zero. It is found that along with increasing the strength of σ z τ z , the gap first reduces, and closes at a critical value, then reopens. After the gap reopens, the pseudospin Chern numbers change from C X+ = C X− = 1 2 to = -= + -C C X X 1 2 . This means that a topological phase transition from the QAH phase to QPSH phase occurs for the X valley. Because the amplitudes of σ z τ z and σ z depend on the strengths of the y and z components of the Zeeman field, the above result indicates that the topological property of X is actually determined by B y and B z . For a fixed B z , there exists a critical value B yc that closes the energy gap. If B y is bigger than B yc , the X point is in the QPSH phase. Otherwise, it is in the QAH phase.
We now consider the effect of the Zeeman field on the Y point. In the absence of the Zeeman field, the Hamiltonian near the Y point is given below Similar to X, for Y point we could obtain the Hamiltonian of the Zeeman field based on symmetry analysis. For simplicity, we also just consider two terms σ z τ z and σ z the same as X point. A difference is that now the term σ z τ z is induced by x component B x rather than y component B y . Indeed, we can deduce the effect of the Zeeman field on Y point from that on X point by symmetry considerations. Because X and Y are related to each other by a rotation of p 2 , the band structure near X has a symmetry-related copy near Y. For example, Zeeman field B z breaking both symmetries M x and M y has the same effect on X and Y, while the effect of Zeeman field B x (B y ) on Y can be deduced from the effect of Zeeman field B y (B x ) on X. Therefore, the topological property of the Y point is determined by B x and B z .
After the effect of Zeeman field  B on the X and Y points has been understood, we can investigate the topological phase transitions of the TCI surface containing both X and Y valleys in the presence of the Zeeman field. Before studying the phase diagram, we first briefly compare our results with a previous work about the QAH effect, and analyze the QPSH effect in our four-band model. Take the perpendicular Zeeman field as an instance. In the two-band model [19], when a perpendicular Zeeman field is applied, every gapped Dirac cone contributes a quantized Hall conductance  , with ± determined by the direction of the Zeeman field. In the four-band model under consideration, as shown above, when only a perpendicular Zeeman field is applied, 2 , with ± determined by the direction of the Zeeman field, so the total Chern number, defined as = + + + is the creation operator on site i with τ = ±1 standing for two orbits, namely, s and p orbits. The first term of H 0 is the Rashba spin-orbit coupling, ( )  s s s s = , , x y x y . The second term is the s − p-orbit coupling with amplitude m andt the orbit different from τ. The third term is the nearest neighbor hopping term, which can open up gaps at (0, 0) and (π, π). It can be easily shown that the lattice Hamiltonian H 0 can be reduced to the first three terms of H X and H Y near (π, 0) and (0, π) except the forth δ term. This is not an important issue as the δ term does not qualitatively influence the topology of the system. H′stands for the spin-and orbit-dependent next nearest neighbor hopping term with t 2 = λ x B 0 /4, where we have set B x = B y = B 0 . This term can open gaps at the X and Y points, and drive the system into the QPSH phase. To study the edge states, we calculate the energy spectrum of a long ribbon with 60 chains. The calculated energy spectrum is shown in figure 3(a). It is clear that there exist four different edge states at a given Fermi energy in the bulk band gap. Through the analysis of the spatial distribution of the wave functions, as shown in figure 3(b), one can find that the edge states represented by solid lines localize near one boundary of the ribbon, while the other two edge states represented by dashed lines localize near the other boundary. Take the edge states indicated by solid lines on one boundary as an example. From the slopes of the red and green lines, it is easy to determine that the two edge states are counterpropagating. We also examine the pseudospin polarizations of their wave functions, the red line being almost fully s-orbit polarized and the green line p-orbit polarized. Therefore, there exist two counterpropagating edge states with opposite pseudospin polarizations on a sample edge in the case C + = −C − = 1, which give rise to no net charge transfer, but can contribute to a net transport of pseudospin. In general, adding terms to the Hamiltonian, which do not close the bulk gap and mix the pseudospin, can gap out the edge states. From the viewpoint of edge states, the QPSH phase will no longer be well defined. However, the nontrivial bulk band topology characterized by the pseudospin Chern numbers will remain intact [29], and can have some observable effects, such as topological pseudospin pumping directly from the bulk of the system, as proposed recently [30].
Besides the pseudospin degree of freedom, there exists the valley degree of freedom in the TCI surface due to the two cones or valleys X and Y. This provides us with a new platform for valleytronics. Valleytronics [31][32][33][34], a technology of manipulating the degree of freedom, to which an inequivalent degenerate state electron near the Fermi level belongs, is a promising candidate for the next generation electronics. The main target of valleytronics is the honeycomb lattice systems. Indeed, various valley-dependent topological phases have been proposed theoretically in honeycomb lattices, such as the quantum valley Hall phase [32][33][34][35], valley-polarized QAH phase [22], spin-valley Hall phase [36], and quantum spin-QAH phase [23]. It is expected that these topological phases can also be achieved on the TCI surface. As shown above, the effects of a Zeeman field on the X and Y valleys of the TCI surface are different, such that valley-dependent topological phase transitions may happen with varying the Zeeman field.
In the above discussions, we show that the topological property of X(Y) is determined by B y (B x ) and B z . This indicates that it is possible to induce topological phase transitions through controlling the orientation of the Zeeman field. Therefore, we fix the magnitude B of the Zeeman field, and change its direction alone to study the topological phase transitions. Specifically, we consider a sphere with radius B, on which each point denotes a unique direction of the Zeeman field. The calculated phase diagram on the sphere is plotted in figure 4, in which different colors represent different topological phases. Yellow, blue, green and brown represent the QAH phase with total Chern number C = 2, QAH phase with total Chern number C = 1, QPSH phase, and insulator phase, respectively. We take figure 4(a) as an instance to illustrate these phases. In the yellow area I, owing to the strong out-of-plane Zeeman field, the pseudospin Chern numbers are 2 , for the X and Y points, respectively. The system is in the QAH phase, with the total Chern number C = 2. In the blue area II, the x component B x of the Zeeman field is bigger than the critical value B xc determined by corresponding B z , and the pseudospin Chern numbers for the Y point change to , as the y component of B y is small. As a result, the total Chern number is C = 1, and the system is in a QAH phase. In the green area III, B x and B y are bigger than the critical values B xc and B yc , as determined by the corresponding small B z , so the pseudospin Chern numbers for both X and Y change to 2 . Now, the total Chern number is 0, but the total pseudospin Chern number is C ± = ±1, and the system is in the QPSH phase. Finally, in the brown area IV, similar to the case in the green area III, the amplitudes of B x and B y are bigger than the critical values determined by the corresponding B z , but B y is negative, so the pseudospin Chern numbers for X and Y are ( ) Since now both total Chern number and total pseudospin Chern number are zero, the system is in an insulator phase.
It is interesting to examine more closely the topological properties of the QAH phase with Chern number C = 1 in the blue area II, and the insulator phase in the brown area IV. Because in the two phases, the pseudospin Chern numbers are different between X and Y. They are valley-dependent topological phases, and may have potential application in valleytronics. For the QAH phase with C = 1, one can see that the X point is in the QAH phase with pseudospin Chern numbers . Therefore, X and Y are in the QPSH states, but contribute to opposite pseudospin Hall conductivities. This phase is very similar to the valley Hall effect, in which upon the application of an electric field, electrons in different valleys will flow to opposite directions transverse to the electric field, giving rise to a net valley Hall current in the bulk [32][33][34][35]. In the present case, the two valleys give rise to opposite pseudospin Hall currents, leading to a net valley-pseudospin Hall current. The two phases, i.e., the QAH phase,with C = 1, and the unusual insulator phase, can be used to generate valley polarization and valley-pseudospin polarization by the Zeeman field, forming the basis for valleytronics.
It is worth noting that the regions with the same color are not always in the same topological phase. They are only in the same kind of topological phase. For example, both the green regions in figure 4(a) are in the QPSH phase, but they are in different topological phases, because the pseudospin Chern numbers in the two regions are different. In the upper area, the pseudospin Chern numbers for X and Y points are ( ) =   C X Y  , only two terns σ z and σ z τ z are considered, and we set μ z = λ y = 1. The inset illustrates the effect of a nonzero σ x τ x term on the phase boundaries. In (c) and (d), all terms that could be induced by the Zeeman field are considered, and we set μ x = μ y = μ z = 1, λ x = λ y = λ z = 0.8, η x = η y = η z = 0.5.