Explicit spacetime symmetry breaking in matter: the reversed Vavilov-\v{C}erenkov radiation

TL;DR

This paper demonstrates that reversed Vavilov-erenkov radiation can occur in magnetoelectric materials when a charge moves near an interface, showing backward radiation in addition to the forward emission.

Contribution

It introduces the concept of reversed Vavilov-erenkov radiation in magnetoelectric media, expanding understanding of spacetime symmetry breaking effects.

Findings

Reversed Vavilov-erenkov radiation occurs in magnetoelectric interfaces.

Backward radiation is observed alongside forward emission.

Angular distribution of radiation is calculated using Green's function.

Abstract

We show that reversed Vavilov-\v{C}erenkov radiation occurs simultaneously with the forward output in naturally existing materials when an electric charge moves with a constant velocity perpendicular to the planar interface between two magnetoelectric media. Using the Green's function in the far-field approximation we calculate the angular distribution of the radiated energy per unit frequency obtaining a non-zero contribution in the backward direction.

Equations

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\frac{1}{4}(k_F{)}_{\kappa \lambda \mu \nu }{F}^{\kappa \lambda }{F}^{\mu \nu }+\frac{1}{2}(k_{AF}{)}^{\kappa }{ϵ}_{\kappa \lambda \mu \nu }{A}^{\lambda }{F}^{\mu \nu }.$$

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\frac{\alpha }{32{\pi }^2}\vartheta (\mathbf{x}){ϵ}_{\kappa \lambda \mu \nu }{F}^{\kappa \lambda }{F}^{\mu \nu },$$

$$\left({\left[{\square }^2\right]}_{\nu }^{\mu }-\tilde{\theta }\delta (z){\epsilon }_{\nu }^{3\mu \alpha }{\partial }_{\alpha }\right)G_{\sigma }^{\nu }(x,x^{\prime })=4\pi {\eta }_{\sigma }^{\mu }{\delta }^4(x-x^{\prime }).$$

$${\mathcal{I}}_1(\omega ,\theta )$$

$${\mathcal{I}}_2(\omega ,\theta )$$

Full text

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1 Introduction

As a motivation for this work let us recall that spacetime-dependent couplings have been identified as possible sources of fundamental Lorentz-invariance violation.[1] Also, let us consider the photon sector of the Standard-Model Extension[2]

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A term that is missing in the Lagrange density (1) is $(k_{F})_{\kappa\lambda\mu\nu}=\theta\epsilon_{\kappa\lambda\mu\nu}$, with the axion coupling $\theta$ being a constant, simply because this contribution is a total derivative. Nevertheless, if we promote the constant $\theta$ to $\theta(x)$ we are able to describe the effective electromagnetic response of a host of interesting physical phenomena where spacetime symmetries are expected to be explicitly broken. In this work we deal with axion electrodynamics

[TABLE]

which describes the electromagnetic response of magnetoelectric materials like topological insulators[3] and Weyl semimetals,[4] for example. Here we summarize the discovery of reversed Vavilov–Čerenkov radiation (RVCR) in naturally existing magnetoelectrics, reported in Ref. \refciteURRU.

In 1968 Veselago [6] proposed that RVCR could be produced in materials characterized by negative permittivity $\epsilon$ and permeability $\mu$. In this case a radiation cone is observed in the backward direction with respect to the velocity of the propagating charge. Since such materials are not found in nature, this proposal gave a major thrust to the design and construction of metamaterials that would provide the required properties in a given range of frequencies.[7]

2 RVCR in axion electrodynamics

We consider two semi-infinite media separated by the interface $\Sigma$ at $z=0$ with the axion field $\vartheta(z)=H(z)\vartheta_{2}+H(-z)\vartheta_{1}$ and $\vartheta_{1}$, $\vartheta_{2}$ constants, where $H(z)$ is the Heaviside function. In order to suppress the transition radiation we take $\epsilon_{1}=\epsilon_{2}=\epsilon$ as a first approximation. We also set $\mu_{1}=\mu_{2}=1$. The modified Maxwell equations resulting from the Lagrange density (2) are those corresponding to a normal material medium with constitutive relations $\mathbf{D}=\epsilon\mathbf{E}-\alpha\vartheta(z)\mathbf{B}/\pi$ and $\mathbf{H}=\mathbf{B}+\alpha\vartheta(z)\mathbf{E}/\pi$. These equations introduce field-dependent effective charge and current densities with support only at the interface $\Sigma$, which produce the magnetoelectric effect,[3] being governed by the parameter ${\tilde{\theta}}=\alpha(\vartheta_{2}-\vartheta_{1})/\pi$. To calculate the radiation fields we rewrite the modified Maxwell equations in terms of the potential $A_{\mu}$ $(c=1)$ and introduce the Green’s function (GF) $G_{\;\;\sigma}^{\nu}(x,x^{\prime})$ by setting $A^{\mu}(x)=\int d^{4}x^{\prime}G_{\;\,\nu}^{\mu}({x},{x}^{\prime})J^{\nu}({x}^{\prime})$, obtaining

[TABLE]

Let us remark that the resulting $\theta$-dependent contribution to the GF is a function of $(|z|+|z^{\prime}|)$, instead of $|z-z^{\prime}|$ as it is in the normal case. This is the mathematical origin of the RVCR. The required GF in the far-field approximation includes integrals of rapidly oscillating functions and its leading contribution is obtained by the stationary-phase approximation.

Now, let us consider a charge $q$ moving at a constant velocity $v\mathbf{\hat{u}}$ with charge and current densities $\varrho(\mathbf{x}^{\prime};\omega)=\frac{q}{v}\delta(x^{\prime})\delta(y^{\prime})e^{i\omega\frac{z^{\prime}}{v}},\quad\mathbf{j}(\mathbf{x}^{\prime};\omega)=\varrho v\mathbf{\hat{u}}$. Here, ${\mathbf{\hat{u}}}$ is the unit vector perpendicular to the interface, directed from region 1 to region 2. We henceforth assume $v>0$ and consider the motion in the interval $z\in(-\zeta,\zeta)$, with $\zeta\gg v/\omega$. In the far-field approximation we calculate the electric field $\mathbf{E}$ starting from the potential $A^{\mu}$ and using the corresponding approximation of the GF obtained from Eq. (3). Then, we are able to calculate the spectral distribution (SD) of the radiation $d^{2}E/d\omega d\Omega=(\mathbf{E}^{\ast}\cdot\mathbf{E})\,nr^{2}/4\pi^{2}$ in the limit $r\rightarrow\infty$. The main point here is that the following integrals, which result from the convolution of the GF with the sources,

[TABLE]

enter in the expression for the electric field. The notation is $\Xi_{-}=\frac{\omega}{v}\left(1-vn\cos\theta\right)$ and $\tilde{\Xi}_{-}=\frac{\omega}{v}\left(1-vn|\cos\theta|\right)$. More importantly, in the limit $\zeta\gg v/\omega$, ($\zeta\rightarrow\infty$), the right-hand side of Eqs. (4) and (5) yields expressions like $\sin(\zeta\,\rho N)/(\rho N)$ which behave as $\pi\delta(\rho N)$. This delta-like behavior means that the contributions of the electric field to the SD are nonzero only when (i) $1\pm vn\cos\theta=0$ and/or (ii) $1\pm vn|\cos\theta|=0$. Following our conventions, the case (i) yields the standard Čerenkov condition $\cos\theta_{C}=1/(nv)$. The second case (ii) opens up the possibility that $\cos\theta$ is negative yielding the reversed Čerenkov cone at $\theta_{\rm R}=\pi-\theta_{C}$. In other words, the terms containing $\tilde{\Xi}_{-}$ make possible the production of radiation in the backward direction. The SD of such radiation is suppressed with respect to the forward output, but nevertheless it is different from zero.

To illustrate our results we choose medium 2 as the topological insulator ${\rm TlBiSe}_{2}$, with $n_{2}=2$ prepared in such a way that ${\tilde{\theta}}=11\alpha$, together with a normal insulator in medium 1 having $n_{1}=n_{2}$. We consider the radiation emitted at the average frequency of $\omega=2.5\,$eV ($500\,$nm) in the Čerenkov spectrum. The SD of the total radiation is shown in Fig. 1. In the left panel of the figure we plot a zoom in the backward direction showing the onset of the RVCR and making evident the high suppression of this new contribution with respect to the forward Vavilov–Čerenkov radiation. The details of the above calculations are reported in Ref. \refciteURRU.

3 Summary

We have considered the radiation produced by an electric charge moving at a constant velocity ${v\mathbf{\hat{u}}}$ perpendicular to the interface $z=0$ between two semi-infinite planar magnetoelectric media with the same refraction index. When $v$ is higher than $1/n$ we discover the emission of RVCR codified in the angular distribution illustrated in Fig. 1. The main characteristics of this RVCR are: (i) The threshold condition $v>1/n$ must be satisfied as in the standard case. (ii) The RVCR occurs for all frequencies in the Čerenkov spectrum and it is always accompanied by the forward Vavilov–Čerenkov radiation with the same frequency. (iii) The SD of the RVCR is suppressed with respect to the forward output by the factor ${\tilde{\theta}}^{2}/(8n^{2})$.[5]

Acknowledgments

OJF has been supported by the doctoral fellowship CONACYT-271523. OJF, LFU, and ORT acknowledge support from the projects: #237503 from CONACYT and

IN103319 from Dirección General de Asuntos del Personal Académico (Universidad Nacional Autónoma de México).

We thank VAK and collaborators for a wonderful meeting.