-coupling in Electrodynamics and the Reversed Vavilov-Cherenkov - - PowerPoint PPT Presentation

θ-coupling in Electrodynamics and the Reversed Vavilov-Cherenkov radiation

• M. Sc. Omar Jesus Franca Santiago

Institute of Nuclear Sciences National Autonomous University of Mexico XXXIII Annual Meeting of the Division of Particles and Fields Mexican Physics Society 26.05.2019 - 28.05.2019

Outline

Introduction to θ-Electrodynamics Green’s Function Approach Reversed Vavilov-Cherenkov Radiation (RVCR) in 3D-TIs Conclusions

Importance of topological terms: Anomalies in QFT Quantum Hall Effect

EFT for MEs: 3D-TIs, antiferromagnets and Weyl

• semimetals. All of them are RHMs.

θ-Electrodynamics

• R. Bertlmann. Anomalies on Quantum Field Theory.

Oxford University Press, Great Bretain, 2000. Qi, Li, Zang, Zhang Science 323 1184 (2009)

¯ E

¯ B

k

• X. L. Qi, T. L. Hughes and S. C. Zhang, Phys. Rev. B 78, 195424 (2008). 

• A. M. Essin, J. E. Moore and V

anderbit, Phys. Rev. Lett. 102, 146805 (2009).

θ(x) = Θ(z)θ1 + Θ(−z)θ2.

Huerta, Zanelli Phys. Rev. D 85 085024 (2012)

θ-Electrodynamics

S[Φ, A] = Z dt d3x  1 8⇡

• "E2 − B2

− ↵ 4⇡2 ✓(x) E · B − %Φ + j · A

• ,

θ-Electrodynamics

Euler-Lagrange Equations give us the following Modified Maxwell’s equations:
 
 
 
 
 
 with r · B = 0 , r ⇥ E + ∂B ∂t = 0 , ✏r · E = 4⇡% + ˜ ✓(z)B · ˆ u ,

r ⇥ B ✏@E @t = 4⇡J + ˜ ✓(z)E ⇥ ˆ u ,

˜ θ = α π (θ2 − θ1) .

θ-Electrodynamics

Boundary Conditions: Potentials ∆E⊥

• Σ = ˜

θB⊥|Σ , ∆Bk

• Σ = −˜

θEk|Σ , ∆Ek|Σ = 0 .

Huerta, Zanelli Phys. Rev. D 85 085024 (2012)

E = rΦ ∂A ∂t , B = r ⇥ A. ∆B⊥|Σ = 0 ,

Green’s Function Approach

Introducing the 4-potential Aμ=(Φ,A) in the standard way, we find: 
 where Jμ=(ρ,j) and For a general solution for Aμ with arbitrary external sources Jμ, we introduce the GF:

L.F. Urrutia et al. Europhysics Letters 113, 60005 (2016); OJF, L. F. Urrutia and Omar Rodríguez- Tzompantzi (On 24.05.2019 accepted to be published by Phys. Rev. D).

[Oµ

ν]xGν σ(x, x0) = 4πηµ σδ(x − x0).

h⇥ ⇤2⇤µ

ν − ˜

θδ(z)ε3µα

ν∂α

i Aν = 4πJµ , ⇥ ⇤2⇤µ

ν =

• ✏⇤2, ⇤2i

j

• .

Green’s Function Approach

Fourier’s transverse and frequencies transform How could we resolve the GF equation? We know 
 
 where We can transform the PDEs system into an algebraic system. [Oµ

ν]k = [Oµ ν]kED + [Oµ ν]k˜

θ .

[Oµ

σ]kEDGσ 0 ν(z, z0; k?ω) = ηµ νδ(z − z0) ,

Gσ

0 ν(z, z0; k?, ω) = ησ ν

ieikz|zz0| 2kz ,

L.F. Urrutia et al. Europhysics Letters 113, 60005 (2016), Phys. Rev. D 94, 085019 (2016).

• C. Grosche, Phys. Rev. Lett. 71, 1 (1993).

Too large and too difficult expressions to work!! We consider the far zone regime of the GF. Using approximation methods (steepest descent method/saddle-point method and stationary phase method) we integrate the GF.

Chew, W eng Cho. W aves and Fields in Inhomogenous Media, IEEE Press Series on Electromagnetic W ave Theory. New Y

• rk, 1990.

Green’s Function Approach


 
 
 
 
 
 where , 
 There are two possible different phases!!

Gµ

˜ θ ν(x, x0; ω) = εµ α3 ν

2˜ θ 4n2 + ˜ θ2 sα | cos ϑ| ei˜

k0r

r ei˜

k0(n?·x0

?+|z0 cos ϑ|) ,

Gµ

˜ θ2 ν(x, x0; ω) =

˜ θ2 4n2 + ˜ θ2 ei˜

k0r

r cos2 ϑCµ

ν(x, n)ei˜ k0(n?·x0

?+|z0 cos ϑ|) ,

Green’s Function Approach

sα = (1/n, ˆ n) n⊥ = (x/r, y/r, 0).

OJF, L. F. Urrutia and Omar Rodríguez-Tzompantzi (On 24.05.2019 accepted to be published by Phys. Rev. D).

˜ k0 = nω ,

¯ Gµ

ED ν(x, x0; ω) = ηµ ν

ei˜

k0r

r ei˜

k0ˆ n·x0,

Reversed Vavilov-Cherenkov Radiation (RVCR)

Charged particle with constant velocity v = vˆ z %(x0; !) = q v (x0)(y0)eiω z0

v ,

j(x0; ω) = qδ(x0)δ(y0)eiω z0

v ˆ

z, where v > 1/p✏ > 0, z0 2 (⇣, ⇣) and ⇣ v/!. Aµ(x) = Z d4x0Gµ

ν(x, x0)Jν(x0) .

Reverse Vavilov-Cherenkov Radiation (RVCR)

E1(x; ω) = − sin ϑiωqei ˜

k0r

vrn [cos φI1(ω, ϑ) + nI2(ω, ϑ)

× 2˜ θ 4n2 + ˜ θ2 ✓ sin φ | cos ϑ| − ˜ θcos φ 2n ◆# ,

E2(x; ω) = − sin ϑiωqei˜

k0r

vrn [sin φI1(ω, ϑ) − nI2(ω, ϑ)

× 2˜ θ 4n2 + ˜ θ2 ✓ cos φ | cos ϑ| + ˜ θsin φ 2n ◆# ,

E3(x; ω) = sin ϑiqωei˜

k0r

vrn  (vn − cos ϑ) I1(ω, ϑ) sin ϑ

− 2˜ θ2 4n2 + ˜ θ2 I2(ω, ϑ) 2 cot ϑ # .

OJF, L. F. Urrutia and Omar Rodríguez-Tzompantzi (On 24.05.2019 accepted to be published by Phys. Rev. D).

Reverse Vavilov-Cherenkov Radiation (RVCR)

I1(ω, ϑ) = 2 sin ⇥ ζ ω

v (1 − vn cos ϑ)

⇤ ζ ω

v (1 − vn cos ϑ)

,

where I2(ω, θ) = sin ⇥ ζ ω

v (1 − vn| cos θ|)

⇤

ω v (1 − vn| cos θ|)

+2i sin2 h

ζ 2 ω v (1 − vn| cos θ|)

i

ω v (1 − vn| cos θ|)

.

OJF, L. F. Urrutia and Omar Rodríguez-Tzompantzi (On 24.05.2019 accepted to be published by Phys. Rev. D). W . K. H. Panofsky and M. Phillips. Classical Electricity and Magnetism. Addison-W esley Publishing, Inc., 2nd ed., 1962, USA.

Reverse Vavilov-Cherenkov Radiation (RVCR)

Angular distribution of the total energy radiated per unit frequency

d2E dωdΩ = nr2 4π2 E∗(x; ω) · E(x; ω). n = 2 ω = 2.48 eV ˜ θ = 0

v = 1/2, ζ = 2.80 eV −1

v = 1/2, ζ = 4830 eV −1

v = 0.9, ζ = 1.0 eV −1

v = 0.9, ζ = 343 eV −1

Reversed Cherenkov in TIs!!!!

Reverse Vavilov-Cherenkov Radiation (RVCR)

n = 2 ω = 2.48 eV ˜ θ = 11α

OJF, L. F. Urrutia and Omar Rodríguez-Tzompantzi (On 24.05.2019 accepted to be published by Phys. Rev. D).

v = 1/2, ζ = 2.80 eV −1

v = 0.9, ζ = 1.0 eV −1

v = 1/2, ζ = 4830 eV −1

v = 0.9, ζ = 343 eV −1

TlBiSe2

Reverse Vavilov-Cherenkov Radiation (RVCR)

Reversed Cherenkov in TIs!!!!

n = 2 ω = 2.48 eV ˜ θ = 11α

v = 1/2, ζ = 4830 eV −1

v = 0.9, ζ = 343 eV −1

OJF, L. F. Urrutia and Omar Rodríguez-Tzompantzi (On 24.05.2019 accepted to be published by Phys. Rev. D).

TlBiSe2

Total radiated energy per unit frequency But only for RCR Effective frequency

Reverse Vavilov-Cherenkov Radiation (RVCR)

dET OT AL dω = q2ωz ✓ 1 − 1 v2n2 ◆ . dERCR dω = q2ωz ✓ 1 − 1 v2n2 ◆ " 1 2 ˜ θ2 4n2 + ˜ θ2 # .

OJF, L. F. Urrutia and Omar Rodríguez-Tzompantzi (On 24.05.2019 accepted to be published by Phys. Rev. D).

ωeff ' ω ˜ θ2 8n2 ,

ωeff ∈ [4 × 10−3, 0.5]meV .

TlBiSe2 [˜ θ = α , ˜ θ = 11α]

• Z. Duan et al., Nature Communications,8, 14901 (2017).

Reversed Vavilov-Cherenkov Radiation (RVCR)

Very nice! But, how about the orders of magnitude? In comparison with other metamaterials it could be measured. ω [eV ] d2ERCR/dtdω [µW/eV ] 2 3.5 × 10−3 4 7.0 × 10−3 6 1.0 × 10−2 8 1.4 × 10−2 For TlBiSe2 with n = 2, ˜ θ = 11α, v = 0.8, and q = √α.

• J. Tao, Q. W

ang, J. Zhang and Y . Luo. Scientific Reports 6, 30704 (2016). OJF, L. F. Urrutia and Omar Rodríguez-Tzompantzi (On 24.05.2019 accepted to be published by Phys. Rev. D).

Conclusions

θ-terms show interesting features to study modifications of standard ED and to model MEs. GF approach together with the far-zone regime allows us to obtain analytical forms of the modified electromagnetic field generated by time-dependent arbitrary sources, in particular to a Cherenkov particle. RVCR in MEs, which are RHMs, could offer an application on Cherenkov detectors.

• P.A.M. Dirac

In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in the case of poetry, it's the exact

• pposite!

Thank you!

Future work

Dipolar radiation in homogeneous and inhomogeneous media. Transition Radiation Perform a hamiltonian analysis of θ-ED in order to study the quantum version of this EFT (θ- QED).

OJF, L. F. Urrutia, Omar Rodríguez-Tzompantzi and Alberto Martín-Ruíz OJF and Omar Rodríguez-Tzompantzi OJF and L. F. Urrutia

Green’s Function Approach

¯ Gµ

ED ν(x, x0; ω) = ηµ ν4π

Z d2k? (2π)2 eik?·R? iei√˜

k2

0k2 ?|zz0|

2 q ˜ k2

0 − k2 ?

,

¯ Gµ

˜ θ ν(x, x0; ω) = iεµ α3 ν

4π˜ θ 4n2 + ˜ θ2 Z d2k? (2π)2 eik?·R?kα ei√˜

k2

0k2 ?(|z|+|z0|)

˜ k2

0 − k2 ?

,

¯ Gµ

˜ θ2 ν(x, x0; ω) =

4π˜ θ2 4n2 + ˜ θ2 Z d2k? (2π)2 eik?·R? × ⇥ kµkν − (ηµ

ν + uµuν) k2⇤ ei√˜ k2

0k2 ?(|z|+|z0|)

2 ⇣ ˜ k2

0 − k2 ?

⌘3/2 .

Green’s Function Approach

Cµ

ν(x, n) =

    sin2 ϑ −x/rn y/rn x/rn −1/n2 y/rn −1/n2 + sin2 ϑ     .

Reversed Cherenkov Radiation (RCR)

d2E1 dωdΩ = nω2q2 4π2 ✓ 1 − 1 v2n2 ◆ I2

1(ω, ϑ),

d2E12 dωdΩ = −nω2q2 2π2 ˜ θ2 4n2 + ˜ θ2 ✓ 1 − 1 v2n2 ◆ I1(ω, ϑ)Re [I2(ω, ϑ)] ,

d2E2 dωdΩ = nω2q2 4π2 ˜ θ2 4n2 + ˜ θ2 ✓ 1 − 1 v2n2 ◆ I2(ω, ϑ)

• 2.