Electric permittivity of the vacuum
in a strong constant electric field
Hidetoshi Taya
RIKEN → Keio U. (from April)HT, C. Ironside, PRD 108, 096005 (2023) [2308.11248]

This talkDiscuss the electric permittivity 𝝐 of the vacuum
in a strong constant electric field
In the pure vacuum 𝒟 = ℰ

In an EM field 𝒟 = 𝜖ℰ ≠ ℰ
ഥ𝑬～～～～

ത due to the vacuum polarization
・ 𝜖 is no longer a const. 𝜖 = 𝜖(𝐸)・many studies since the early days,[Heisenberg-Euler (1936)] [Toll 1952] [Klein-Nigam (1964)] [Baier-Breitenlohner (1967)] …
Review: [King-Heinzl 2015] [Ejlli et al. (PVLAS) 2020] …

but is still worthwhile to be investigated ⇒ 3 motivations

Motivations (1/3)(1) The current understanding is limited to weak/slow regime
The most famous formula = based on Euler-Heisenberg Lagrangian
𝜕ℒEuler−Heisenberg
𝒟=−𝜕ℰ

⇒

𝛼 𝑒𝐸ത
𝜖=45𝜋 𝑚22×ቊ6 (∥)
2 (⊥)[Baier-Breitenlohner (1967)]

Motivations (1/3)(1) The current understanding is limited to weak/slow regime
The most famous formula = based on Euler-Heisenberg Lagrangian
𝜕ℒEuler−Heisenberg𝒟=−𝜕ℰ⇒𝛼 𝑒𝐸ത𝜖=45𝜋 𝑚22×ቊ

6 (∥)
2 (⊥)

[Baier-Breitenlohner (1967)]

ഥ ≪ 𝒎𝟐
Problem 1: Valid only in the weak limit 𝒆𝑬power corrections could be included 𝑒𝐸ത 𝑛 , BUT
・is factorially divergent ⇒ does not necessarily improve the formula・non-pert. factor like 𝑒 1/𝑒𝐸ത can never be included

cf. [Heinzl-Schroder (2006)]

Problem 2: Valid only in the slow limit 𝝎 ≪ 𝒎・not possible to discuss 𝜔 dependence

cf. [King-Heinzl-Blackburn (2023)]

・the physics must be different above the pair-production threshold 𝜔 > 2𝑚Problem 3: Neglecting the imaginary part 𝐈𝐦 𝓛𝐄𝐮𝐥𝐞𝐫−𝐇𝐞𝐢𝐬𝐞𝐧𝐛𝐞𝐫𝐠
𝜖 (in the coordinate space) must be real, so one must set Im ℒEuler−Heisenberg⇒ pair production and “non-equilibrium-ness” of E field are completely dismissed

Q: What happens if I go beyond those limitations ?

Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ=𝟎QED vacuum at 𝑬

← positive energy states← mass gap ~ 2m

← Dirac sea

Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ≠𝟎
QED vacuum at 𝑬
E field direction

Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ≠𝟎
QED vacuum at 𝑬
E field direction

electrons coming𝜓 ∝ 𝑒 −i∫ d𝑧 𝑃Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ≠𝟎
QED vacuum at 𝑬
E field direction

electrons coming𝜓 ∝ 𝑒 −i∫ d𝑧 𝑃
∝ 𝑒 +i∫ d𝑧 𝑃

reflected by the gap

Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
ഥ≠𝟎
QED vacuum at 𝑬
E field direction

electrons coming𝜓 ∝ 𝑒 −i∫ d𝑧 𝑃
∝ 𝑒 +i∫ d𝑧 𝑃

reflected by the gap
2∝ |# 𝑒 +i∫ d𝑧 𝑃 +# 𝑒 −i∫ d𝑧 𝑃 ቚ ∝ cos ∫ d𝑧 𝑃interfered electron distribution

Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
Observable imprintഥ≠𝟎
QED vacuum at 𝑬Ex.) spectrum of the dynamically-assisted Schwinger(= pair prod. in const 𝐸ത + fast probe ℰ)

E field direction[HT (2019)] [Huang-HT (2020)]⇒electrons coming𝜓 ∝ 𝑒 −i∫ d𝑧 𝑃
∝ 𝑒 +i∫ d𝑧 𝑃

reflected by the gap
2∝ |# 𝑒 +i∫ d𝑧 𝑃 +# 𝑒 −i∫ d𝑧 𝑃 ቚ ∝ cos ∫ d𝑧 𝑃interfered electron distribution

Exact numericalresults

Analytic(Furry picture)
Oscillation

probe frequency 𝜔/𝑚Motivations (2/3)(2) As a signature of non-trivial QED vacuum structure in E field
The QED vacuum (= the Dirac sea) has a non-trivial electron dist. in an E field,
which can leave observable imprints
Observable imprintഥ≠𝟎
QED vacuum at 𝑬Ex.) spectrum of the dynamically-assisted Schwinger(= pair prod. in const 𝐸ത + fast probe ℰ)

E field direction[HT (2019)] [Huang-HT (2020)]⇒electrons coming𝜓 ∝ 𝑒 −i∫ d𝑧 𝑃
∝ 𝑒 +i∫ d𝑧 𝑃

reflected by the gap
2∝ |# 𝑒 +i∫ d𝑧 𝑃 +# 𝑒 −i∫ d𝑧 𝑃 ቚ ∝ cos ∫ d𝑧 𝑃interfered electron distribution

Exact numericalresults

Analytic(Furry picture)
Oscillation

probe frequency 𝜔/𝑚

Q: What happens to the electric permittivity ?
Note: The motivation (1) (in particular, going beyond 𝜔 ≪ 𝑚) is important to achieve thisMotivations (3/3)(3) Pursue analogy between strong-field QED and semicond. phys・ Ground-state structure of semicond. = The QED vacuum⇒ the QED vacuum should response against external field in a similar way
to a semiconductor and vice versa

Motivations (3/3)(3) Pursue analogy between strong-field QED and semicond. phys・ Ground-state structure of semicond. = The QED vacuum⇒ the QED vacuum should response against external field in a similar way
to a semiconductor and vice versa[Yacoby (1966)][Seraphin-Hess (1965)]with GeRe Δ𝜖with SiIm Δ𝜖Δ𝜖 ≔ 𝜖 𝑒𝐸ത ≠ 0 − 𝜖(𝑒𝐸ത = 0)・ Nontrivial oscillating change in 𝜖 (i.e., motivation (2)) has already been observed
more than 50 yrs ago in semicond. physics !
(⇒ the Franz-Keldysh effect and electroreflectance) [Franz (1958)] [Keldysh (1958)]

probe frequency 𝜔probe frequency 𝜔

Q: Natural to expect this change in QED. To what extent is this true ?

This talkDiscuss the electric permittivity 𝝐 of the vacuum
in a strong constant electric field

I. Introduction(1) The current understanding is limited to weak/slow regime(2) As a signature of non-trivial QED vacuum structure in E field(3) Pursue analogy between strong-field QED and semicond. phys

II. Theory Linear response theory based on in-in formalism of QFT+ Kramers-Kronig rel. + a “phenomenological” IR regularization

III. Results Will give positive statements to the motivations (1) - (3)IV. Summary

Theory (1/5)
Linear response theory based on in-in formalism of QFT

Theory (1/5)
Linear response theory based on in-in formalism of QFT
Setup: QED in the presence of a constant strong field 𝐸ത + a weak spatially homo. probe ℰ(𝑡)Theory (1/5)
Linear response theory based on in-in formalism of QFT
Setup: QED in the presence of a constant strong field 𝐸ത + a weak spatially homo. probe ℰ(𝑡)Step 1: Definitions of 𝒟 and 𝜖ത ℰ) = 𝐸 + 𝑃0 (𝐸)ത +𝑃1 (𝐸)ℰത +⋯𝒟・ Total flux 𝐷 = 𝐸 + 𝑃(𝐸,ത + (1 +𝑃1 𝐸ത )ℰ + ⋯ത
= 𝐸ത + 𝑃0 (𝐸)⇒ 𝜖 = 1 +𝑃1 (𝐸)𝐸ത + ℰ

Theory (1/5)
Linear response theory based on in-in formalism of QFT
Setup: QED in the presence of a constant strong field 𝐸ത + a weak spatially homo. probe ℰ(𝑡)Step 1: Definitions of 𝒟 and 𝜖ത ℰ) = 𝐸 + 𝑃0 (𝐸)ത +𝑃1 (𝐸)ℰത +⋯𝒟・ Total flux 𝐷 = 𝐸 + 𝑃(𝐸,ത + (1 +𝑃1 𝐸ത )ℰ + ⋯ത
= 𝐸ത + 𝑃0 (𝐸)⇒ 𝜖 = 1 +𝑃1 (𝐸)𝐸ത + ℰ
Step 2: Calculate 𝑃1
ത 𝐸,ത ℰ)𝛾 𝜇 𝜓(𝐸,ത ℰ)|0; in・Ampere law: −𝑃ሶ = 𝐽 = 0; in|𝜓(ത= 0; in|𝜓ത0 𝐸ത 𝛾 𝜇 𝜓0 (𝐸)|0;
in

𝑃1

ത
+ 0; in|𝜓ത1 𝐸ത 𝛾 𝜇 𝜓0 𝐸ത + 𝜓ത0 𝐸ത 𝛾 𝜇 𝜓1 (𝐸)|0;in × ℰ + 𝒪(ℰ 2 )・Not in-out amplitude, but in-in !

cf. [Copinger-Fukushima (2018)]⇒ crucial when pair creating (or in non-equil.): |0; outۧ = |0; inۧ + (pair states like |𝑒 + 𝑒 − ; inۧ)

Theory (1/5)
Linear response theory based on in-in formalism of QFT
Setup: QED in the presence of a constant strong field 𝐸ത + a weak spatially homo. probe ℰ(𝑡)Step 1: Definitions of 𝒟 and 𝜖ത ℰ) = 𝐸 + 𝑃0 (𝐸)ത +𝑃1 (𝐸)ℰത +⋯𝒟・ Total flux 𝐷 = 𝐸 + 𝑃(𝐸,ത + (1 +𝑃1 𝐸ത )ℰ + ⋯ത
= 𝐸ത + 𝑃0 (𝐸)⇒ 𝜖 = 1 +𝑃1 (𝐸)𝐸ത + ℰ
Step 2: Calculate 𝑃1
ത 𝐸,ത ℰ)𝛾 𝜇 𝜓(𝐸,ത ℰ)|0; in・Ampere law: −𝑃ሶ = 𝐽 = 0; in|𝜓(ത= 0; in|𝜓ത0 𝐸ത 𝛾 𝜇 𝜓0 (𝐸)|0;in

𝑃1

ത
+ 0; in|𝜓ത1 𝐸ത 𝛾 𝜇 𝜓0 𝐸ത + 𝜓ത0 𝐸ത 𝛾 𝜇 𝜓1 (𝐸)|0;in × ℰ + 𝒪(ℰ 2 )・Not in-out amplitude, but in-in !

cf. [Copinger-Fukushima (2018)]⇒ crucial when pair creating (or in non-equil.): |0; outۧ = |0; inۧ + (pair states like |𝑒 + 𝑒 − ; inۧ)
Step 3: Calculate 𝜓0 and 𝜓1
・Solve Dirac eq. in E field:

𝜓1/ − 𝑒(𝐴/ − 𝑚 𝜓 = 0 ⇒ 𝜓 = 𝜓0 𝐸ത + 𝑆R 𝐸ത 𝑒𝒜𝜓/ҧ + 𝒜)/ 0 𝐸ത + 𝒪(ℰ 2 )𝑖𝜕Theory (2/5)
Linear response theory based on in-in formalism of QFT
Step 4: Collect everything + massaging …
+𝑆𝒑,𝑠1 +∞𝜖𝑖𝑗 𝜔 = 1 + 2 න d𝜏 ei𝜔𝜏 Θ 𝜏 Πij (𝜏) ≃ 1 +𝜔 −∞𝛾𝑖𝛾𝑗−𝑆𝒑,𝑠where

d3 𝒑 𝑖 −𝜏𝜏 𝑗 +𝜏𝜏Πij 𝜏 ≔ 2𝑒 Im tr ෍ න𝛾𝑆(+,−)𝛾𝑆(−,+)
𝒑,𝑠𝒑,𝑠2𝜋 32 22 22𝑠,𝑠′±±±𝑆𝒑,𝑠≔ 0; in|𝜓ത0,𝒑,𝑠𝜓0,𝒑,𝑠|0; inw/+−
𝜓0 = ෍(𝜓0,𝒑,𝑠+ 𝜓0,𝒑,𝑠)𝒑,𝑠Point 1: No Feynman propagator (∵ in-in calculation )Point 2: Manifestly causal 𝜖 𝑡 ∝ 𝑆R (𝑡) ∝ Θ(𝑡)Theory (2/5)
Linear response theory based on in-in formalism of QFT
Step 4: Collect everything + massaging …
+𝑆𝒑,𝑠1 +∞𝜖𝑖𝑗 𝜔 = 1 + 2 න d𝜏 ei𝜔𝜏 Θ 𝜏 Πij (𝜏) ≃ 1 +
𝜔 −∞

𝛾𝑖

𝛾𝑗−𝑆𝒑,𝑠where

d3 𝒑 𝑖 −𝜏𝜏 𝑗 +𝜏𝜏
Πij 𝜏 ≔ 2𝑒 Im tr ෍ න𝛾
𝑆(+,−)𝛾𝑆(−,+)
𝒑,𝑠𝒑,𝑠2𝜋 32 22 22𝑠,𝑠′±±±𝑆𝒑,𝑠≔ 0; in|𝜓ത0,𝒑,𝑠𝜓0,𝒑,𝑠|0; inw/+−𝜓0 = ෍(𝜓0,𝒑,𝑠+ 𝜓0,𝒑,𝑠)𝒑,𝑠Point 1: No Feynman propagator (∵ in-in calculation )Point 2: Manifestly causal 𝜖 𝑡 ∝ 𝑆R (𝑡) ∝ Θ(𝑡)Step 5: Do the integrations ! ⇒ But it is not straightforward
Problem 1: UV divergence (∵ loop diagram)
Problem 2: IR divergence (∵ the LO treatment in ℰ breaks down,
since the non-linearity parameter 𝜉 ≔ 𝑒𝒜 Τ𝑚 ≃ 𝑒ℰ Τ𝜔 → ∞ )Theory (3/5)
Linear response theory based on in-in formalism of QFT
Step 6: Use Kramers-Kronig relation

Theory (3/5)
Linear response theory based on in-in formalism of QFT
Step 6: Use Kramers-Kronig relation

Idea: (1) Only the real part is UV divergent, and the imaginary part is finite and calculable
2Im=cf. Cutkosky rule

Theory (3/5)
Linear response theory based on in-in formalism of QFT
Step 6: Use Kramers-Kronig relation

Idea: (1) Only the real part is UV divergent, and the imaginary part is finite and calculable
2Im=
cf. Cutkosky rule(2) Due to causality, Re and Im are related with each other(Kramers-Kronig relation)

𝜖𝑖𝑗 𝜔[Toll (1960)] See also [Heinzl, Schroeder (2006)] [Borysov et al. (2022)] for nBW[Aspnes (1967)] for semicond.

+∞
111 +∞′i𝜔𝜏Re𝜖(𝜔)=P.
V.
නd𝜔
Im 𝜖(𝜔′ )= 1 + 2 ∫−∞ d𝜏 e Θ 𝜏 Πij (𝜏) ⇒𝜔𝜋𝜔′ − 𝜔−∞Causality = the step function is the essence (for in-in response functions)(not unitarity, unlike the optical theorem for in-out amplitudes)∴ It is sufficient to calculate the imaginary part, which does not suffer from UV div.

Theory (4/5)
Linear response theory based on in-in formalism of QFT
Step 7: Introduce a counter term and match it with Schwinger formula

Theory (4/5)
Linear response theory based on in-in formalism of QFT
Step 7: Introduce a counter term and match it with Schwinger formula

Idea: (1) In general, resummation gives a counter term Δ𝜖⇒ 𝜖 → 𝜖reg = 𝜖 + Δ𝜖(2) Resum. not easy ⇒ determine Δ𝜖 phenomenologically by matching 𝜖reg w/ smth known

Theory (4/5)
Linear response theory based on in-in formalism of QFT
Step 7: Introduce a counter term and match it with Schwinger formula

Idea: (1) In general, resummation gives a counter term Δ𝜖⇒ 𝜖 → 𝜖reg = 𝜖 + Δ𝜖(2) Resum. not easy ⇒ determine Δ𝜖 phenomenologically by matching 𝜖reg w/ smth known(3) Use relationship between Im 𝜖 and the Schwinger effect (by 𝐸 = 𝐸ത + ℰ )Intuitive explanation: analog of electromagnetism in matterd𝑈lossd𝒟 1=ℰ= 𝜔ℰ 2 Im 𝜖reg
d𝑡d𝑡 2・Microscopically, the decay of a probe is caused by the pair production
・Dielectric energy loss (= decay of a probe)⇒ Energy for pair prod.
⇒d𝑈ppd𝑡=𝜔𝑁 ℰ≠0 −𝑁(ℰ=0)

𝑁 ℰ ≠ 0 − 𝑁(ℰ = 0) 1 2= ℰ Im 𝜖reg
𝑉𝑇2𝑉𝑇

must equal to 𝑈loss

cf. see Landau-Lefshitz

Theory (4/5)
Linear response theory based on in-in formalism of QFT
Step 7: Introduce a counter term and match it with Schwinger formula

Idea: (1) In general, resummation gives a counter term Δ𝜖⇒ 𝜖 → 𝜖reg = 𝜖 + Δ𝜖(2) Resum. not easy ⇒ determine Δ𝜖 phenomenologically by matching 𝜖reg w/ smth known(3) Use relationship between Im 𝜖 and the Schwinger effect (by 𝐸 = 𝐸ത + ℰ )Intuitive explanation: analog of electromagnetism in mattercf. see Landau-Lefshitz

d𝑈lossd𝒟 1=ℰ= 𝜔ℰ 2 Im 𝜖reg
d𝑡d𝑡 2・Microscopically, the decay of a probe is caused by the pair production
・Dielectric energy loss (= decay of a probe)⇒ Energy for pair prod.
⇒d𝑈ppd𝑡=𝜔

𝑁 ℰ≠0 −𝑁(ℰ=0)
𝑉𝑇

must equal to 𝑈loss

𝑁 ℰ ≠ 0 − 𝑁(ℰ = 0) 1 2= ℰ Im 𝜖reg
𝑉𝑇2(4) LHS at 𝜔 → 0 can be calculated with the Schwinger formula, so Δ𝜖 can be fixed𝑁Schwinger 𝐸ത + ℰ − 𝑁Schwinger 𝐸ത ∝ exp −𝜋𝑚2𝑒 𝐸ത + ℰ𝑚2− exp −𝜋𝑒 𝐸ത

𝑚2= … × exp −𝜋× ℰ2𝑒 𝐸ത

Theory (5/5)
Linear response theory based on in-in formalism of QFT

1 +∞𝜖𝑖𝑗 𝜔 = 1 + 2 න d𝜏 ei𝜔𝜏 Θ 𝜏 Πij (𝜏) ≃ 1 +𝜔 −∞+𝑆𝒑,𝑠𝛾𝑖

𝛾𝑗−𝑆𝒑,𝑠Schwinger 𝑁KKIm 𝜖Re 𝜖This talkDiscuss the electric permittivity 𝝐 of the vacuum
in a strong constant electric field

I. Introduction(1) The current understanding is limited to weak/slow regime(2) As a signature of non-trivial QED vacuum structure in E field(3) Pursue analogy between strong-field QED and semicond. physII. Theory Linear response theory based on in-in formalism of QFT+ Kramers-Kronig rel. + a “phenomenological” IR regularization

III. Results Will give positive statements to the motivations (1) - (3)IV. Summary

Results (1/4)ഥ ≠ 𝟎 − 𝝐(𝑬ഥ = 𝟎)
Imaginary part of the change 𝚫𝝐 = 𝝐 𝑬～～～～

Probe ℰ

Probe ℰ

Strong 𝐸ത

Strong 𝐸ത

・ Oscillation, as expected from the tilted vacuum
・ Essentially the same as what have observed in semiconductor physics
・ Birefringent (Im Δ𝜖∥ ≠ Im Δ𝜖⊥ ) but basically the same・ Non-vanishing even at 𝜔 → 0 due to the non-perturbative pair production (Schwinger)
𝑚2𝑚2𝑚2∵ Im 𝜖(𝜔 → 0) ∝ (𝑁Schwinger 𝐸ത + ℰ − 𝑁Schwinger 𝐸ത ) ∝ (exp −𝜋 𝑒(𝐸+ℰ)− exp[−𝜋 𝑒𝐸ത ]) = (finite) × exp −𝜋 𝑒 𝐸തതResults (2/4)ഥ ≠ 𝟎 − 𝝐(𝑬ഥ = 𝟎)
Real part of the change 𝚫𝝐 = 𝝐 𝑬～～

Probe ℰ～～

Probe ℰ

Strong 𝐸ത

Strong 𝐸ത

・ Again oscillation, which is again consistent with semi-conductor
・ Logarithmically divergent at 𝜔 → 0 due to the non-pert. pair prod.
𝑚2∵ Re 𝜖 0 = 𝜋 P. V. ∫−∞ d𝜔′ 𝜔′ Im 𝜖 𝜔′ ∼ 𝜋 ∫−∞ d𝜔′ 𝜔′ Im 𝜖 0 ∼ log div. × exp −𝜋 𝑒𝐸ത
1+∞

11+∞

1

⇒ a “matter” effect, which needs to be subtracted to compare with
the Euler-Heisenberg result

Results (3/4)
Comparison w/ EH result around 𝝎 ≈ 𝟎Strong 𝐸ത

Probe ℰ～～～～

Probe ℰ

Strong 𝐸ത

・ After the log subtraction, the result is consistent with the EH result
・ The agreement becomes very good for weak fields 𝑒 𝐸ത Τ𝑚2 < 0.2~but EH underestimates about O(>10%) for strong fields

𝛼 𝑒 𝐸ത
Δ𝜖EH =45𝜋 𝑚22×ቊ6 (∥)
2 (⊥)

Results (4/4)
Behaviors of the peaks

Numerical fit says (similar numbers for Im 𝜖⊥ and Re 𝜖∥,⊥ )where 𝐼 = 𝐸ത 2 /2 is the focused intensity

・much larger than EH e.g., Δ𝜖EH = 𝑂(10−7 ) for 𝐼 = 𝑂(1 × 1023 W/cm2) (PW laser)⇒ High-frequency probe is useful to study 𝜖 (or vacuum birefringence, in general)・weak E dependence Δ𝜖 ∝ 𝐼

1 1~5 6⇒ the peaks of Δ𝜖 can still be large for subcritical fieldse.g., only one-order smaller Δ𝜖 = 𝑂 1 × 10−5 ≫ Δ𝜖EH at GW scale

This talkDiscuss the electric permittivity 𝝐 of the vacuum
in a strong constant electric field

I. Introduction(1) The current understanding is limited to weak/slow regime(2) As a signature of non-trivial QED vacuum structure in E field(3) Pursue analogy between strong-field QED and semicond. phys

II. Theory Linear response theory based on in-in formalism of QFT+ Kramers-Kronig rel. + a “phenomenological” IR regularization

III. Results Will give positive statements to the motivations (1) - (3)IV. Summary

Summary
Discussed the electric permittivity 𝝐 of the vacuum
in a strong constant electric field
3 motivations and answers to them:(1) The current understanding is limited to weak/slow regime⇒ Succeeded in developing such a theory:
Linear response theory based on in-in formalism of QFT+ Kramers-Kronig rel. + a “phenomenological” IR regularization(2) As a signature of non-trivial QED vacuum structure in E field(3) Pursue analogy between strong-field QED and semicond. phys⇒ Yes: a characteristic oscillating structure in 𝜖 and is analogous to semicond.