Time-reversal-breaking induced quantum spin Hall effect (2024)

Model Hamiltonian

We consider the model as shown in Fig. 1(a). On a square lattice, there exist magnetic fluxes with directions alternating square by square. The magnetic fluxes induce an accompanying Peierls phase factor γ in the hopping elements along the arrow²³. Due to the existence of the magnetic fluxes, the translation symmetry is broken, and the lattice is divided into two sublattices denoted by A and B. The lattice constant is taken to be d = 1 in the following calculations. For such a two-dimension compound lattice, the primitive lattice vectors can be chosen to be a₁ = (1, 1) and a₂ = (1, −1). The corresponding primitive reciprocal lattice vectors are b₁ = (π, π) and b₂ = (π, −π).

The Hamiltonian of the system under consideration is given by

Here, is the electron creation operator on site i with α = A, B standing for A, B sublattices. The first term is the usual nearest-neighbor hopping term with being the sublattice different from α and the angular bracket in 〈ij〉 standing for nearest-neighbor sites. The second term is the intrinsic SOC with coupling strength λ_SO, where σ is the spin Pauli matrix vector, i and j are two next nearest-neighbor sites, k are their common nearest-neighbor sites, and vector d_ik points from k to i. The last term stands for the Rashba SOC with coupling strength λ_R. In addition, a Peierls phase factor appears in all of the hopping terms due to the staggered magnetic fluxes. Here, we adopt a symmetric gauge, and the Peierls phase satisfies , . The Peierls phase will be confined in the range .

It is clear that when the Peierls phase γ is nonzero, the TR symmetry is broken. As we will show below, it is the TR symmetry breaking that induces the QSH effect in the present case of a square lattice. Different from the honeycomb lattice of the Kane-Mele model, in which there exists only one common nearest neighbor between a pair of next nearest neighbor sites, there are two common nearest neighbors between a pair of next nearest neighbor sites in the square lattice. Thus, in the intrinsic SOC term, there exist two hopping paths connecting a pair of next nearest neighbor sites, and the total intrinsic SOC is the sum of them. If the Peierls phase is zero, the intrinsic SOC vanishes, as the cross products d_kj × d_ik from the two paths cancel. Take A₁, A₂ in Fig. 1(a) for example. The intrinsic SOC, coming from the sum of the two paths A₂ → B₁ → A₁, A₂ → B₂ → A₁, can be expressed as , where and represent the unit vectors in the x and y directions, respectively. It is clear that the sum in the square brackets is zero. However, the situation is different in the presence of a Peierls phase of hopping. Taking a nonzero Peierls phase into account, we have

One can see that only if the Peierls phase γ ≠ 0 or , there exists a nonzero intrinsic SOC, which is crucial for inducing the QSH effect.

Edge states

By making a Fourier transformation, we obtain the Hamiltonian in momentum space as

where the base vectors are chosen as {c_A↑, c_B↑, c_A↓, c_B↓}, and

Solving the energy eigenequation and by some algebra, it is found that for with , there is a finite energy gap of magnitude between the conduction and valence bands. For , the energy gap vanishes, and the conduction and valence bands cross at points in the momentum space.

Due to the existence of the nonzero intrinsic SOC, the system is possible in the QSH state, when the band gap is open. First, it is clear that the system is in the QSH state when λ_R = 0. In this case, the electron spin is conserved, and the system can be divided into two spin sectors. As long as γ does not take the value 0 or π/2, the contributions to the intrinsic SOC from different paths do not cancel, and act as spin-dependent magnetic fluxes coupled to the electron momentum. The two spin sectors of the QSH system behave like two independent QH systems without Laudau levels. When λ_R ≠ 0, the spin conservation is destroyed, the system can no longer be divided into two QH systems, and unconventional topological invariants, the Z₂ index or spin Chern numbers are needed to classify the system. Here, we use the spin Chern numbers to characterize the topological quantum phase. Employing the standard method^18,19 to calculate the spin Chern numbers, we obtain C_± = ±1 for , which corresponds to a topological QSH insulator.

According to bulk-edge correspondence, the spin Chern numbers C_± = ±1 indicate that there are a pair of edge modes counterpropagating at the open boundary of the system, which are spin polarized. It is worth pointing out that for the Kane-Mele model, when a Zeeman field, a TR-symmetry-breaking term, is present, there appears a small gap in the spectrum of the counter-propagating edge states due to the mixing of the two edge states on the same boundary. In the present model, the TR symmetry is also broken due to the staggered magnetic fluxes, so it is necessary to examine the behavior of edge states. To study the spectrum of the edge states, we consider an infinitely long strip geometry running along the x direction with width (N_y − 1)d. Open boundary conditions are imposed at the two edges of the strip. The system has translational invariance along the x direction, so that the x component of the momentum k_x is a good quantum number. The energy spectra corresponding to the QSH phase are shown in Fig. 2(a) and (b) for different Peierls phases and , respectively. One can notice a significant distinction between the edge states displayed in the two figures. In the case of , as shown in Fig. 2(b), it is apparent that there is a energy gap in the edge modes. This is to be expected, because the TR symmetry is broken, and the two edge states on the same boundary are mixed. However, for , the edge states are gapless. The two edge modes at the same sample edge cross each other at , as shown in Fig. 2(a). In general, in low dimensions, the band degeneracy is protected by some symmetry. For example, in the Kane-Mele model, the gapless edge states are protected by the TR symmetry. In the present case, there should be certain symmetry in the system to protect the gapless nature of the edge states for . We will search such a symmetry in the next section.

Symmetry protection of the gapless edge states

In order to search the symmetry, we first examine the characteristics of the edge states localized near one boundary. Considering the energy-momentum relation and the spin polarizations of the edge states, we have and , where α = x, y, z, and the subscript is the edge state index. Owing to with π being the reciprocal lattice vector of the strip geometry, one can find that this is similar to Kramers’ theorem of TR. Therefore, we conjecture that the symmetry protecting the gapless edge states consists of the TR. In addition, the fact that the edge states are gapless only when , indicates that the symmetry contains a factor closely relating to . We find the corresponding symmetry operator to be , where is a operator that transforms the accompanying phase in the hopping coefficients γ to , and Θ = iσ_yK represents the TR operator for spin- fermions, with K being the complex conjugation operator. For , the model is invariant under the composite transformation .

A Bloch function of the electrons has the form . The symmetry operator ΘP _{− π/2} acts on the Bloch function as follows

Because is the symmetry operator of the system, Ψ′_k′(r) must be a Bloch wave function of the system. Thus, we obtain k′ = −k.

From Eq. (5), it is easy to show that the operator has the effect to transform electron wave vector as k → −k = k′. If k′ = k + K, where K is the reciprocal lattice vector, then k is a -invariant point in momentum space. If G is a -invariant point, then we have . Due to , Ψ_G(r) and Ψ′_G(r) are both the eigenstates of Hamiltonian H and have the same eigenenergy E(G). In order to further understand the role of , we study the effect of on the Bloch function Ψ_G(r) based on Eq. (5), and find

We should note that when the second operator acts on Ψ′_G(r), which transforms the accompanying phase in the hopping coefficients γ′ to rather than because the TR reverses the direction of the magnetic fluxes. Finally, we obtain at all the -invariant points. It is known that if a system is invariant under the action of an antiunitary operator and the square of the operator is not equal to 1, there must be degeneracy protected by this antiunitary operator. Thus, the bands must be degenerate at the -invariant points in the Brillouin zone. For a strip geometry, just the k_x is a good quantum number, so there exist only two -invariant points k_x = 0 and . At one given edge, there are two dispersion branches cross each other at the -invariant point . However, at k_x = 0 they merge into the bulk and pair with the edge states of the other boundary.

From the above discussion, one can see that the symmetry is essentially a TR. The factor is a consequence of the symmetric gauge that we have chosen. Due to the freedom of gauge transformation, for the staggered magnetic fluxes shown in Fig. 1(a), we can choose different gauges, namely, the accompanying phases in the hopping terms can be chosen in different manners. This causes the Hamiltonian to be in different forms, and have different symmetry operators. If we choose a proper gauge, the factor in the symmetry operator can be removed. Specifically, if we choose the gauge shown in Fig. 1(b), it is clear that when the accompanying phase , which corresponds to in Fig. 1(a), the symmetry of the system will be characterized by the ordinary TR operator.

The effect of a Zeeman field

It is well known that QSH effect is not stable to magnetic perturbations, which break the TR symmetry. In the Kane-Mele model, a uniform Zeeman field will gap out the edge states of the QSH phase. Now we also consider the effect of a Zeeman field with strength g in the present system. The Hamiltonian of the Zeeman field is given by . It is clear that the Zeeman field H_Z violates the symmetry , no matter what value the accompanying phase γ takes. The energy spectrum for the strip geometry with is plotted in Fig. 3(a). To our surprise, when the system is in the QSH phase with C_± = ±1, even if g ≠ 0, the edge states are gapless as long as . One can easily see that the Zeeman field does not lift the degeneracy of the two edge states localized near a given boundary but shifts the position of the degenerate point through raising the energy of one edge state and reducing the energy of the other edge state. This means that the Zeeman field does not mix the two edge states, so does not cause backscattering of the edge states, and the transport in the edge states is still dissipationless at zero temperature. Similarly to the case of g = 0, however, once the accompanying phase γ deviates from , the edge states open a small gap as shown in Fig. 3(b).

As discussed above, the degeneracy in low dimensions is usually protected by certain symmetry. However, in the presence of the Zeeman field, we have not found the symmetry protecting the gapless edge states. Similarly, we calculate the spin polarizations of the edge states, and find that the relations no longer hold. Through numeric calculation, it is found that the degenerate points of the edge states localized near the two boundaries approximately satisfy , . The degenerate points are not at high-symmetry points, and shift with changing the strength of the Zeeman field. Because the edge states and degenerate points can no longer provide any useful clue, it is difficult to find the hidden symmetry.

Lastly, we make a comparison with previous works, refs 23 and 24, which also describe the robust QSH phase under TR symmetry breaking. In both the works, the symmetry protecting the gapless edge states is ΘT_1/2, with Θ the TR, and T_1/2 a primitive-lattice translation. Owing to the primitive-lattice translation, the edge states protected by ΘT_1/2 are no longer robust against nonmagnetic or magnetic disorder. In contrast, in our work, the symmetry protecting the gapless edge states does not depend on the primitive-lattice translation, so the edge states are robust against nonmagnetic disorder. More interestingly, the fact that the edge states in our model remain gapless in the presence of a Zeeman field and spin-flipping term indicates that the edge states are also robust against magnetic disorder.