Appendix A: Connection between photonic bands in the tetragonal and face-centered cubic Brillouin zones

Here, we illustrate the equivalence of photonic band in the scheme of tetragonal (TET) and face-centered-cubic (FCC) unit-cells under the case with $l_x=l_y=0.5$. For simplicity, we assume that the all lattice constants along the three directions are 1, i.e., $b_x=b_z=1$. As mentioned in the main text, the FCC unit-cell is the primitive unit-cell with only two logs, while the TET unit-cell has four logs. For the TET scheme, the lattice vectors are defined as follows:

While for the FCC scheme, we choose the lattice vexctors as follows:

Thus $|{\vec a}_1|=|{\vec a}_2|=\frac {\sqrt {3}}{2}$, $|{\vec {a}_3}|=1$, and the volume of the FCC unit-cell is $\Omega =|\vec {a}_3 \cdot (\vec {a}_1 \times \vec {a}_2)|=\frac {1}{2}$. It is known to all that the first Brillouin zone reduced by half when the unit-cell is doubled. In order to match the high symmetry points between the TET and FCC Brillouin zone, we need to rotate the unit-cell with 45 degree around $z-axis$, i.e., multiple the lattice vectors with rotation matrix $R$,

Therefore, the new lattice vectors are:

Finally, the new reciprocal lattice basis of FCC unit-cell can be derived as:

and the correspondence of the high symmetry points between FCC and TET Brillouin zone are list as follows: point $Z_0$, $X_0$, $W_0$ in the Brillouin zone under the FCC scheme refer to the point $\Gamma$, $M$, $A$ in the Brillouin zone under the TET scheme, respectively.

Figure 5 shows the photonic bands below the fundamental gaps of woodpile in scheme of FCC and TET unit-cells, respectively. It should be noted there are four bands below the fundamental gap in the TET scheme, while there are only two bands in FCC scheme. In spite of the difference of the amount of the bands in two schemes, we remarked that the equivalence of these two schemes can be verified by band folding analysis. The fourfold degeneracy at the $Z$ point in the scheme of TET unit-cell originates from the band folding.

Appendix B: Other Kramers degeneracies

The Kramers degeneracy on the $k_i=\pi (i=x,y)$ plane can be understood via the construction of the anti-unitary operators $\Theta _x\equiv S_x*{\cal T}: (x,y,z, t)\rightarrow \left (x+\frac {1}{2},-y+\frac {1}{2},-z-\frac {1}{4}, -t\right )$ and $\Theta _y\equiv S_y*{\cal T}: (x,y,z,t) \rightarrow \left (-x+\frac {1}{2},y+\frac {1}{2},-z+\frac {1}{4}, -t\right )$, respectively, where $S_{x} : (x,y,z)\rightarrow \left (x+\frac {1}{2},-y+\frac {1}{2},-z-\frac {1}{4}\right )$ and $S_y : (x,y,z) \rightarrow \left (-x+\frac {1}{2},y+\frac {1}{2},-z+\frac {1}{4}\right )$ are the two-fold screw symmetries. Let us consider the anti-unitary operator $\Theta _x$ as an example. $\Theta _x$ transforms ${\vec k}$ to $(-k_x, k_y, k_z)$ is a symmetry operator on the $k_x=\pi$ plane. The square of this operator is $\left (\Theta _{x}\right )^{2}: (x,y,z,t) \rightarrow (x+1,y,z,t)$, which yields $\left (\Theta _{x}\right )^{2} =e^{ik_x}$ for all photonic Bloch states(including both the electric field ${\vec E}$ and the magnetic field ${\vec H}$). Hence, on the $k_x=\pi$ plane we have $\Theta _x^{2}=-1$ which explains the double degeneracy on the $k_x=\pi$ plane. Specifically, there exist the following commutation relations between anti-unitary operator $\Theta _{x}$ and operator $M_x$ for the $k_x=\pi$ plane,

Thus for an eigenstate of $M_x$ with eigenvalue $m_x$, labeled as $| {m_x}\rangle$, we have

Hence $\Theta _x| {m_x}\rangle$ is also an eigenstate of $M_x$ with opposite eigenvalue. These are the two degenerate Bloch states on the $k_x=\pi$ plane.

In a similar way, the anti-unitary operator $\Theta _y$ is a symmetry operator on the $k_y=\pi$ plane. The square of this operator $\left (\Theta _{y}\right )^{2}: (x,y,z,t) \rightarrow (x,y+1,z,t)$ yields $\Theta _y^{2} = e^{i k_y}$ for all Bloch states. Thus on the $k_y=\pi$ plane we have $\Theta _y^{2} =-1$, which yields the Kramers degeneracy.

Moreover, we find that $\Theta _{gx}\equiv G_x*{\cal T} : (x, y, z, t) \to (-x, y + \frac {1}{2}, z + \frac {1}{2}, -t)$ and $\Theta _{gy}\equiv G_y*{\cal T} : (x, y, z, t) \to (x + \frac {1}{2}, -y, z + \frac {1}{2}, -t)$ can also yield Kramers degeneracy, where $G_x : (x, y, z) \to (-x, y + \frac {1}{2}, z + \frac {1}{2})$ and $G_y : (x, y, z) \to (x + \frac {1}{2}, - y , z + \frac {1}{2})$ are two glide symmetries. Since $\Theta _{gx}$ transforms ${\vec k}$ into $(k_x,-k_y,-k_z)$, it is a symmetry operator only for the four lines: $k_y, k_z=0, \pi$. The square of this operator $\left (\Theta _{y}\right )^{2}: (x,y,z,t) \rightarrow (x,y+1,z+1,t)$ yields $\Theta _{gx}^{2}=e^{i(k_y+k_z)}$ for all Bloch states. Therefore it leads to double degeneracy for the two lines: $(k_x, 0, \pi )$ and $(k_x, \pi, 0)$. Similarly, $\Theta _{gy}$ is a symmetry operator for the four lines: $k_x, k_z=0,\pi$. The square of this operator $\left (\Theta _{y}\right )^{2}: (x,y,z,t) \rightarrow (x,y+1,z+1,t)$ yields $\Theta _{gy}^{2}=e^{i(k_x+k_z)}$ which reuslt in Kramers degeneracy on the two lines: $(0, k_y, \pi )$ and $(\pi, k_y, 0)$. Notice that these lines crossing at the $Z$ point $(0,0,\pi )$, where both $\Theta _{gx}$ and $\Theta _{gy}$ are effective.

Appendix C: Proof of the fourfold degeneracy on the $M$-$A$ line

The fourfold degeneracy on the $M$-$A$ line can be understood as follows: Any Bloch state on this line can be labeled with the eigenvalues $m_x$ and $m_y$ of the two mirror operators $M_x$ and $M_y$, respectively. We shall prove that $|{m_x,m_y}\rangle$, $\Theta _z|{m_x,m_y}\rangle$ ($\Theta _z\equiv G_z*{\cal T}$), $S_\pi |{m_x,m_y}\rangle$, and $\Theta _zS_\pi |{m_x,m_y}\rangle$ are distinct from each other. We find that such degeneracy is essentially related to the following commutation relationships on the $M$-$A$ line,

where $[A, B]_{\pm } = AB \pm BA$. Thus for an eigenstate labeled with $m_x$ and $m_y$ of mirror operator $M_x$ and $M_y$, we have

From these relations, we find that: (i) $\Theta _z |{m_x,m_y}\rangle$ is also an eigenstates of $M_x$ with opposite eigenvalue, i.e., $\Theta _z|{m_x,m_y}\rangle = | {-m_x,m_y}\rangle$; (ii) $S_{\pi } |{m_x,m_y}\rangle$ is also an eigenstates of $M_x (M_y)$ with opposite eigenvalue, i.e., $S_\pi |{m_x,m_y}\rangle=| {-m_x,-m_y}\rangle$. (ii) $\Theta _z S_\pi$ is also an eigenstates of $M_y$ with opposite eigenvalue, i.e., $\Theta _z S_\pi |{m_x,m_y}\rangle = | {m_x,-m_y}\rangle$. It is evident that these four states, i.e., $|{m_x,m_y}\rangle$, $\Theta _z|{m_x,m_y}\rangle$, $S_\pi |{m_x,m_y}\rangle$, and $\Theta _zS_\pi |{m_x,m_y}\rangle$ are distinct from each other and they are related by symmetry operators. Therefore, they are degenerate states.

Appendix D: ${\vec k}\cdot {\vec P}$ Hamiltonian of the Dirac nodal line

The photonic Hamiltonian is obtained by applying the ${\vec k}\cdot {\vec P}$ method to the Maxwell equation,

The photonic Hamiltonian $\hat {{\cal H}}_{EM}\equiv c^{2} \boldsymbol {\nabla }\times \frac {1}{\varepsilon }\boldsymbol {\nabla }\times$ around a high symmetry ${\vec K}$ is obtained as follows. We first expand the Bloch states at a wavevector ${\vec k}$ in the basis formed by the four Bloch states at the high symmetry point, i.e., (${\vec q}\equiv {\vec k}-{\vec K}$)

where ${\vec H}_{n^{\prime }}$ is the magnetic field at the ${\vec q}=0$ point (i.e., the high symmetry point) and $C_n$ are the coefficients to be solved by diagonalizing the Hamiltonian (and normalized as $\sum _n|C_n|^{2}=1)$. The magnetic fields are normalized such that (UC stands for unit-cell)

The Hamiltiona is written in the basis of the Bloch states ${\vec H}_{n^{\prime }}$, and we find that

Direct calculation yields,

We shall use the Maxwell equation to simplify the above results

where $\omega _n=\omega _0$ at the degenerate point, $\epsilon _{\nu \alpha \mu }$ is the Levi-Civita tensor, ${\vec E}_{n}$ is the electric field of the Bloch states at ${\vec q}=0$ satisfying the normalization condition of

Using Eq. (24), we find that

where $\alpha =(x,y,z)$ and ${\vec n}_{\alpha }$ is the unit vector along the $\alpha$ direction. And

The photonic Hamiltonian $\hat {{\cal H}}_{EM}$ is connected with the simulated fermion-like as follows, ($\hbar \equiv 1$)

Therefore, near the degenerate point, we have

where $\cdots$ represents higher order terms, and

One can easily verify that $\hat {v}^{\alpha }$ and $\hat {w}_{\alpha,\beta }$ are Hermitian operators.

We now examine the constraints on the matrix element of $\hat {v}$ imposed by the mirror and/or glide symmetry. Explicitly, we have

If, say, there is a mirror or glide symmetry, labeled as $\hat {F}_{x}$, the mirror operation acts on the electric and magnetic fields as follows:

Thus, the velocity matrix element $v_{n,n^{\prime }}^{\alpha }$ is nonzero only when the $n$ and $n^{\prime }$ states carry opposite mirror eigenvalue $m_\alpha$ (or glide eigenvalue $g_\alpha$).

The invariance of the Hamiltonian under a symmetry operation ${\cal S}$ implies that

For example, the time-reversal symmetry symmetry ${\cal T}=-{\cal K}$ yields that the ${\vec k}\cdot {\vec P}$ around a time-reversal invariant momentum has

Hence the matrix element of the $q$ linear terms are purely imaginary. One can prove that the $q$ quadratic terms are purely real.

We now derive the ${\vec k}\cdot {\vec P}$ Hamiltonian for the Dirac nodal line. Around a point on the $M$-$A$ line, the four degenerate states are chosen as the $|{m_x,m_y}\rangle$ for $m_x,m_y=\pm 1$. In the basis of $(| {1,1}\rangle, |{-1,-1}\rangle, |{-1,1}\rangle, |{1,-1}\rangle)^{T}$ the $q$ linear Hamiltonian is written as

where $a_i$ and $b_i$ are the ${\vec k}\cdot {\vec P}$ coefficients which can be calculated from the photonic Bloch functions ${\vec H}_{n}$ at the degeneracy point $(\pi,\pi,k_z)$ (hence these coefficients are $k_z$ dependent). Those coefficients are restricted by the symmetry via Eq. (33). Two symmetry operations are relevant: $\Theta _z$ and $S_\pi$. In the chosen basis, these operations are manifested as the following matrices:

Their effects on the wavevectors are

We thus obtain from Eq. (33) that

It is convenient to define $v=\sqrt {|b_1|^{2}+|b_2|^{2}}=\sqrt {|a_1|^{2}+|a_2|^{2}}$, $\gamma _x=a_1/v$, and $\gamma _y=a_2/v$. In the new basis of $(| {\uparrow,p}\rangle, | {\downarrow,p}\rangle, | {\uparrow,a}\rangle, | {\downarrow,a}\rangle)^{T}$, the ${\vec k}\cdot {\vec P}$ Hamiltonian can be further simplified as

The above Hamiltonian applies for the whole $M$-$A$ line, where the coefficients $\omega _0$, $v$, $\gamma _x$ and $\gamma _y$ are $k_z$-dependent. At the M or A point, the $S_4$ symmetry also holds, thus there will be additional constraints on the coefficients. Let’s consider such constraints in the original basis for the Hamiltonian Eq. (35). $S_4$ is manifested as $-\sigma _0$ in the even-parity doublet, whereas $S_4=-i\sigma _y$ in the odd-parity doublet. Imposing (33) we find that for the M and A points, $\gamma _x=\gamma _y$. In addition, for these points, the time-reversal symmetry dictates that $\gamma _x=\gamma _y$ are purely imaginary coefficients. Therefore, at these points the Dirac point is doubly degenerate in the $k_x$-$k_y$ plane, while away from these points the dispersion in the $k_x$-$k_y$ plane is generally non-degenerate. The double degeneracy for generic points on the MA line (except the M and A points) are restored only for $q_x=0$ or $q_y=0$ (i.e., at the $k_x=\pi$ or $k_y=\pi$ plane), where the nonsymmorphic screw symmetry ensures double degeneracy.

For the quadratic degeneracy at the $Z$ point, the eigenstates can be labeled as $| {m_x,m_y,g_z}\rangle$ for $m_x=-m_y=\pm 1, g_z=\pm 1$. Using the basis of $(| {-1,1,1}\rangle, | {1,-1,1}\rangle, | {-1,1,-1}\rangle, | {1,-1,-1}\rangle)^{T}$, the $q$ linear term comes simply as $v_z q_z\tau _y$, because it is finite only between states with opposite $g_z$. Considering the $S_{\frac {\pi }{2}}$ symmetry, the higher order terms are

The above Hamiltonian recovers Eq. (8) in the main text when written in the basis of spin states that carrying angular momentum. When the $S_{\frac {\pi }{2}}$ symmetry is broken the degeneracy between states with different $m_x$ at $q_z=0$ is split by a constant term $\Delta _zf_0\sigma _z$ with $\Delta _z$ characterizing the strength of the perturbation.

Appendix E: Proof of the fourfold quadratic degeneracy on the $Z$ point

The $S_{\frac {\pi }{2}}$ operator transforms the eigenstate of $M_x$ with the eigenvalue $m_x$ at ${\vec k}$ to a Bloch state at ${\vec k}^{\prime }$ as the eigenstate of $M_y$ with the same eigenvalue, i.e.,

Thus $\Theta _{\frac {\pi }{2}}$ is a symmetry operator only for the $Z$ and $A$ points. Analogous to the Kramers degeneracy, the above yields fourfold degeneracy at the $Z$ point. The four degenerate states are $| {\Psi }\rangle$, $\Theta _{\frac {\pi }{2}}| {\Psi }\rangle$, $(\Theta _{\frac {\pi }{2}})^{2}| {\Psi }\rangle$, and $(\Theta _{\frac {\pi }{2}})^{3}| {\Psi }\rangle$.

The four degenerate states can be labeled by the eigenvalue of $M_x$, $M_y$ and $\tilde {G}_z=e^{i\frac {\pi }{4}}G_z$. We find that

Thus $| {\Psi }\rangle$ and $\Theta _{\frac {\pi }{2}}| {\Psi }\rangle$ carry distinct mirror eigenvalues but the same $\tilde {G}_z$ eigenvalue, whereas $| {\Psi }\rangle$ and $(\Theta _{\frac {\pi }{2}})^{2}| {\Psi }\rangle$ carry the same mirror eigenvalues but different $\tilde {G}_z$ eigenvalues. For simplify, we label the Bloch states, which is the eigenstates of $M_x$, $M_y$ and $G_z$ with eigenvalue $m_x$, $m_y$, and $g_z$, respectively, as $| {m_x,m_y,g_z}\rangle$. According to the above commute (anticommute) relationships, we find that $\Theta _{\frac {\pi }{2}}| {\Psi }\rangle=| {m_y,m_x,g_z}\rangle$, $(\Theta _{\frac {\pi }{2}})^{2}| {\Psi }\rangle=| {m_x,m_y,-g_z}\rangle$, and $(\Theta _{\frac {\pi }{2}})^{3}| {\Psi }\rangle=| {m_y,m_x,-g_z}\rangle$. It is obviously that these four states, i.e., $| {m_x,m_y,g_z}\rangle$, $| {m_y,m_x,g_z}\rangle$, $| {m_x,m_y,-g_z}\rangle$, and $| {m_y,m_x,-g_z}\rangle$ are distinct from each other and they are related by symmetry operations. Therefore, we prove the fourfold degeneracy at $Z$ point.

Appendix F: Varying geometries and materials

Here we show the emergent Dirac physics are robust to the material and geometry of the photonic crystals. We calculate the photonic bands along the high symmetry lines with three different geometric/material parameter settings. The results of the photonic bands are shown in Fig. 6, where the Dirac nodal line and the quadratic point holds for all parameters. This is because such double degeneracy are guranteed the lattice symmetry. This also indicates the stableness of the Dirac points and implies that such a symmetry-guided method can also be effective in other classicl/bosonic systems.

Appendix G: Optical properties of type-II Dirac points

The type-II DPs offer special band structures that may enable in the manipulation of light in ways that cannot be achieved in uniform dielectric materials. The refraction properties can be determined by matching the frequency and the wavevector parallel to the interface. Here, we consider refraction on the $(001)$ surface where light is injected from air (above the PC). The wavevector in the air is determined by the frequency and angle of incidence ($\theta _i$ and $\phi _i$) via

The wavevector $k_x$ and $k_y$ remains the same in the PC, the quantity to be found is the wavevector along $z$ direction in the PC. We shall denote the wavevector in the PC as ${\vec q}$. So far we have

For the Dirac points, $q_z$ is obtained by solving the equation,

for the $\pm$ branches with $\tau _z=-1$ as required by that the group velocity along the $z$ direction should be negative. The above equation can be solved straightforwardly.

The group velocity along the $x$ and $y$ directions are then,

The refraction angle $\phi _o$ is then determined as

for the $\pm$ branches. The crucial physics is that around the type-II Dirac point the $v_{g,x}$ becomes significant for the two branches and they are of opposite sign, while the $v_{g,y}$ do not change significantly. This yields a large variation of the refraction angle across the Dirac point, such variation goes to opposite direction for the two branches.