Weyl semimetals in optical lattices: moving and merging of Weyl points, and hidden symmetry at Weyl points

We propose to realize Weyl semimetals in a cubic optical lattice. We find that there exist three distinct Weyl semimetal phases in the cubic optical lattice for different parameter ranges. One of them has two pairs of Weyl points and the other two have one pair of Weyl points in the Brillouin zone. For a slab geometry with (010) surfaces, the Fermi arcs connecting the projections of Weyl points with opposite topological charges on the surface Brillouin zone is presented. By adjusting the parameters, the Weyl points can move in the Brillouin zone. Interestingly, for two pairs of Weyl points, as one pair of them meet and annihilate, the originial two Fermi arcs coneect into one. As the remaining Weyl points annihilate further, the Fermi arc vanishes and a gap is opened. Furthermore, we find that there always exists a hidden symmetry at Weyl points, regardless of anywhere they located in the Brillouin zone. The hidden symmetry has an antiunitary operator with its square being −1.

Scientific RepoRts | 6:33512 | DOI: 10.1038/srep33512 two dimensions, gapless topological phases were proposed in honeycomb optical lattices 21,22 and square optical lattices [23][24][25] . The important progress is the realization of topological semimetals in honeycomb optical lattices 26 . In three dimensions, Weyl semimetal were proposed to realized in optical lattices [27][28][29][30] . In order to engineer the topological phases in optical lattices, sometimes, the hopping-accompanying phase, i.e., the Peirls phase, is required. In experiments, the hopping-accompanying phase has been realized with periodic lattice shaking 31,32 and laser-assisted tunneling techniques [33][34][35][36] . Another important progress in experiments is the measurement of Zak phase of topological Bloch bands in optical lattices 37 , which provides a path to detect topological characters in optical lattices.
In this paper, we design a cubic optical lattice trapping cold fermionic atoms, which can be realized based the laser-tunnelling technique [33][34][35][36] . In different parameter ranges, the system supports three classes of Weyl semimetals, one of which has two pairs of Weyl points in the Brillouin zone, the other two have one pair of Weyl points in the Brillouin zone. By adjusting the parameters, we can study the moving and merging of Weyl points. When Weyl points with opposite topological charges meet together, they annihilate and a topological phase transition happens. We also investigate the Fermi arc of surface states of a (010) slab. Fermi arcs connect the projections of Weyl poionts on the surface Brillouin zone and evolve with the moving of Weyl points. For the Weyl semimetal phase with two pairs of Weyl points, there are two Fermi arcs connect projections of Weyl points with opposite charges on the surface Brillouin zone. When a pair of Weyl points annihilate, the two Fermi arcs link into one single Fermi arc connecting the projections of the remaining Weyl points. We find that the band degeneracy at Weyl points implies a hidden symmetry that has an antiunitary operator with its square being − 1. Based on a mapping method, we discover the hidden symmetry at each Weyl point in the Brillouin zone and discuss its relation with topological phase transitions.

Results
Weyl semimetals in optical lattices. Here, we consider a cubic optical lattice as shown in Fig. 1, where the arrows represent the hopping-accompanying phase. The hopping-accompanying phase is π/2 for the hopping along the y axis and π for the z axis. Due to the appearing of the hopping-accompanying phases, the translation symmetry is broken. Thus the lattice is divided into two sublattices, i.e. sublattices A and B. Assuming the distance between the nearest lattice sites being 1, we define the primitive lattice vectors as a 1 = (1, − 1, 0), a 2 = (1, 1, 0), and a 3 = (0, 0, 1). The primitive reciprocal lattice vectors are b 1 = (π, − π, 0), b 2 = (π, π, 0), and b 3 = (0, 0, 2π). Besides the hopping between nearest lattice sites, we also consider the diagonal hopping in the x− y plane and a staggered potential. The corresponding Hamiltonian is  Figure 1. Schematic of the cubic optical lattice. Here, the blue and green balls represent sublattices A and B, respectively; the single arrows and double arrows denote π/2 and π phases along with the hopping, respectively.

a a t e b b Hc
where a i and b i are the annihilation operators destructing a particle at a lattice site of sublattice A and B, respectively; t x and t y represent the amplitudes of hopping along the x and y directions, respectively; t xy denotes the amplitude of hopping along the diagonal direction; v represents the magnitude of the staggered on-site potential. This optical lattice can be realized through the laser-assisted tunneling technique, which has been applied in several experiments [33][34][35][36] .
Taking the Fourier's transformation on Equations (1), (2) and (3), we rewritten the Hamiltonian as x x x y y y z z x y z with α = v/2t z and β = 2t xy /t z being the dimensionless parameters and σ x , σ y and σ z being the Pauli matrices defined in the sublattice space. Diagonalizing Equation (4), we obtain the corresponding dispersion relation as with m = 2t z (α + β sin k x sin k y ). From this dispersion relation, we can see that two bands touch at some points W i in the Brillouin zone in some parameter ranges. Near the touching points, the dispersion relation has the linear form as x x x y y y z z z with p = k − W i . Around the the touching points, the chirality can be defined as ij which is also the topological charge at Weyl points. Thus, the touching points are Weyl points and, correspondingly, the system is a Weyl semimetal phase. According to the number of Weyl points in different parameter ranges, we can classify the system into four phases: (i) When α β + < 1 and α β − < 1 are satisfied, there are four distinct points π π α β = ± + W ( /2, /2, arccos( )) 1,2 and π π α β = − ± − W ( /2, /2, arccos( )) 3,4 in the Brillouin zone. Since there are two pairs of Weyl points in the Brillouin zone, we term this phase as WSM2 phase.
(ii) For the case α β + < 1 and α β − > 1, only the pair W 1,2 exists. Thus, we term this phase as WSM1a phase. (iii) For the case α β + > 1 and α β − < 1, where the Weyl points W 3,4 still remain. We term this new phase as WSM1b phase, which is different from the WSM1a phase. (iv) For α β + > 1 and α β − > 1, no Weyl point exists and a gap opens, so the system is a band insulator. The phase diagram is shown as in Fig. 2.

Moving and merging of Weyl points, topological phase transition, and Fermi arcs of surface states.
Here, we investigate moving and merging of Weyl points along with varying of the dimensionless parameters α and β. Merging of Weyl points and annihilations of topological charges lead to topological phase transitions. In our model, there are four kinds of topological phase transitions such as (i) transition from the WSM2 phase to the MSM1a phase, (ii) transition from the MSM2 phase to the MSM1b phase, (iii) transition from the MSM1a phase to the band insulator phase, and (iv) transition from the MSM1b phase to the band insulator phase. The WMS2 phase has two pairs of Weyl points W 1,2 and W 3,4 with topological charges C 1,2 = ± 1 and C 3,4 = ± 1, as shown in Fig. 3(a). When we keep α + β invariant and increase α − β, the Weyl points W 3,4 move towards each other and W 1,2 stay at the original positions. When α − β increases to 1, the Weyl points meet at (π/2, − π/2, 0) in the Brillouin zone and merge, as shown in Fig. 3(b). When α − β further increases more than 1, the Weyl points W 3,4 annihilate and only W 1,2 remain, the system from the MSM2 phase turns into the MSM1a phase, i.e., topological phase transition (i) happens. Topological phase transition (i) can also occur through the other type of moving and merging of Weyl points. Starting from the WSM2 phase, we keep α + β invariant and decrease α − β, the Weyl points W 3,4 move away from each other and W 1,2 stay at their starting positions. When α− β decreases to − 1, the Weyl points W 3,4 arrive at (π/2, − π/2, ± π), which are identical points in the Brillouin zone, i.e., W 3,4 meet and merge, as shown in Fig. 3(c). When α− β is less than − 1, W 3,4 annihilate and topological phase transition (i) happens. When it arrives at the MSM1a phase, there exist only one pair of Weyl points W 1,2 , which have opposite topological charges, in the Brillouin zone, as shown in Fig. 3(d). Similarly, there are two types of moving and merging of Weyl points to realize topological phase transition (ii), i.e., the transition from the WSM2 phase to the WSM1b phase. We can vary the value of α + β and keep α− β invariant. When α + β increases to 1 or − 1, the Weyl points W 1,2 meet and merge at the center or the surface of the Brillouin zone, and W 3,4 still remain. When |α + β| is greater than 1, the Weyl points W 1,2 annihilate and a topological phase transition from the MSM2 phase into MSM1b phase happens, as shown in Fig. 3(e). For the MSM1a phase, we can also increase |α + β| to 1, the remaining Weyl points W 1,2 meet and merge at the center or the surface of the Brillouin zone, as shown in Fig. 3(f). If we further increase |α + β| greater than 1, the remaining Weyl points W 1,2 annihilate and a gap opens, topological phase transition (iii) happens. For the MSM1b phase, if we increase |α − β| to 1, the remaining Weyl points W 3,4 meet and merge at the surface or corner of the Brillouin zone. If we further increase |α − β| greater than 1, the remaining Weyl points W 3,4 annihilate and a gap opens, so topological phase transition (iv) happens. In all the topological phase transitions, it is found that topological charges respect a conservation law and they are only created and annihilated in pairs.
In order to further study the characters of Weyl semimetals and topological phase transitions, we calculate the surface states of a slab geometry with (010) surfaces and investigate the evolution of Fermi arcs along with the moving and merging of Weyl points. In Fig. 4, we show the spectral function of the surface states at zero energy. The spectral function can be calculated through the formula Green function of the system. The projections of Weyl points W 1,2,3,4 on the surface Brillouin zone are denoted as ∼ W 1,2,3,4 . Figure 4(a) shows that, in the WSM2 phase, there are two Fermi arcs in the surface Brilllouin zone, which connect points ∼ W 1 and ∼ W 3 , ∼ W 2 and ∼ W 4 , respectively. Since Weyl points W 1 and W 3 , W 2 and W 4 have opposite topological charges, we conclude that Fermi arcs connect the projections of Weyl points with opposite topological charges on the surface Brillouin zone. When |α− β| increases to 1, ∼ W 3 and ∼ W 4 meet and merge at the side boundary or the corners of the surface Brillouin zone, thereby two Fermi arcs combine into one single Fermi arc, as shown in Fig. 4(b,c), which corresponds to topological phase transition (i). When |α− β| increases greater than 1, the system in the MSM1a phase, the Fermi arc connects the projections of the remaining Weyl points ∼ W 1,2 , as shown in Fig. 4(d). Similarly, for the MSM1b phase, there exists a Fermi arc connect the points ∼ W 3 and ∼ W 4 in the surface Brillouin zone, as shown in Fig. 4(e). When the transition from the MSM1a phase or the MSM1b phase to the band insulator phase the happen, the Fermi arc firstly shrink into a point, as shown in Fig. 4(f), and finally disappears.
Hidden symmetry at Weyl points. Here, we build the hidden symmetry at Weyl points. For convenience to construct the hidden symmetry, we suppose the case with the Hamiltonian H 0 as Eq. (1) where K is the complex conjugate operator; T x is a translation operator that moves the lattice along the x direction by a unit vector; σx is the Pauli matrix representing the sublattice exchange; π e ( ) i i z is a local U(1) gauge transformation. It is easy to prove that the symmetry operator ϒ is antiunitary, and its square is equal to ϒ =T x 2 2 . By setting α = 0 and β = 0, the Bloch Hamiltonian of original model can be obtain from Eq. (4) as The symmetry operator ϒ can be considered as a self-mapping of the original model defined as with i = 1, 2 for two sublattices and R n being a lattice vector. Performing the symmetr y transformation on the Bloch function (10) leads to ϒΨ = Ψ′ ′ r r ( ) ( ) . If the condition ′ = + k k G, where G is the reciprocal lattice vector, is satisfied, k is a ϒ-invariant point. In the Brillouin zone, the distinct ϒ-invariant points are π π π = ± M ( /2, /2, z being the shift of the z-component of the wave vector k due to the mapping Ω α,β . If ′ = + k k G is satisfied, k is a Λ α β , -invariant point. In the B r i l l o u i n z o n e , t h e d i s t i n c t Λ α β , -i nv a r i a nt p o i nt s a r e π π α β = ± + P ( /2, /2, arccos( )) 1,2 , π π α β = − ± − P ( /2, /2, arccos( )) 3,4 , α = ± Q (0, 0, arccos ) 1,2 and π α = ± Q (0, , arccos ) 3,4 . From the definition of the operator Λ α β , , we can verify Λ = Ω ϒ Ω We can interpret the above results in an intuitive way. The mapping Ω α,β from the Brillouin zone of the modified model into that of the original model is not surjective, which can be seen in Fig. 5. For the WSM2 phase, i.e. α β + < 1 and α β − < 1, the image of the mapping for the Brillouin zone of the modified model covers the degenerate ϒ-invariant points M 1,2 and M 3,4 in the Brillouin zone of the original model, as shown in Fig. 5(a). Therefore, there are always two pairs of Λ α β , -invariant points P 1,2 and P 3,4 , where the Weyl points locate, map into the degenerate ϒ-invariant points M 1,2 and M 3,4 . When we increase α β − to 1, P 3 Fig. 5(f), which means that the four Λ α β , -invariant P 1,2 and P 3,4 merge as two points. Therefore, the two pairs of Weyl points simultaneously merge at the edge of the Brillouin zone of the modified model.

Discussion
In summary, we have proposed a scheme to realize Weyl semimetals in a cubic optical lattice. There exist three Weyl semimetal phases, such as the WSM2, WSM1a, and WSM1b phases, for different parameter ranges. In the Brillouin zone, there are two pairs of Weyl points for the WSM2 phase while there is one pair of Weyl points for the MSM1a and MSM1b phases. The Weyl points move along with varying of the parameters. When the Weyl points with opposite topological charges meet, they merge and annihilate, which leads to a topological phase transition. The spectral functions of surface states at zero energy for a slab with (010) surfaces have been calculated. Fermi arcs appear to connect the projection of the Weyl points with opposite topological charges on the surface Brillouin zone. There are two Fermi arcs in the WSM2 phase and there is one in the MSM1a and MSM1b phases. When the phase transition from the WSM2 phase to the MSM1a or MSM1b phase happens, the two Fermi arcs combine into one Fermi arc. For the phase transition from the MSM1a or MSM1b phase to the band insulator phase, the Fermi arc shrinks into a point, then disappears. We also found that there exist hidden symmetries at all of Weyl points. These hidden symmetries have an antiunitary operator with its square being − 1. Based on the mapping method 25 , we constructed hidden symmetries at all of Weyl points. Our work deepens our understanding of Weyl semimetals on the point view of symmetry.