Introduction

In last decade, topological matters have become an important branch of condensed matter physics1,2. Previously, the studies mainly concentrate on gapped systems, such as integer quantum Hall insualtors3 and quantum anomalous Hall insulator4, topological insulators5, chiral topological superfluids6, helical topological superfluids or superconductors7, and so on. Recently, physicists pay much attention on the topological characters of gapless systems, which were dubbed as topological semimetals. Generally, topological semimetals include Weyl semimetals8,9,10,11,12, Dirac semimetals13,14, topological nodal-line semimetals10. For Weyl semimetals, the materials have band structures with band-touching nodal points in momentum space, where the isolated band degeneracy occurs. Near these touching points, the dispersion relation is linear and can be described by a massless two-component Weyl Hamiltonian. At the nodal points, there exist singularities of a Berry field. Integrating the Berry field on the surface enclosing the singular point in momentum space, one obtain a Chern number, i.e., a topological charge. Thus, the band-touching nodal points can be considered as monopoles in momentum space. Due to the Nielsen-Ninomya theorem, the nodal points with opposite topological charges appear in pairs. The meeting of opposite charges in momentum space can lead to annihilation of nodal point pairs. The opposite topological charges can be separated from each other in momentum space so that they cannot be destroyed by the mutual annihilation if the time-reversal symmetry or inversion symmetry is broken. Time reversal symmetry breaking Weyl semimetals were firstly predicted in pyrochlore iridates8 and HgCr2Se49. Recently, inversion symmetry breaking Weyl semimetals were discovered in TaAs family15,16,17,18,19. In the theoretical aspect, recently, Ganeshan and Das Sarma presented a method to construct a Weyl semimetal by stacking one-dimensional Aubry-Andre-Harper lattice with tight-binding models with nontrivial topology20, which provides a theoretical connection between the commensurate Aubry-Andre-Harper model in one dimension and Weyl semimetals in three dimensions.

It is a difficult task to investigate moving and merging of Weyl points and topological phase transitions in real solid materials, the parameters of which can not be tuned in a wide ranges. Fortunately, the high controllability and tunability, and large number of mature detection techniques of cold atoms in optical lattices make them a platform to stimulate many interesting physics in condensed matters. Therefore, it is intriguing to study moving and merging of Weyl points, and topological phase transitions in optical lattices. In recent years, many schemes have been proposed to realize various topological semimetals with neutral atoms in optical lattices. In two dimensions, gapless topological phases were proposed in honeycomb optical lattices21,22 and square optical lattices23,24,25. The important progress is the realization of topological semimetals in honeycomb optical lattices26. In three dimensions, Weyl semimetal were proposed to realized in optical lattices27,28,29,30. 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 shaking31,32 and laser-assisted tunneling techniques33,34,35,36. Another important progress in experiments is the measurement of Zak phase of topological Bloch bands in optical lattices37, 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 technique33,34,35,36. 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 a1 = (1, −1, 0), a2 = (1, 1, 0), and a3 = (0, 0, 1). The primitive reciprocal lattice vectors are b1 = (π, −π, 0), b2 = (π, π, 0), and b3 = (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 H = H0 + Hd + Hs with

Figure 1
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.

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and

and

where ai and bi are the annihilation operators destructing a particle at a lattice site of sublattice A and B, respectively; tx and ty represent the amplitudes of hopping along the x and y directions, respectively; txy 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 experiments33,34,35,36.

Taking the Fourier’s transformation on Equations (1), (2) and (3), we rewritten the Hamiltonian as , where is the corresponding Bloch Hamiltonian as

with α = v/2tz and β = 2txy/tz being the dimensionless parameters and σx, σy and σz being the Pauli matrices defined in the sublattice space. Diagonalizing Equation (4), we obtain the corresponding dispersion relation as

with m = 2tz(α + β sin kx sin ky). From this dispersion relation, we can see that two bands touch at some points Wi in the Brillouin zone in some parameter ranges. Near the touching points, the dispersion relation has the linear form as

with p = k − Wi. Around the the touching points, the chirality can be defined as

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 and are satisfied, there are four distinct points and 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 and , only the pair exists. Thus, we term this phase as WSM1a phase. (iii) For the case and , where the Weyl points W3,4 still remain. We term this new phase as WSM1b phase, which is different from the WSM1a phase. (iv) For and , no Weyl point exists and a gap opens, so the system is a band insulator. The phase diagram is shown as in Fig. 2.

Figure 2
figure 2

Schematic of the phase diagram.

Here, WSM2 (blue) denotes the Weyl semimetal phase with two pairs of Weyl points in the Brillouin zone; WSW1a (yellow) and WSW1b (green) denote the two semimetal phases with a pair of Weyl points in the Brillouin zone; BI denotes the band insulator phase.

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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 W1,2 and W3,4 with topological charges C1,2 = ±1 and C3,4 = ±1, as shown in Fig. 3(a). When we keep α + β invariant and increase α − β, the Weyl points W3,4 move towards each other and W1,2 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 W3,4 annihilate and only W1,2 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 W3,4 move away from each other and W1,2 stay at their starting positions. When α−β decreases to −1, the Weyl points W3,4 arrive at (π/2, −π/2, ±π), which are identical points in the Brillouin zone, i.e., W3,4 meet and merge, as shown in Fig. 3(c). When α−β is less than −1, W3,4 annihilate and topological phase transition (i) happens. When it arrives at the MSM1a phase, there exist only one pair of Weyl points W1,2, 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 W1,2 meet and merge at the center or the surface of the Brillouin zone, and W3,4 still remain. When |α + β| is greater than 1, the Weyl points W1,2 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 W1,2 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 W1,2 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 W3,4 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 W3,4 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.

Figure 3
figure 3

Weyl points in Weyl semimetal phases for.

(a) The WSM2 phase with α = 0 and β = 0, (b) the boundary between the MSM2 phase and the MSM1a phase with α = 0.5 and β = −0.5, (c) the boundary between the MSM2 phase and the MSM1a phase with α = −0.5 and β = 0.5, (d) the MSM1a phase with α = 0.8 and β = −0.8, (e) the MSM1b phase α = 0.8 and β = 0.8, (f) the boundary between the MSM1a phase and the band insulator phase with α = 1.3 and β = −0.3. For all of cases, we have set tx = ty = tz = t. The yellow bulks represent the Brillouin zone; the red and green balls represent the Weyl points with positive and negative topological charges (also denoted by all-out and all-in arrows), respectively; the half red and half green balls represent merged Weyl points.

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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 , where Gr(E) is the retarded Green function of the system. The projections of Weyl points W1,2,3,4 on the surface Brillouin zone are denoted as . Figure 4(a) shows that, in the WSM2 phase, there are two Fermi arcs in the surface Brilllouin zone, which connect points and , and , respectively. Since Weyl points W1 and W3, W2 and W4 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, and 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 , as shown in Fig. 4(d). Similarly, for the MSM1b phase, there exists a Fermi arc connect the points and 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.

Figure 4
figure 4

The spectral functions of surface states with E = 0 on the (010) surface Brillouin zone for.

(a) The WSM2 phase with α = 0 and β = 0, (b) the transition from the MSM2 phase to the MSM1a phase with α = 0.5 and β = −0.5, (c) the transition from the MSM2 phase to the MSM1a phase with α = −0.5 and β = 0.5, (d) the MSM1a phase with α = 0.8 and β = −0.8, (e) the MSM1b phase α = 0.8 and β = 0.8, (f) the transition from the MSM1a phase to the band insulator phase with α = 1.3 and β = −0.3. For all of cases, we have set tx = ty = tz = t. Here, the rectangles represent the (010) surface Brillouin zone and kxy is the component on the xy plane in momentum space for the wavevectors on the surface Brillouin zone. The red and green circles represent the projections of Weyl points with positive and negative topological charges on the surface Brillouin zone; the half red and half green circles represent the projections of merged Weyl points on the surface Brillouin zone.

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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 H0 as Eq. (1) as the original model and the total model H = H0 + Hd + Hs as the modified model.

Hidden symmetry at Weyl points of the original model

In the following, we will show that the Weyl points in the original model are protected by a hidden symmetry. For the original model, the lattice is invariant under the operation defined as

where K is the complex conjugate operator; 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; is a local U(1) gauge transformation. It is easy to prove that the symmetry operator is antiunitary, and its square is equal to .

By setting α = 0 and β = 0, the Bloch Hamiltonian of original model can be obtain from Eq. (4) as

The symmetry operator can be considered as a self-mapping of the original model defined as

where and are the Bloch functions of the original model. We suppose that the Bloch functions of the square lattice model have the form as

with with i = 1, 2 for two sublattices and Rn being a lattice vector. Performing the symmetry transformation on the Bloch function (10) leads to with . If the condition , where G is the reciprocal lattice vector, is satisfied, k is a -invariant point. In the Brillouin zone, the distinct -invariant points are , , , and . The square of the operator can be written in the form as in the Bloch representation. It is easy to verify that the points Mi(i = 1, 2, 3, 4), while at the points Ni(i = 1, 2, 3, 4). Considering the antiunitarity of the operator , based on Kramers theorem, we can conclude that there must be band degeneracies at the -invariant points Mi(i = 1, 2, 3, 4), which are just the Weyl points Wi(i = 1, 2, 3, 4) in the MSM2 phase with α = 0 and β = 0. There, a hidden symmetry with its square of operators being −1 exists at the Weyl points of the original model.

Hidden symmetry at Weyl points of the modified model

It is easy to verify that the hidden symmetry is violated in the modified model. However, with the mapping Ωα,β from the modified model into the original model defined in section Methods, we can find the hidden symmetry in the modified model. Based on the mapping Ωα,β, we define an operation , which can be regarded as a self-mapping of the modified model as

Performing the above operation on the Bloch function of the modified model, we have , where with being the shift of the z-component of the wave vector k due to the mapping Ωα,β. If is satisfied, k is a -invariant point. In the Brillouin zone, the distinct -invariant points are , , and .

From the definition of the operator , we can verify , which acts on the Bloch function as . Substituting the -invariant points and into the above equation, we find at , while at . Since is an antiunitary operator, based on Kramers theorem, there must exist band degeneracies at the -invariant points , which are just the Weyl points of the WSM2 phase for and . For and , P3,4 do not exist, there are only the -invariant points P1,2, which correspond to the Weyl points W1,2 of the WSM1a phase. Similarly, for and , P1,2 do not exist, there are only the -invariant points P3,4, which correspond to the Weyl points W3,4 of the WSM1b phase. For the case and , all the -invariant points P1,2,3,4 do not exist, so there are not Weyl points, which corresponds to a band insulator phase.

We can interpret the above results in an intuitive way. The mapping Ωα,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. and , the image of the mapping for the Brillouin zone of the modified model covers the degenerate -invariant points M1,2 and M3,4 in the Brillouin zone of the original model, as shown in Fig. 5(a). Therefore, there are always two pairs of -invariant points P1,2 and P3,4, where the Weyl points locate, map into the degenerate -invariant points M1,2 and M3,4. When we increase to 1, P3 and P4 become the same point of the Brillouin zone of the modified model, which maps into the points M3,4 in the Brillouin zone of the original model as shown in Fig. 5(b), so the corresponding two Weyl points merge and a phase transition from the WSM2 phase to the WSM1a phase occurs. When and , the image of mapping for the Brillouin zone of the modified model only covers the degenerate -invariant points M1,2 in the Brillouin zone of the original model as shown in Fig. 5(c). There exists a pair of Weyl points P1,2 in the Brillouin zone of the modified model mapping into the degenerate -invariant points M1,2, which corresponds to the WSM1a phase. Similarly, when and , the image of the mapping for the Brillouin zone of the modified model covers M1,2 and M3,4 while M1,2 locate at the edge of the image and the same point in the Brillouin zone of the modified model maps into M1,2. The degenerate -invariant points P1,2 merge at the edge of the Brillouin zone of the modified model while P3,4 still exist. This case corresponds to the phase boundary between the WSM2 and WSM1b phases. When and , the image of the mapping for the Brillouin zone of the modified model only covers the degenerate -invariant points M3,4 as shown in Fig. 5(d). Correspondingly, there exists a pair of Weyl points P3,4 in the Brillouin zone of the modified model mapping into the degenerate -invariant points M3,4, which corresponds to the WSM1b phase. When and , the image of mapping for the Brillouin zone of the modified model does not cover any degenerate -invariant points in the Brillouin zone of the original model as shown in Fig. 5(e). Therefore, there does not exist any Weyl point in the Brillouin zone of the modified model and a gap opens, which corresponds to the band insulator phase. When and , a direct phase transition between the MSM2 phase and the band insulator phase occurs, where two pairs of Weyl points merge simultaneously. For this case, the edge of the image of the mapping for the Brillouin zone of the modified model covers the degenerate -invariant points M1,2 and M3,4 in the Brillouin zone of the original model as shown in Fig. 5(f), which means that the four -invariant P1,2 and P3,4 merge as two points. Therefore, the two pairs of Weyl points simultaneously merge at the edge of the Brillouin zone of the modified model.

Figure 5
figure 5

The mapping from the Brillouin zone of the modified model into the Brillouin zone of the original model for.

(a) α = −0.3, β = 0.3; (b) α = 1, β = −0.8; (c) α = 1, β = 0.8; (d) α = 2, β = 0; (e) α = 0.6, β = −0.4; (f) α = 1, β = 0;. Here, the yellow bulk represents the Brillouin zone of the original model; the green balls mark the Weyl points of the original model; the blue part represent the image of the mapping in the Brillouin zone of the original model for the Brillouin zone of the modified model.

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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 method25, we constructed hidden symmetries at all of Weyl points. Our work deepens our understanding of Weyl semimetals on the point view of symmetry.

Methods

The mapping from the modified model into the original model

We can define a mapping from the modified model into the original model as25

where represents the Bloch function of the modified model. The concrete form of the mapping depends on the dimensionless parameters with and . For this mapping, we have with

where . Replacing k in equation (4) with K via Equations (11), (12), and (13), we obtain

with , , and , which is just the Bloch Hamiltonian (9) of the original model.

Additional Information

How to cite this article: Hou, J.-M. and Chen, W. Weyl semimetals in optical lattices: moving and merging of Weyl points, and hidden symmetry at Weyl points. Sci. Rep. 6, 33512; doi: 10.1038/srep33512 (2016).