Hidden symmetry and protection of Dirac points on the honeycomb lattice

The honeycomb lattice possesses a novel energy band structure, which is characterized by two distinct Dirac points in the Brillouin zone, dominating most of the physical properties of the honeycomb structure materials. However, up till now, the origin of the Dirac points is unclear yet. Here, we discover a hidden symmetry on the honeycomb lattice and prove that the existence of Dirac points is exactly protected by such hidden symmetry. Furthermore, the moving and merging of the Dirac points and a quantum phase transition, which have been theoretically predicted and experimentally observed on the honeycomb lattice, can also be perfectly explained by the parameter dependent evolution of the hidden symmetry.

general honeycomb lattice model with the bond angle θ can be well described by the Bloch Hamiltonian as (the unit bond length is adopted) where t 1 and t 2 denote the amplitudes of hopping as sketched in Fig. 1(a); σ x and σ y are the pauli matrices defined in the sublattice space (for detail see supplementary information). Diagonalizing equation (1), we obtain the dispersion relation as Concretely, when θ π = /6, the lattice is the ideal honeycomb lattice, such as graphene, which has the Bloch Hamiltonian as When θ = 0, the lattice reduces to the brick-wall lattice as shown in Fig. 1(b), which has the corresponding Bloch Hamiltonian as We find that, for the ideal honeycomb lattice and the brick-wall lattice with the parameters = t t 1 2 , the band structures are both gapless and have the Dirac points at π (± / , ) 4 3 3 0 and π (± / , ) 2 3 0 in the The arrows represent a hoppingaccompanying π phase; t x and t y represent the amplitudes of hopping. In (a), (b,c), the blue and green balls represent the lattice sites in sublattices A and B, respectively. (d) The dispersion relation for the honeycomb lattice model with θ π = /6 and = = t t t 1 2 . (e) The dispersion relation for the brick-wall lattice model with = = t t t 1 2 . (f) The dispersion relation for the square lattice model with = = t t t x y . Brillouin zones (for the definitions of the Brillouin zones see Methods) as show in Fig. 1(d,e), respectively. In order to find the hidden symmetry behind the the honeycomb lattice, an auxiliary square lattice with a hopping-accompanying π phase is introduced as well, as shown in Fig. 1(c), which has an intrinsic relation with the honeycomb lattice. The Bloch Hamiltonian for the square lattice can be written as where t x and t y represent the amplitudes of hopping along the horizontal and vertical directions, respectively (for detail see supplementary information). The corresponding dispersion relation is which is gapless and has Dirac points at π (± / , ) 2 0 in the Brillouin zone as shown in Fig. 1(f). Geometrically, the square lattice can transform into the ideal honeycomb lattice continuously in two steps. First, the square lattice changes into the brick-wall lattice when the amplitude of hopping with a π phase is tuned to zero, and then reaches the ideal honeycomb lattice by a deformation of the bond angle θ from 0 to π/6, which can be understood with the help of Fig. 1(a-c). Besides the intuitive relation between these lattice structures in the real space, their band structures also strongly correlate to each other. The energy bands are all characterized by two linear Dirac cones in the Brillouin zone as shown in Fig. 1(d-f). More importantly, these Dirac points are able to evolve continuously into each other with the variation of the lattice parameters. For the general honeycomb lattice with β = 1 (β is defined as the hopping amplitude ratio β = / ) t t 2 1 , the corresponding Dirac points locate at π θ (± / , ) 2 3 cos 0 in the Brillouin zone. As a result, when the bond angle θ varies from π/6 to 0, the lattice first changes from the ideal honeycomb lattice into the brick-wall lattice, inducing a shift of the Dirac points from π (± / , ) 4 3 3 0 to π (± / , ) 2 3 0 , as shown in Fig. 1(d,e). Starting from the brick-wall lattice, the square lattice can be obtained by turning on the amplitude of hopping with a π phase from 0 to −t y . Accordingly, the corresponding Dirac points evolve from π (± / , ) 2 3 0 to π (± / , ) 2 0 , as shown in Fig. 1(f). It is impressive that during the whole evolution of the lattice, the Dirac points are always stable without any gap opening. We will show that this property can be perfectly explained by the protection of the hidden symmetry of the lattice structures.
Hidden symmetry on the auxiliary square lattice and protection of Dirac points. Firstly, we consider the auxiliary square lattice as shown in Fig. 1(c). One can verify that the square lattice is invariant under the action of the operator defined as where T x is a translation operator that moves the lattice along the horizontal direction by a unit vector x ; K is the complex conjugate operator; σ x is the Pauli matrix representing the sublattice exchange; ( ) π e i i y is a local ( ) U 1 gauge transformation and i y is the y component of the coordinate of lattice sites. Figure 2 schematically shows the invariance of the square lattice under the action of the hidden symmetry ϒ . This kind of transformation invariance indicates a hidden symmetry of the square lattice 24 . It is easy to prove that the symmetry operator ϒ is antiunitary, and its square is equal to ϒ =T x 2 2 . Mathematically, the hidden symmetry operator ϒ can be considered as a self-mapping of the square lattice model defined as Comparing equation (10) with equation (9), we have π , which can be regarded as the transformation of the wave vector under the action of the operator ϒ . If ′ = + k k K m s , where K m s is a reciprocal lattice vector for the square lattice, then we can say that k is a ϒ -invariant point. In the Brillouin zone, the ϒ -invariant points are π = (± / , ) i i is unconstrained for equation (11). Therefore, we arrive at the conclusion that the system must be degenerate at points , M 1 2 , which are just the locations of the Dirac points as shown in Fig. 1(f). From the above discussion, one can see that the Dirac points on the auxiliary square lattice are exactly protected by the hidden symmetry ϒ .
Mapping from the honeycomb lattice into the square lattice. In this section, we define a mapping from the honeycomb lattice into the auxiliary square lattice. In order to interpret this mapping in an intuitive way, we divide it into two mappings. The first one is the mapping from the general honeycomb lattice model with the bond angle θ into the brick-wall lattice model and the second one is the mapping from the brick-wall lattice model with the hopping amplitude ratio β into the square lattice model. In the following, we explain these in detail.
The mapping ω 1,θ from the honeycomb lattice into the brick-wall lattice. The general honeycomb lattice model with the bond angle θ is equivalent with the brick-wall lattice model in some sense. To manifest this equivalence, we define a mapping ω θ , 1 from the general honeycomb lattice model into the brick-wall lattice model as are the Bloch functions of the honeycomb lattice with the bond angle θ and the brick-wall lattice, respectively. To find the explicit form of the mapping, we take a transformation on the Bloch Hamiltonian (equation (1) , we take the limit as the definition of θ S k . After the transformation, we then obtain which is just the Bloch Hamiltonian (equation (4) . This mapping is one-to-one and surjective. Thus, we can regard this mapping as a kind of equivalence. The explicit form of the mapping depends on the bond angle θ. When θ = 0, this mapping is an identity mapping.
The mapping ω θ , 1 gives a one-to-one correspondence between the Brillouin zones of the honeycomb lattice with the bond angle θ and the brick-wall lattice. That is to say, for some wave vector k in the Brillouin zone of the honeycomb lattice, the Bloch Hamiltonian  ( ) . The mapping from the Brillouin zone of the honeycomb lattice with θ π = /6 into that of the brick-wall lattice is schematically shown in Fig. 3.
The mapping ω 2,β from the brick-wall lattice into the square lattice. Similarly, we can define a mapping from the brick-wall lattice model to the square lattice model as , which is just the Bloch Hamiltonian of the square lattice model. The explicit form of this mapping depends on the hopping amplitude ratio β.
This mapping is not surjective. That is to say, the image of the mapping for the Brillouin zone of the brick-wall lattice just covers part of the Brillouin zone of the square lattice. The mapping for the wave vectors is schematically shown in Fig. 4. In Fig. 4, the left panel shows the Brillouin zone of the brick-wall lattice. In order to clearly manifest the mapping from the brick-wall lattice model to the square lattice model, we first map the Brillouin zone of the brick-wall lattice into the reciprocal space of the square lattice, not restricted in the Brillouin zone, as shown in the middle panel of Fig. 4. The image of the Brillouin zone of the brick-wall lattice in the reciprocal space of the square lattice looks like a butterfly. The left and right halves of the Brillouin zone of the brick-wall lattice map into the left and right wings of the butterfly, respectively. If we restrict the image of the mapping in the Brillouin zone of the square lattice, then the butterfly-like image is equivalent to that as shown in the right panel of Fig. 4.
The composite mapping Ω θ,β . The above two mapping can be combined into a composite mapping ω ω Ω = , where the transformation matrix θ S k . Second, replacing the wave vectors k in ′ ( ) θ k with K via equations (21) and (22), one obtains the Bloch Hamiltonian as   For θ π = /6 case, such as graphene, the symmetric Brillouin zone is hexagon, i.e., the area enclosed by the black lines in Fig. 6(a). An alternative Brillouin zone equivalent to the symmetric Brillouin zone is a diamond, i.e., the yellow shaded area in Fig. 6(a). In our work, for convenience, we always use the diamond Brillouin zone for the honeycomb lattice. . The primitive reciprocal lattice vectors are π π = ( , ) b 1 and π π = ( , − ) b 2 . The square lattice has a square Brillouin zone as shown in Fig. 6(b). The brick-wall lattice can be considered a special honeycomb lattice with the bond angle θ = 0. The primitive lattice vectors become = ( , ) a 1 1 1 and = ( , − ) a 1 1 2 . The primitive reciprocal lattice vectors are π π = ( , ) b 1 and π π = ( , − ) b 2 . The corresponding Brillouin zone turns into a square, which is the same with that of the square lattice as shown in Fig. 6(b).