Nonlocal topological electromagnetic phases of matter

In (2+1)-dimensional materials, nonlocal topological electromagnetic phases are defined as atomic-scale media which host photonic monopoles in the bulk band structure and respect bosonic symmetries (e.g., time reversal ${\mathcal{T}}^2=+1$). Additionally, they support topologically protected spin-1 edge states, which are fundamentally different than spin-$\frac{1}{2}$ and pseudo-spin-$\frac{1}{2}$ edge states arising in fermionic and pseudofermionic systems. The striking feature of the edge state is that all electric and magnetic field components vanish at the boundary, in stark contrast to analogs of Jackiw-Rebbi domain wall states. This surprising open boundary solution of Maxwell's equations, dubbed the quantum gyroelectric effect [Phys. Rev. A 98, 023842 (2018)], is the supersymmetric partner of the topological Dirac edge state where the spinor wave function completely vanishes at the boundary. The defining feature of such phases is the presence of temporal and spatial dispersion in conductivity (the linear response function). In this paper, we generalize these topological electromagnetic phases beyond the continuum approximation to the exact lattice field theory of a periodic atomic crystal. To accomplish this, we put forth the concept of microscopic photonic band structure of solids, analogous to the traditional theory of electronic band structure. Our definition of topological invariants utilizes optical Bloch modes and can be applied to naturally occurring crystalline materials. For the photon propagating within a periodic atomic crystal, our theory shows that besides the Chern invariant $\mathfrak{C}\in \mathbb{Z}$, there are also symmetry-protected topological (SPT) invariants $\nu \in {\mathbb{Z}}_N$ which are related to the cyclic point group $C_N$ of the crystal $\nu =\mathfrak{C}modN$. Due to the rotational symmetries of light $\mathcal{R}\left(2\pi \right)=+1$, these SPT phases are manifestly bosonic and behave very differently from their fermionic counterparts $\mathcal{R}\left(2\pi \right)=-1$ encountered in conventional condensed-matter systems. Remarkably, the nontrivial bosonic phases $\nu \ne 0$ are determined entirely from rotational (spin-1) eigenvalues of the photon at high-symmetry points in the Brillouin zone. Our work accelerates progress toward the discovery of bosonic phases of matter where the electromagnetic field within an atomic crystal exhibits topological properties.

Figure 1

Brillouin zone of each cyclic point group $C_N$. (a), (b), (c), (d), and (e) correspond to $N=2,3,4,6$, and $\infty$, respectively. Due to rotational symmetry, the total Brillouin zone is equivalent to $N$ copies of the irreducible Brillouin zone ($\mathrm{IBZ}$), which is represented by the blue quadrant. For continuous symmetry $N=\infty$, this is simply a line. The yellow circles label high-symmetry points $R{\mathbf{k}}_i={\mathbf{k}}_i$ where the crystal Hamiltonian is invariant under a certain rotation $\widehat{\mathcal{R}}$. At these specific momenta, a Bloch photonic wave function $\mathcal{R}{\tilde{f}}_{{\mathbf{k}}_i}\left(R^{-1}\mathbf{r}\right)=\eta \left({\mathbf{k}}_i\right){\tilde{f}}_{{\mathbf{k}}_i}\left(\mathbf{r}\right)$ is an eigenstate of an $N$-fold rotation $\eta \left({\mathbf{k}}_i\right)={\eta }_N\left({\mathbf{k}}_i\right)=\left[i\frac{2\pi }{N}{m}_N\left({\mathbf{k}}_i\right)\right]$ such that the photon possesses quantized integer eigenvalues $m_N\left({\mathbf{k}}_i\right)\in {\mathbb{Z}}_N$. Since $m_N$ are discrete quantum numbers, their values cannot vary continuously if the crystal symmetry is preserved; they can only be changed at a topological phase transition.

Figure 2

The collection of spin-1 (bosonic) charges for the $C_4$ point group. (a) Fourfold rotations ${\left({\mathcal{R}}_4\right)}^4=+1$; there are four unique eigenvalues ${\eta }_4=exp\left[i\frac{2\pi }{4}{m}_4\right]$ corresponding to the roots of unity ${\left({\eta }_4\right)}^4=+1$. These represent the modulo 4 integers $m_4\in {\mathbb{Z}}_4$. Note that $m_4=3=-1$ can also be interpreted as a left-handed eigenstate. (b) Bosonic inversion ${\left({\mathcal{R}}_2\right)}^2=+1$; there are two unique eigenvalues ${\eta }_2=exp\left[i\frac{2\pi }{2}{m}_2\right]$ corresponding to the roots of unity ${\left({\eta }_2\right)}^2=+1$. These represent the modulo 2 integers $m_2\in {\mathbb{Z}}_2$.

Figure 3

The collection of spin-$\frac{1}{2}$ (fermionic) charges for the $C_4$ point group. (a) Fourfold rotations ${\left({\mathcal{R}}_4\right)}^4=-1$; there are four unique eigenvalues ${\zeta }_4=exp\left[i\frac{2\pi }{4}{m}_4\right]$ corresponding to the roots of negative unity ${\left({\zeta }_4\right)}^4=-1$. These represent the modulo 4 half-integers $m_4\in {\mathbb{Z}}_4+\frac{1}{2}$. Note that $m_4=\frac{7}{2}=-\frac{1}{2}$ can be interpreted as a spin-down fermion while $m_4=\frac{3}{2}=\frac{1}{2}+1$ and $m_4=\frac{5}{2}=-\frac{1}{2}+3$ constitute a fermion plus a boson. (b) Fermionic inversion ${\left({\mathcal{R}}_2\right)}^2=-1$; there are two unique eigenvalues ${\zeta }_2=exp\left[i\frac{2\pi }{2}{m}_2\right]$ corresponding to the roots of negative unity ${\left({\zeta }_2\right)}^2=-1$. These represent the modulo 2 half-integers $m_2\in {\mathbb{Z}}_2+\frac{1}{2}$. Note that $m_2=\frac{3}{2}=-\frac{1}{2}$ can also be interpreted as a spin-down fermion under modulo 2.

Figure 4

Examples of SPT bosonic phases in a crystal with $C_4$ symmetry. These phases are characterized by their SPT invariant $\nu =m_4\left(\mathrm{\Gamma }\right)+m_4\left(M\right)+2m_2\left(Y\right)mod4$, which determines the electromagnetic Chern number up to a multiple of 4. Here, $m_4\in {\mathbb{Z}}_4$ and $m_2\in {\mathbb{Z}}_2$ are modulo integers. (a), (b), (c), and (d) correspond to SPT bosonic phases of $\nu =3,2,1$, and 0, respectively. For bosons, we simply add up all the integer charges within the irreducible Brillouin zone. For instance, the $\nu =2$ phase has eigenvalues of $m_4\left(\mathrm{\Gamma }\right)=1$ at the center and $m_4\left(M\right)=3=-1$ at the vertices, with inversion eigenvalues of $m_2\left(Y\right)=m_2\left(X\right)=1$ at the edge centers: $\nu =1+3+2\times 1=2$.

Figure 5

Examples of SPT fermionic phases in a crystal with $C_4$ symmetry. These phases are characterized by their SPT invariant $\nu =m_4\left(\mathrm{\Gamma }\right)+m_4\left(M\right)+2m_2\left(Y\right)+2mod4$, which determines the electronic Chern number up to a multiple of 4. In this case, $m_4\in {\mathbb{Z}}_4+\frac{1}{2}$ and $m_2\in {\mathbb{Z}}_2+\frac{1}{2}$ are modulo half-integers. (a), (b), (c), and (d) correspond to SPT fermionic phases of $\nu =3,2,1$, and 0, respectively. The problem is more complicated for fermions because the charges are fractional and we must also account for the antisymmetric phases of a spinor wave function. As an example, the $\nu =2$ phase has eigenvalues of $m_4\left(\mathrm{\Gamma }\right)=m_4\left(M\right)=\frac{1}{2}$ at the center and vertices, with inversion eigenvalues of $m_2\left(Y\right)=m_2\left(X\right)=\frac{3}{2}=-\frac{1}{2}$ at the edge centers: $\nu =\frac{1}{2}+\frac{1}{2}+2\times \frac{3}{2}+2=2$.