Topological phase transitions with and without energy gap closing

Topological phase transitions in a three-dimensional (3D) topological insulator (TI) with an exchange field of strength $g$ are studied by calculating spin Chern numbers $C^\pm(k_z)$ with momentum $k_z$ as a parameter. When $|g|$ exceeds a critical value $g_c$, a transition of the 3D TI into a Weyl semimetal occurs, where two Weyl points appear as critical points separating $k_z$ regions with different first Chern numbers. For $|g|<g_c$, $C^\pm(k_z)$ undergo a transition from $\pm 1$ to 0 with increasing $|k_z|$ to a critical value $k_z^{\tiny C}$. Correspondingly, surface states exist for $|k_z|<k_z^{\tiny C}$, and vanish for $|k_z| \ge k_z^{\tiny C}$. The transition at $|k_z| = k_z^{\tiny C}$ is acompanied by closing of spin spectrum gap rather than energy gap.

The quantum Hall (QH) effect [1,2] in a twodimensional (2D) electron gas under a strong magnetic field provided the first example of topological state of matter in condensed matter physics, which cannot be described by the Landau theory of symmetry breaking. Thouless, Kohmoto, Nightingale, and Nijs (TKNN) revealed that the essential character of a QH insulator, different from an ordinary insulator, is a topological invariant of occupied electron states [3] or many-body wavefunctions [4]. They related the Hall conductivity of the system to the first Chern number (or TKNN number), which is quantized when the Fermi level lies in an energy gap between Landau levels. In such systems, topological phase transitions can happen only by closing the energy gap. Gapless edge states must appear on the boundary between a QH insulator and an ordinary insulator, which is ensured by the topological invariant. Interestingly, Haldane [5] proposed a spinless electron model on a 2D honeycomb lattice with staggered magnetic fluxes to realize the topological QH effect without Landau levels.
The quantum spin Hall (QSH) effect was first theoretically predicted by Kane and Mele [6] and by Bernevig and Zhang [7], and then experimentally observed in HgTe quantum wells. [8,9] Unlike the QH systems, where timereversal (TR) symmetry must be broken, the QSH systems preserve the TR symmetry. The main ingredient is the existence of strong spin-orbit coupling, which acts as spin-dependent magnetic fluxes coupled to the electron momentum. The QSH state is characterized by a bulk band gap and gapless helical edge states on the sample boundary. [6][7][8][9][10][11] The existence of the edge states is due to nontrivial topological properties of bulk energy bands. However, the bulk band topology of the QSH systems cannot be classified by the first Chern number, which always vanishes. Instead, it is classified by new topological invariants, namely, the Z 2 index [12] or the spin Chern numbers. [13][14][15] For TR-invariant systems, both Z 2 and spin Chern numbers were found to give an equivalent description. [14,15] The robustness of the Z 2 index relies on the presence of the TR symmetry. In contrast, the spin Chern numbers remain to be integer-quantized, independent of any symmetry, as long as both the band gap and spin spectrum gap stay open. [14] They are also different from the first Chern number for the QH state, which is protected by the bulk energy gap alone. The spin Chern numbers have been employed to study the TR-symmetry-broken QSH effect. [16] The QSH system is an example of the 2D topological insulators (TIs). Its generalization to higher dimension led to the birth of 3D TIs. [17][18][19][20][21] A 3D TI has a bulk band gap and surface states on the sample boundary. The metallic surface states provide a unique platform for realizing some exotic physical phenomena, such as Majorana fermions [22] and topological magnetoelectric effect. [23,24] The 3D TIs have been experimentally observed in Bi 1−x Sb x , Bi 2 Te 3 , and Bi 2 Se 3 materials, [25][26][27][28][29][30] which greatly stimulates the research in this field. The 3D TIs with TR symmetry are usually classified by four Z 2 indices, [17,18] and are divided into two general classes: strong and weak TIs, depending on the sum of the four Z 2 indices. In the presence of disorder, while the weak TIs are unstable, the strong TIs remain to be robust. The Z 2 indices are essentially defined only on the TR-symmetric planes in the Brillouin zone, and do not provide information about the distribution of surface states in the full momentum space. When the TR symmetry is broken, the Z 2 indices become invalid. Therefore, a more general characterization scheme for the bulk band topology, which does not rely on any symmetry and can provide more information about the distribution of surface states, is highly desirable.
In this Letter, for a 3D TI with an exchange field of strength g, we consider a momentum component, e.g., k z , as a parameter, and analytically calculate spin Chern numbers C ± (k z ) for the effective 2D system. The phase diagram for C ± (k z ) obtained can describe the systematic evolution of the bulk band topology of the 3D TI with varying parameters, and provide more information about the surface states. For small g, C ± (k z ) take values ±1 for |k z | < k C z , and undergo a transition to 0 at |k z | = k C z .
Correspondingly, on a sample surface parallel to the zaxis, helical surface states exist in the region |k z | < k C z , and disappear for |k z | ≥ k C z . At |k z | = k C z , the spin spectrum gap closes, but the energy gap remains open, which is distinct from a usual topological phase transition, where the energy gap always collapses. When |g| is greater than a critical value g c , a topological transition of the 3D TI into a Weyl semimetal occurs. Two Weyl points appear as critical points separating a QH phase of the effective 2D system for |k z | < k W z from an ordinary insulator phase for |k z | > k W z , indicating that their appearance is topological rather than accidental. Chiral surface states existing in the region |k z | < k W z give rise to the Fermi arcs.
Let us start from the effective Hamiltonian proposed in Ref. [19]: which was used to describe the strong TI of Bi 2 Se 3 . Here, σ m and τ m (m = x, y or z) denote the Pauli matrices in spin and orbital spaces, and y is the mass term expanded to the second order. In the last term, we include an exchange field of strength g, in order to study the TR-symmetry-broken effect on topological properties of the TI. For convenience, the momentum is set to be dimensionless, by properly choosing the units of parameters in the model, namely, Making a unitary transformation H = U † HU with The eigenstates of Hamiltonian (1) can be easily solved by first diagonalizing the operator in the square bracket.
The four eigenenergies are obtained as where λ(k) = M 2 (k) + A 2 1 k 2 z , and subscripts ± indicate two valence (conduction) bands with superscript v (c). The electron wavefunctions in the valence bands are given by The basic idea of our theoretical calculation is explained as follows. We consider one of the momentum components, e.g., k z as a parameter. For a given k z , Eq.
(1) is equivalent to a 2D system, for which spin Chern numbers C ± (k z ) can be defined. For a semi-infinite sample of the 3D TI with its surface parallel to the z axis, k z remains to be a good quantum number. Correspondingly, nonzero C ± (k z ) indicate that edge states with the given k z must appear on the edge of the effective 2D system. The edge states at various k z essentially form surface states of the 3D sample. Therefore, the characteristics of the surface states can be determined from the calculation of the k z -dependent spin Chern numbers C ± (k z ).
The spin Chern numbers for the effective 2D system are calculated in a standard way, which has been described in details in previous works. [15,16] By studying a special case of k z = 0, we find that the topological properties of Eq. (1) can be described by the spin Chern numbers C ± (k z ) associated with τ z . Here, τ z corresponds to U τ z U † = τ z σ z in the original Hamiltonian H, and so can be considered as a spin operator, measuring the difference of spin polarization between the two orbits. The eigenstates of projected spin operator P τ z P need to be calculated first, where P is the projection operator into the valence bands. Since P τ z P commutes with momentum operator, its eigenstates can be obtained at each momentum k separately. The eigenvalues of P τ z P are given by The corresponding eigenfunctions are denoted by Ψ ± (k), whose expressions are lengthy and will not be written out here. The spin Chern numbers are just the Chern numbers of the two spin sectors formed by Ψ ± (k), i.e., where ∇ 2 is the 2D Laplace operator acting on (k x , k y ). By some algebra, C ± (k z ) are derived to be with P (k z ) = M 0 − B 1 k 2 z and Q(k z ) = P 2 (k z ) + A 2 1 k 2 z . Equations and vanish otherwise. C ± (0) play a role similar to the Z 2 index. Nonzero C ± (0) ensure that surface states exist in the vicinity of k z = 0 on a surface parallel to the z axis. Without loss of generality, we will focus on the parameter region of B 2 > 0 and M 0 > 0, to which Bi 2 Se 3 belongs. We wish to emphasize here that when k z is considered as a parameter, the effective 2D Hamiltonian (1) breaks the TR symmetry for any k z = 0, even if g = 0, as its TR couterpart is at −k z . Therefore, while the k z -dependent spin Chern numbers given by Eq. (5) remain to be valid at any k z , a Z 2 index cannot be defined for any k z = 0. Eq. (5) allows us to extract more information about the basic characteristics of the surface states.
A typical phase diagram for the spin Chern numbers in the k z versus g plane, as determined by Eq. (5), is plotted in Fig. 1. For simplicity, A 2 is taken to be the unit of energy. For small |g|, C ± (k z ) = ±1 at small |k z |, corresponding to a QSH phase of the effective 2D system, and drop to 0 with increasing |k z | to a critical value k C z = M 0 /B 1 , as indicated by the dotted lines. The system becomes an ordinary insulator for |k z | > k C z . When |g| is greater than a critical value g c , the effective 2D system enters a QH phase with a nonzero total (first) Chern number C(k z ) ≡ C + (k z ) + C − (k z ) = 1 if g > 0, and −1 if g < 0. The boundary enclosing this phase is determined by equation g = ±Q(k z ), as indicated by the solid curves. The critical exchange field is given by g c = min[Q(k z )], and g c ≃ 0.1 for the parameters used in Fig. 1.
It is interesting to see how the energy gap ∆ E (k z ) = min[E c ± (k) − E v ± (k)]| kz and the spin spectrum gap ∆ τ (k z ) = min[ξ + (k) − ξ − (k)]| kz behave on the boundary (solid and dashed lines) between different phases. From Eqs. (2) and (4), we find that on the dotted lines in Fig.  1, the spin spectrum gap closes at k x = k y = 0, but the energy gap remains open. On the contrary, on the solid boundary lines, the energy gap closes at k x = k y = 0, but the spin spectrum gap remains open. ∆ E and ∆ τ as functions of k z for several values of g are plotted in Figs. 2(a) and 2(b), respectively. We notice that at g = 0.14 and k z = k C z = 0.2, the energy gap and spin spectrum gap vanish simultaneously. This is because the dotted and solid boundary lines in Fig. 1 intersect just at that point.
To study the surface states directly, we construct a tight binding model on a cubic lattice with two spins and two orbits on each site, which recovers the Hamiltonian Eq. (1) in the continuum limit. A semi-infinite sample with its surface parallel to the y-z plane is considered, where k y and k z remain to be good quantum numbers. The calculated energy spectrum for g = 0.05 is plotted as a function of k y for four different values of k z in Fig.  3. Although g = 0 breaks the TR symmetry, the surface states remain to be gapless at k z = 0, because τ z in Eq.
(1) is conserved at k z = 0. From Fig. 3, it is found that for k z = 0, k C z /3, and 2k C z /3, surface states always exist in the bulk energy gap, but no surface states appear at k z = k C z . To see the evolution of surface states with k z more clearly, we define a maximum level spacing δE between the surface states and bulk states, as illustrated in Fig. 3. In Fig. 4(a), δE is plotted as a function of k z . One can see that δE decreases with increasing k z , and drops to nearly 0 at k z ≥ k C z . Therefore, we conclude