Chiral magnetic effect of light

TL;DR

This paper explores a photonic analog of the chiral magnetic effect, demonstrating how magnetoelectric tensors influence light propagation and cause observable transverse displacements, akin to effects seen in chiral fermions.

Contribution

It introduces the concept of a chiral magnetic effect of light, linking magnetoelectric tensor properties to anomalous wave packet shifts and extending analysis beyond geometric optics.

Findings

Magnetoelectric tensors act as a 'vector potential' for light.

Anomalous wave packet shifts occur due to Berry curvature effects.

Transverse displacements are observable in experiments with magnetoelectric materials.

Abstract

We study a photonic analog of the chiral magnetic (vortical) effect. We discuss that the vector component of magnetoelectric tensors plays a role of "vector potential," and its rotation is understood as "magnetic field" of a light. Using the geometrical optics approximation, we show that "magnetic fields" cause an anomalous shift of a wave packet of a light through an interplay with the Berry curvature of photons. The mechanism is the same as that of the chiral magnetic (vortical) effect of a chiral fermion, so that we term the anomalous shift "chiral magnetic effect of a light." We further study the chiral magnetic effect of a light beyond geometric optics by directly solving the transmission problem of a wave packet at a surface of a magnetoelectric material. We show that the experimental signal of the chiral magnetic effect of a light is the nonvanishing of transverse displacements for the beam normally incident to a magnetoelectric material.

Equations

$$4\pi \mathbf{P}$$

$$4\pi \mathbf{M}$$

$${\nabla }_x\cdot \left(ϵ\mathbf{E}-\mathbf{g}\times \mathbf{H}\right)$$

$${\nabla }_x\cdot \left(\mu \mathbf{H}+\mathbf{g}\times \mathbf{E}\right)$$

$${\nabla }_x\times \mathbf{E}+\frac{1}{c}{\partial }_t\left(\mu \mathbf{H}+\mathbf{g}\times \mathbf{E}\right)$$

$${\nabla }_x\times \mathbf{H}-\frac{1}{c}{\partial }_t\left(ϵ\mathbf{E}-\mathbf{g}\times \mathbf{H}\right)$$

$$\mathbf{D}\times {\mu }^{-1}\left(\mathbf{D}\times \tilde{\mathbf{E}}\right)-ϵ\tilde{\mathbf{E}}$$

$$\mathbf{D}\times {ϵ}^{-1}\left(\mathbf{D}\times \tilde{\mathbf{H}}\right)-\mu \tilde{\mathbf{H}}$$

$$[D_i,D_j]=-i\text{lambdabar}{ϵ}_{ijk}{\mathbf{b}}_k,$$

$$[{\hat{\mathbf{x}}}_i,{\hat{\mathbf{x}}}_j]=i\text{lambdabar}{ϵ}_{ijk}{\mathbf{\Omega }}_k,$$

$${\dot{\mathbf{x}}}_c$$

$${\dot{\mathbf{p}}}_c$$

$$|{\dot{\eta }}_c⟩$$

$$\mathbf{\Lambda }=\frac{1}{p\tan \theta }{\mathbf{e}}_{\phi }{\sigma }_z,\mathbf{\Omega }=-\frac{\mathbf{p}}{p^3}{\sigma }_z,$$

$$\sqrt{J}{\dot{\mathbf{x}}}_{\chi }$$

$$\sqrt{J}{\dot{\mathbf{p}}}_{\chi }$$

$${\Psi }_{\mathrm{initial}}=\frac{\Psi (y)}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)=\int d{p}_y{e}^{i{p}_{y}y}\frac{\Phi (p_y)}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right).$$

$${\Psi }_{\mathrm{final}}$$

$${\delta }_{\pm }^{\mathrm{H}}$$

$${\delta }_{\pm }^{\mathrm{V}}$$

$${\Theta }_{\pm }=\int dl\frac{d{\mathbf{p}}_{\pm }}{dl}\cdot [\mathbf{\Lambda }{]}_{\pm \pm }=\int d{\mathbf{p}}_{\pm }\cdot [\mathbf{\Lambda }{]}_{\pm \pm },$$

Full text

I Introduction

The Berry phase Berry (1984) and Berry curvature have attracted a lot of interests in subdisciplines of physics such as condensed matter physics, nuclear physics, and particle physics. They characterize topology of wave functions in momentum space and explain many properties of topological materials such as topological insulators Qi et al. (2008); Hasan and Kane (2010), topological superconductors Qi and Zhang (2011), and Weyl/Dirac semimetals Zahid Hasan et al. (2017); Armitage et al. (2017) as well as those of relativistic Weyl/Dirac fermions such as quarks and neutrinos.

It has been known that photons have a nonzero Berry curvature, which originates from the massless and helical nature of them Chiao and Wu (1986); Tomita and Chiao (1986); Berry (1987). The Berry curvature of photons leads to novel effects, which cannot be explained by the standard geometric optics according to Fermat’s principle. The famous example is the Hall effect of light Onoda et al. (2004, 2006), which is also known as the optical Magnus effect Dooghin et al. (1992); Liberman and Zel’dovich (1992); Duval et al. (2006, 2007). A spatially varying refractive index can be understood as an “electric field” of a light, and causes transverse shifts of a wave packet of a light. This deviation from Snell’s law in a finite beam has been confirmed by experiments Bliokh and Bliokh (2006); Hosten and Kwiat (2008). The phenomena can be explained by considering the anomalous group velocity of a light due to the interplay between “electric fields” and Berry curvature, and the mechanism is the same as that of the anomalous Hall effect in electron systems Nagaosa et al. (2010); Xiao et al. (2010).

Such an analogy between geometric optics and semiclassical dynamics of electrons is not complete yet. This is because photons do not couple with magnetic fields, and no Lorentz force appears in the standard geometric optics. So far, it has been indicated to use inhomogeneous optical magnetoelectric effect as “magnetic fields” of a light in Ref. Sawada and Nagaosa (2005). There have also been works to study the physics of photons under “magnetic fields” in the context of photonic crystals Haldane and Raghu (2008); Raghu and Haldane (2008); Wang et al. (2008); Hafezi et al. (2011); Fang et al. (2012); Hafezi et al. (2013); Lu et al. (2014). However a magnetic analog of anomalous Hall effect, that is, the exotic phenomenon originated from the interplay between “magnetic fields” and Berry curvature Stephanov and Yin (2012); Son and Yamamoto (2012); Chen et al. (2013); Son and Spivak (2013), which is referred to as the chiral magnetic (vortical) effect in the physics of chiral fermions Fukushima et al. (2008); Vilenkin (1979); Son and Surówka (2009), has not been discussed.

In this paper, we study a photonic analog of the chiral magnetic (vortical) effect Fukushima et al. (2008); Vilenkin (1979); Son and Surówka (2009). We discuss that rotation of the vector component of magnetoelectric tensors behaves as “magnetic fields” of a light Sawada and Nagaosa (2005), and causes anomalous shifts of a wave packet of a light along the direction parallel to it. In terms of geometrical optics, the mechanism is the same as that of the chiral magnetic (vortical) effect of chiral fermions Stephanov and Yin (2012); Son and Yamamoto (2012); Chen et al. (2013); Son and Spivak (2013), and thus we term the helicity-dependent shifts “chiral magnetic effect of a light.” We show that the helical shifts due to the chiral magnetic effect of a light arise beyond the geometrical optics approximation. We also discuss an analog of the spectral flow Son and Spivak (2013); Nielsen and Ninomiya (1983) in geometric optics.

The same effect arising in rotating frame was discussed in Refs. Zyuzin (2017); Avkhadiev and Sadofyev (2017); Yamamoto (2017). References Zyuzin (2017); Avkhadiev and Sadofyev (2017) calculated helicity current of photons only in equilibrium states at finite temperature. Reference Yamamoto (2017) calculated helicity current using the kinetic theory with the Berry curvature correction, which is, in principle, applicable to nonequilibrium states. However the wave packet dynamics was not studied in Ref. Yamamoto (2017), which is relevant in photonics. In addition, our proposal does not consider noninertial frame under rotation, but the effective metric is generated by materials. It may be tested in magnetoelectric materials such as LiCoPO4, TbPO4 Fiebig (2005); Rivera (2009), and ZnCr2Se4 spinel with a conical spiral state Spaldin et al. (2008). “Magnetic fields” can experimentally be generated at a surface of magenetoelectirc materials, or by preparing inhomogeneous ordering such as domain walls Sawada and Nagaosa (2005).

II geodesic equation of light

We discuss the propagation of a monochromatic electromagnetic wave with frequency $\omega$ in anisotropic, inhomogeneous, and lossless mediums exhibiting linear magnetoelectric effect in Gaussian units:

[TABLE]

where $\bm{P}$ and $\bm{M}$ ($\bm{E}$ and $\bm{H}$) are polarization, and magnetization (electric and magnetic fields), respectively. $\chi_{e}$, $\chi_{\mu}$, and $\bm{g}$ represent electric susceptibility, magnetic susceptibility, and a vector component of magnetoelectric tensors Fiebig (2005); Spaldin et al. (2008). We first assume that $\bm{g}$ slowly varies on space, and show that $\nabla_{x}\times\bm{g}$ plays a role of “magnetic field” in the geodesic equation of a light.

By taking the polarization and magnetization in Eqs. (1) and (2) into account, the Maxwell equations read

[TABLE]

where $\epsilon$ and $\mu$ are the dielectric permittivity and magnetic permeability, and $c$ is the speed of light in vacuum. Those equations become the same as the Maxwell equations in rotating frame Landau and Lifshitz (1975) if $\bm{g}=\bm{\Omega}\times\bm{x}/2$ with angular velocity $\bm{\Omega}$, and the centrifugal force terms [${\cal O}(\bm{\Omega}^{2}$) terms] are neglected. For the monochromatic wave electric and magnetic fields, $\bm{E}=\tilde{\bm{E}}(\omega)e^{-i\omega t}$ and $\bm{H}=\tilde{\bm{H}}(\omega)e^{-i\omega t}$, we obtain the eigen equations:

[TABLE]

where $\bm{D}=\lambdabar\nabla_{x}-i\bm{g}$, and $\lambdabar=\lambda/(2\pi)=c/\omega$. Because of the inhomogeneous magnetoelectric effect, $\bm{D}$ no longer commute with each other and satisfy

[TABLE]

where $\bm{b}=\nabla_{x}\times\bm{g}$, and $\epsilon_{ijk}$ is the totally antisymmetric tensor. The noncommutativity leads to the Coriolis force in the dynamics of electromagnetic waves.

We consider the eikonal approximation in the linear order of $\lambdabar$, and derive the geodesic equation of a light with the Berry curvature correction. When $\bm{g}=0$, and the anisotropy and inhomogeneity of $\epsilon$ and $\mu$ are small and treated perturbatively, as performed e.g., in Refs. Bliokh and Freilikher (2005); Bliokh et al. (2007, 2008), by introducing dimensionless momentum operator $\hat{\bm{q}}=-i\lambdabar\nabla_{x}$ and rewriting Eq. (7) into the Schödinger-type equation ${\cal H}\tilde{\bm{E}}=0$, we can diagonalize the $3\times 3$ matrix ${\cal H}$, and obtain three eigenvectors. Two of them correspond to transverse modes, and the other corresponds to the longitudinal (resonant) mode. Then when the system is off-resonant ($\epsilon\neq 0$) Bliokh et al. (2007), by neglecting the resonant mode, we can introduce the Berry connection $\bm{\Lambda}$ and Berry curvature $\bm{\Omega}$ to describe the noncommutative dynamics in the projected space spanned by the transverse modes in the same way with quantum mechanics:

[TABLE]

where $\hat{\bm{x}}=i\lambdabar\nabla_{q}+\bm{\Lambda}$ are covariant coordinate operators Bliokh and Freilikher (2005); Bliokh et al. (2007). In the geodesic equation, the noncommutavity is implemented as the spin-orbit coupling Bliokh and Freilikher (2005); Bliokh et al. (2007, 2008).

Now we take the effect of $\bm{g}$ into account based on the derivative expansion. This can be done in the same way with the vector potential in the wave packet dynamics in electron systems Sundaram and Niu (1999); Culcer et al. (2005); Shindou and Balents (2008); Hayata and Hidaka (2017). The modification is straightforward: We replace the canonical momentum $\bm{q}=-i\lambdabar\nabla_{x}$ by the covariant momentum $\bm{p}=-i\bm{D}$ as $\bm{q}=\bm{p}+\bm{g}$ in the effective Lagrangian given in Ref. Bliokh et al. (2008). For simplicity, we hereafter consider a locally isotropic medium in which the dynamics becomes Abelian Onoda et al. (2004, 2006); Bliokh and Freilikher (2005); Bliokh et al. (2007, 2008). The geodesic equation reads, in the linear order of $\lambdabar$ and $\bm{g}$,

[TABLE]

where $\bm{x}_{c}$ and $\bm{p}_{c}$ are the coordinates and dimensionless wave vectors of a ray of light. $|\eta_{c}\rangle=(\eta_{+},\eta_{-})^{t}$ represents polarization states of lights. We use right-handed (+) and left-handed (-) circularly polarized waves as a basis of polarization. The dot means the derivative with respect to the ray length $l$, namely, the derivative along the trajectory, not time Bliokh and Freilikher (2005); Bliokh et al. (2007, 2008). $n(\bm{x}_{c})$ is isotropic and slowly varying refractive index, and $\bm{e}=\nabla_{x_{c}}n(\bm{x}_{c})$. $\bm{\Lambda}$ and $\bm{\Omega}=\nabla_{\bm{p}_{c}}\times\bm{\Lambda}-i\bm{\Lambda}\times\bm{\Lambda}$ are the Berry connection and Berry curvature, and given explicitly as Onoda et al. (2004, 2006); Bliokh and Freilikher (2005); Bliokh et al. (2007, 2008)

[TABLE]

where $p$, $\theta$ and $\varphi$ are spherical coordinates in $\bm{p}$ space, and $\sigma_{z}$ is the Pauli matrix. In the above geodesic equation, $\bm{b}$ is the same as the magnetic field in the classical equation of motion of electrons, so that we term $\bm{b}$ “magnetic field.” However it originates from the effective metric induced by magnetoelectric materials, and strictly speaking, is equal to gravitomagnetic field Hidaka et al. (2013). The same is true for “electric field” $\bm{e}=\nabla_{x_{c}}n(\bm{x}_{c})$, and it is equal to gravitoelectric field Hidaka et al. (2013).

We note that we neglected a self-rotation of a wave packet and the associated “Zeeman energy,” which may not be negligible in a beam with intrinsic angular momentum such as optical vortices. We also note that the same idea to use the optical magnetoelectric effect as “vector potential” of a light has been indicated in Ref. Sawada and Nagaosa (2005) (For the explicit relation to ours, see 111See Supplemental Material at [URL will be inserted by publisher] for the derivation of the optical magnetoelectric effect). However, they employed the conventional Fermat’s variational principle with optical Lorentz force, and topological phenomena caused by “magnetic fields” were not discussed. Such topological phenomena including the chiral magnetic effect can be analyzed only by taking the effect of the Berry curvature of photons into account, and are the main subject of this paper. Moreover, we will analyze the chiral magnetic effect of a light beyond the geometrical optics approximation.

III Chiral magnetic effect of light

We discuss anomalous shift of a wave packet of light caused by the interplay between “magnetic fields” and Berry curvature. The geodesic equation in Eqs. (11) and (12) is diagonal for right- and left-handed polarizations, and we obtain

[TABLE]

where $\sqrt{J}=1-\chi\bm{b}\cdot\lambdabar\bm{\Omega}_{++}$, and $\chi=\pm 1$ for right-handed (left-handed) polarization. The second term in Eq. (15) leads to the optical Hall effect Onoda et al. (2004, 2006); Dooghin et al. (1992); Liberman and Zel’dovich (1992). The last terms in Eqs. (15) and (16) lead to the analog of the chiral magnetic effect Stephanov and Yin (2012); Son and Yamamoto (2012); Chen et al. (2013); Son and Spivak (2013) and the spectral flow Son and Spivak (2013); Nielsen and Ninomiya (1983).

Let us first discuss the chiral magnetic effect of a light. We consider the propagation of a wave packet of a light. As in Eq. (15), the anomalous group velocity parallel to $\bm{b}$ is generated by the Berry curvature. Since the sign is opposite between the right- and left-handed polarizations, those states propagate along the opposite direction parallel to $\bm{b}$. We term the helicity-dependent shift of a wave packet chiral magnetic effect of a light since $\bm{b}$ in the geodesic equation is the same as the magnetic field in the classical equation of motion of electrons Son and Spivak (2013).

To confirm the idea, we consider a physical set up similar to Ref. Sawada and Nagaosa (2005), and numerically calculate the trajectories of wavepackets propagating through thin films at $\bm{b}\lambda=(0,0,2\times 10^{-3})$ 222We estimate the gradient $b=(\nabla\times\bm{t})_{z}$ as $b\lambda\sim\alpha\lambda/L\sim 2\times 10^{-3}$, where $\alpha=0.2$, $L=50\mu$m, and $\lambda=500$nm are typical magnitude of the magnetoelectric tensors Rivera (2009), thickness of a sample, and wavelength of an incident light, respectively. and uniform $n=0.50$, $2.0$ ($\bm{e}=0$). We assume that the sample is infinitely large in the $yz$ plane and thin along the $x$ direction. Then we consider the incident light along the $x$ direction with right-handed polarization. Without $\bm{b}$ field, the light propagates along the $x$ direction. In the presence of $\bm{b}$ field, shifts along the $y$ and $z$ directions occur because of the “Lorentz force,” and chiral magnetic effect of a light. The numerical results are shown in Fig. 1. The quadratic curves due to the Lorentz force are independent of the polarization, and reproduce the result of Ref. Sawada and Nagaosa (2005). In addition, we find the linear and helical shift parallel to $\bm{b}$. This is the consequence of the chiral magnetic effect of a light, and can be analyzed only by taking the Berry curvature correction into account. From the slope, we estimate the shift of the wave packet as $\delta z=\pm 0.05\mu$m for right-handed (left-handed) polarization when the sample thickness is $x=50\mu$m and $n=0.50$.

For quantitative prediction to experiments, we consider a realistic physical set up based on an experimental work Hosten and Kwiat (2008), and compute the transverse displacements of a wave packet refracted at a surface between vacuum and a uniform magnetoelectric material. We consider a wave packet incident with angle $\theta_{\rm I}$ to the $z=0$ surface between vacuum ($z<0$) and a half-infinite magnetoelectric material ($z>0$) with $\bm{g}=(g,0,0)$ [$g>0$] (see Fig. 3). At a sharp surface, $n$ and $\bm{g}$ rapidly change like the step function, and the delta function-like “electric field” $\bm{e}=\nabla n\sim\delta(z)\hat{z}$ and “magnetic field” $\bm{b}=\nabla_{x}\times\bm{g}\sim\delta(z)\hat{y}$ are induced at a surface. Then, from the analysis based on the Berry curvature, we expect shifts of a wave packet due to the spin Hall/chiral magnetic effect of a light. However, when $n$ and $\bm{g}$ rapidly change, we no longer employ the geometrical optics approximation and need to directly solve the Maxwell equations. As will be shown, the shift parallel to $\bm{b}$ still arises, and the above analysis based on the Berry curvature is qualitatively correct. Following Ref. Hosten and Kwiat (2008), we consider a wavepacket with finite distribution along the $y$ direction. For a horizontally-polarized wave, the amplitude reads Hosten and Kwiat (2008)

[TABLE]

The right- and left-handed states rotate in an opposite way upon transmission to satisfy the transversality condition, which leads to the transverse displacement of a central position via the spin-orbit coupling Hosten and Kwiat (2008) as schematically shown in Fig. 3:

[TABLE]

where $t_{p}$ is the Fresnel coefficient. Similar arguments hold for a vertical polarization and lead to the transverse displacement shown below. By solving the transmission problem with the polarization in Eq. (1) and magnetization in Eq. (2), the displacements of the spin components ($\pm$) for horizontally and vertically polarized waves are given as

[TABLE]

where $\tilde{\theta}_{\rm I}=\theta_{\rm I}+g\cos\theta_{\rm I}/(n^{2}-1)$, and $\lambda$, $\theta_{\rm I}$, $\tilde{\theta}_{\rm T}$, $t_{s}$, and $t_{p}$ are the wave length of a light in incident medium, the incident and refraction angles, and the Fresnel coefficients, respectively 333See Supplemental Material at [URL will be inserted by publisher] for computational details. $\tilde{\theta}_{\rm T}$ is related with $\theta_{\rm I}$ by Snell’s law as $\sin\tilde{\theta}_{\rm T}=\sin\tilde{\theta}_{\rm I}/n$. The chiral magnetic effect can be computed as the spin Hall effect in the boosted frame, so that the transverse displacements in Eqs. (19) and (20) have the same form with Ref. Hosten and Kwiat (2008) except that the incident angle is modified by the magnetoelectric effect.

We show the displacements of the right-handed component ($+$) at zero and nonzero $g$ in Fig. 3. Wave packets experience the displacements due to the chiral magnetic effect only at nonzero $g$. We find that the key signal of the chiral magnetic effect is the nonvanishing of $\delta^{\rm H,V}_{\pm}$ at $\theta_{\rm I}=0$, namely, when the beam is normally incident to the material. This can be qualitatively understood from the equation of motion (15). The spin Hall effect is originated from $\chi\bm{\Omega}_{++}\times\bm{e}\sim-\chi\hat{\bm{p}}\times\hat{z}$, and vanishes when $\bm{p}$ is normal to a surface. On the other hand, the chiral magnetic effect is originated from $-\chi\bm{b}(\bm{\Omega}_{++}\cdot\bm{p})\sim\chi\hat{y}$, and is approximately independent of the direction of $\bm{p}$. When $\lambda=633$nm Hosten and Kwiat (2008), the intercepts are estimated as $\delta^{\rm H}_{+}(\theta_{\rm I}=0)=5.4$nm, and $\delta^{\rm V}_{+}(\theta_{\rm I}=0)=3.6$nm, which are observable shifts via quantum weak measurements Hosten and Kwiat (2008).

Finally, we discuss a photonic analog of the spectral flow, which is also referred to as the Adler-Jackiw anomaly. In chiral fermions, when the pseudosclar product of electromagnetic fields $\bm{E}\cdot\bm{B}$ is nonzero, the excitation from the left-handed fermions to right-handed fermions occurs and the chirality imbalance is dynamically generated Son and Spivak (2013); Nielsen and Ninomiya (1983). The phenomena has been utilized an experimental signal of Weyl or Dirac points in transport phenomena in condensed matter materials Son and Spivak (2013); Huang et al. (2015). We expect an analogous effect when $\bm{e}\cdot\bm{b}\neq 0$. As in Eq. (16), $\bm{e}\cdot\bm{b}$ appears in $\dot{\bm{p}}$, so that lights are accelerated/decelated depending on their helicities. From Eq. (13), the integration of $\dot{\bm{p}}\cdot\bm{\Lambda}$ over the trajectories,

[TABLE]

contributes to the phase of the outstate as $|\eta_{c}^{\rm out}\rangle=[e^{i\Theta_{+}}\eta_{+}^{\rm in},e^{i\Theta_{-}}\eta_{-}^{\rm in}]^{t}$, with the initial polarization $|\eta_{c}^{\rm in}\rangle=[\eta_{+}^{\rm in},\eta_{-}^{\rm in}]^{t}$. Therefore, when $\bm{e}\cdot\bm{b}\neq 0$, phase difference arises ($\Theta_{+}\neq\Theta_{-}$) because of the aforementioned helicity-dependent acceleration/deceleration. This is one approach to discuss a photonic analog of the spectral flow. A similar phase shift due to inhomogeneous magnetoelectric effect has been discussed in topological insulators Qi et al. (2008); Maciejko et al. (2010); Tse and MacDonald (2010); Valdés Aguilar et al. (2012); Okada et al. (2016), which involve the scalar component of magnetoelectric tensors Fiebig (2005); Spaldin et al. (2008).

IV Summary

We have studied the photonic analog of the chiral magnetic (vortical) effect. We discuss that rotation of the vector component of magnetoelectric tensors $\bm{b}=\nabla\times\bm{g}$ behaves as “magnetic field” of a light Sawada and Nagaosa (2005). The interplay between “magnetic fields” and Berry curvature of photons causes helical shifts of a wave packet along the direction parallel to “magnetic fields.” This is the chiral magnetic effect of a light in geometric optics. We have confirmed the chiral magnetic effect of a light arises even when geometric optics breaks down, by directly solving the transmission problem of a wave packet at a surface of a magnetoelectric material. We show that the signal of the chiral magnetic effect is the nonvanishing of displacements for the beam normally incident to the material.

There are several generalizations of our paper. One direction is photonic crystals with artificaial magnetic fields Haldane and Raghu (2008); Raghu and Haldane (2008); Wang et al. (2008); Hafezi et al. (2011); Fang et al. (2012); Hafezi et al. (2013); Lu et al. (2014), in which the chiral magnetic effect of a light might be enhanced like the optical Hall effect Onoda et al. (2004, 2006). Another direction is a generalization to include another form of the Berry curvature, which involves temporal derivative, and is understood as emergent electric fields in momentum space Xiao et al. (2010). It causes anomalous transport effects such as the Thouless pumping Thouless (1983). It is interesting to discuss a photonic analog of the Thouless pumping, which may give a sizable effect due to the massless nature of photons, as in the case of Weyl semimetals Ishizuka et al. (2016).

Acknowledgements.

The author thanks K. Fukushima, Y. Hidaka, and S. Nakamura for useful comments. This work was supported by JSPS Grant-in-Aid for Scientific Research (No: JP16J02240).